By Harry F. Tasset


“You never change things by fighting the existing reality. To change something, build a new model to make the existing reality obsolete.” Buckminster Fuller


“Science must constantly rid itself of dogmas, because they are always a sign of human incompleteness and retard the development of science” Ida Noddack, Discoverer of Rhenium, 75


This paper is meant to be a study of the periodic table of the elements and the possibility it has errors that need a closer look. I do not pretend to be a chemist or a physicist but I would like to add one historical note: Charles Janet who was one of the periodic table’s most admired contributors was neither a chemist or a scientist. His Left Step Periodic table is studied in most modern chemistry textbooks. One of my goals here is to show that mirror fermions and other dark matter particles exist as here-to-for un-named elements of my new Periodic Table of Everything or PTOE. Having spent more than 50 years of personal study, finding a place for these un-named/missing elements after 150 years of historical neglect was difficult, to say the least. Like Charles Janet, I am in the twilight of my life and not assured of additional years especially with the covid 19 virus pandemic. My other goal is to distribute this information as quickly and safely as possible because the record needs to be corrected and after 150 years the world needs to catch up with this new technology as soon as possible. I owe an abiding gratitude to a man by the name of Frank (Tony) Smith. As you will see, his E8/Cl(8) physics and explanation made the proof and conclusion of my Periodic Table of Elements possible.




I began to study of the Periodic Table of the Elements after my father, Everett J. Tasset, who was a self made inventor and engineer, told me about his idea of the Expanding Earth Theory. I really wanted my father’s idea to be true but at that time there was no explanation of how the Earth could have grown to over ⅔ of its original size. Knowing that the driving force behind the expansion had to be some type of interior fission, black hole or undiscovered dark matter process, I began my search. After studying this problem for around 3 years, I realized that the current academic periodic table of the time had a glaring error. There were 8 missing spaces. I saw this when I looked at Dmitri Mendeleev’s periodic table from 1871. (See Figure 1)




Fig 1


If one takes the time to re-construct his table with the modern elements discovered after 1917 placed into their assigned spaces, it becomes apparent that there are 8 spaces that remain un-named, 4 elements at the end of the Lanthanoid series and 4 elements at the end of the Actinoid series. (see figure 2) This seemed to be an impossibility to me in 1969. After all, how could scientists have overlooked this error for over a century?  Look carefully below at the next 5 tables from historical figures, beginning with Mendeleev. I have made no erasures nor added elements except for those discovered later and then only in their proper place.



Fig2 (the red type are those elements that were discovered after 1917, the un-named/missing elements are marked as Nihilium 1,2,3 etc.) I chose the name Nihilium from the Latin term meaning “Void” referring to the annihilation/creation operators involved in their creation.. There are six element spaces that have double entries (Red and Black) because at that time there was confusion about the correct position of some of the rare earths like Erbium, Terbium etc. The name of Radon at that time was Niton. I named the two un-named/un-discovered noble gases Eka-Xenon and Eka-Niton in keeping with Mendeleev’s original terminology.


In 1969 I copyrighted what I called, The HarmonAtomic Table with the concept of 8 missing/un-named elements, (see figure 3) below.


Fig3 (the asterisks denote those elements with unassigned names. Note: some of the heavy elements beyond 102 had not been synthesized at that time)


I had all but given up on this UNCONVENTIONAL idea but I then discovered I was not the only person to notice these missing elements. Dr. Isaac Asimov, who was a biochemist from Columbia University and author teaching at Boston University School of Medicine, wrote a text in which he suggests that there are indeed 8 missing/un-named element spaces. (see empty blocks in figure 4 below…please note that he did not continue empty spaces with the other 3 additional missing Actinides e.g. red arrow)




Fig 4 (Isaac Asimov, Physical Sciences, 1960, pp. 154-155)

Another noted scientist and author , Ida Noddack, who won the Nobel prize for her discovery of Rhenium, also published her table with the 8 un-named elements. (see Figure 5)



Fig 5 (Ida Noddack table, 1920, I have marked the un-named elements Nih 1,2,3, Etc.

As with the 2 other Mendeleev tables, I have only modified her table with the as yet undiscovered/un-isolated elements)


My quest to make logical sense of the periodic table’s natural order continued with this table from Professor Eric Scerri, UCLA. Notice that in the two rare earth columns there are eight (8) unfilled spaces. These 8 spaces represent the eight (8) un-synthesized elements of my PTOE. (Just imagine moving the two rows of rare earths directly into the table above, occupying the same empty spaces available. That makes it easier to see the 8 unfilled spaces marked in yellow. Follow the vertical arrows. I have deleted the Uus and Uuo elements which originally appeared below At and Rn, had not been discovered in 2008 and would have been properly placed after the Actinides. Technically, the Lanthanides would slip into the row of 14 elements after Ba and the Actinides directly beneath them. As with the other tables above and below the 8 deficit spaces still show up.) The added 8 missing elements would make both periods 18 elements long like the rest of the tables. ( in no way am I implying here that Dr Scerri approves of or indorses my theory of the 8 missing elements. I only use this table to demonstrate the 8 element quantity difference between the 14 element rare earths periods and the 18 element periods in the standard body of the table)




Notice that the periods of the main body of Dr. Noddack’s , Dr. Asimov’s, and Dr. Eric Scerri’s tables are all exactly 18 elements long, like my PTOE. Here is a link to a 1920 table from A.W. Stewart in which he presents an 18 element periodicity, https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=1075. Looking at figure 6 or any periodic table for that matter, one can start the count from Hydrogen to Argon for 18 elements and continue the same throughout the remaining elements all ending in noble gases. Also, as noted by many periodic table historians and philosophers, Mendeleev intended for his table to have an 18 element sequence. If you are still not convinced, ask yourself one simple question: since the periodic sequence above them are all 18 elements long, why would nature leave a gap of 4 elements in each of the 14 element long rare earth’s and then continue on as if the hole wasn’t there? Ida Noddack suggested in her writings that all the missing elements would someday be discovered……”We just have to look harder.”


After almost 50 years of research, I realized that I had also made an error. I had not increased the atomic number of my elements from 120 to the true count of 128 after accounting for the missing 8 elements. Each successive element would have to be advanced 4 spaces and atomic numbers starting with Ytterbium 70 (now 74) and by 8 for those starting with Nobelium 102, (now 110). My table differs from most in that it has 7 periods of 18 elements each, starting with Lithium, for a total of 126 and 128 with H and He. Similar elements are in groups of 7 vertically, i.e. Li, Sc, Y, La, Lu, Ac, and Lr. To most of you reading this paper the idea My table can also be arranged in the legacy standard form by beginning each period with the Alkaline Metal i.e. K, Rb etc.(see fig 6, my Periodic Table of Everything with the new count of 128) The table contains a lot of information as will be shown later but one numerical oddity that I can’t explain are the sum of the differences between opposing diagonal columns that add up to 72, coincidently the new element 72, is the point at which 4 of the 8 anomalous elements seem to go missing. Go to element 44, Ruthenium, draw a 33° diagonal to 84 or Mercury and the difference is 40. Then do the same thing for the adjacent element Technetium 43 and Lutetium 75 which is 32. You can also add the sum of the differences on the same sides with opposing diagonals e.g. 40 + 32 at Cadmium 48 and Lanthanum 57. It always adds-up to 72! This only happens when one correctly advances the atomic numbers of the new periodic table to 128. There are many more of these “coincidences” as you will see. To many of you reading this paper the idea of adding 8 additional atomic numbers to the middle of the table is considered nonsensical but I hope to show you with a reasoned approach, that this is indeed both possible and necessary. The below table may be an optimal form for its display.(Fig 6)



Fig 6 (Harry Tasset PTOE, 2019)


Those of you who are familiar with the atomic number triad principle of Mendeleev will notice the above table is constructed so that all triads are indeed mathematically true, not only that but every element in the table is a part of more than 42 perfect atomic number triads. Some examples of the many triads: In vertical columns we have Li 3 Sc 21 Y 39 (3+39 ÷ 2 = 21), Kr 36 Xe 54 Nihil 72 (36+72÷2 = 54), Rn 90 Nihil 108 Og 126 (90+126÷2=108), Y 39 La 57 Lu 75 (39+75÷2=57). This new triad table rule applies to all elements except Hydrogen and Helium. Here is a link to an article by Dr. Eric Scerri arguing for the use of atomic number triads where he makes the following statement that seems to support my new view:    “…I claim that the use of atomic number triads, is capable of completing a ‘global aspect’ of the periodic table which reference to electronic structure fails to do.” He goes on to make an argument for using the structure of the nucleus as a model for electron configuration and therefore the periodic table itself.



Ida Noddack spent many years of her life arguing for just this nucleus idea claiming that we could reach a whole new understanding of isotopes as well as the periodic table.

Also, all of the Noble gases have atomic numbers (x10) in the sequence of a circular angular momentum from 180° to 1260°. This finding ties the atomic numbers to the electron structure and defines the Madelung Rule which is basic to Chemistry. Each Noble Gas atomic number ends with the 18th element in each period. (see above table) The atomic numbers have been properly advanced by 8 protons to 126 because we are looking at the table as a theoretical construct during the big bang nucleosynthesis and other star forming processes. I will be discussing these two new findings in detail. My new periodic table of triads and the noble gases’ universal connection to angular momentum are both groundbreaking findings. They are only the beginning of more incredible findings, as you will see below.




Fig 6a (right step periodic table with nuclides.  Credit Harry F. Tasset, 2021 and Wikipedia)


The above figure shows my right-step periodic table with a black and white copy of the Chart of Nuclides.  The right-step table was created when I tried to make sense of the Hafnium and missing element **72 (normally the atomic number of Hf).  The diagonal structure occurred naturally when I placed those 2 elements atop one another (along with the other 3 missing elements).  I immediately thought of the diagonal nuclide chart.  The right step table began to make more sense when I did this,  especially when comparing it to the chart of nuclides.  The prominent open spaces of the nuclide chart around nuclides 72-74 and 106-109 can be seen in the chart and the right-step table below it.  These are the fibrations (twists) that I will be expanding on in the following pages.  I found it remarkable that the sizes of the chart and the table needed very little ‘tweaking’ to align the two.  I am not that familiar with the chart of nuclides but I am seeing a close approximation between it and the missing elements.  I will show a direct pictorial and mathematical relationship below with a new formula for the addition of the 8 additional elements.


It was only after many hours and days of looking into my atomic number error in figure 3 that I found Garret Lisi’s TOE (Theory Of Everything) and his colorful E(8) Coxeter plane Gossett polytope constructed with the exceptional Lie algebras. It was the Star of David symbol in my Fig 6 above that lead me to Lisi’s beautiful symbol with the Star of David in the middle. (see fig 7 below) I will be discussing related Lie, Clifford and other algebras of E(8), Cl(8) and how they fit into and provide mathematical substance to my theory. Each of Lisi’s polytope vertices represent one or more of the 18 fundamental particles/force carriers of the standard model.


Fig 7 (by Garrett Lisi)

It is the above image that some of today’s physicists believe can explain the GUT (Grand Unified Theory). All the colorful points represent the individual groups of subatomic particles in nuclear physics. The E(8) model contains 248 vertices and within these are the 128 special vertices that match the true number of chemical elements in my PTOE. This was more than just a coincidence. The physics and mathematics become very clear at this point. (I will be putting links to the mathematical formulas of my theory here on this web site) The resulting AQFT (Algebraic Quantum Field Theory) is a part of applied quantum particle physics. The underlying basic particles are shown in the following diagram from Wikipedia on which I have placed a capital G for the graviton. (the graviton, a force carrier which is theoretical, makes the total 18 in figure 8 which coincides with the 18 element structure of my periodic table, PTOE).


Figure 8 (Wikipedia)

Since the above 18 fundamental particles/force carriers are the building blocks of all atoms and the atomic elements of our periodic table are the building blocks of all physical matter, then the two have a direct relationship with each other. This is in fact what my Periodic Table of Everything polytope below shows. (see Figure 9) The fact that the 18 primary particles coincide with the 18 element periods just may be serendipity, there may be a deep underlying order. What you are seeing here is a 2 dimensional representation of a 7 dimensional E(7) with each of the 128 vertices representing a baryogenic process of elemental creation. (this will be explained below) The groups of 7 radial elements have been color coded to indicate their place within the group of elements with similar chemical properties. There are 7 circular periods of 18 elements each for a total of 126 (a 2_31 Gosset plane polytope). The elements Hydrogen and Helium (center) are polar vertices (antipodal) making the chemical element count 128. (This number 128 is prominent in many of the algebraic calculations in appendix 1, Tony (Frank) Smith’s physics calculations.) Keep in mind while viewing the below figure that you are looking at a 2D image of a 7 dimension polytope that has 126 vertices, 2016 edges, 10080 triangular faces and 20160 Tetrahedral cells. This all sounds incredibly complicated until you see it animated in all dimensions. Here is a link to J. Gregory Moxness’ amazing E8 rotation based on Garrett Lisi’s E8 physics model: https://youtu.be/LzpbDttfh9g


Fig 9 (My PTOE, inscribed onto a 2_32 coxeter plane polytope)


In the above figure the 128 chemical elements are formed into 7 concentric circles with a period of 18 chemical elements (with 2 polar elements). Each circular period has chemical elements, color coded in a group of 7 with similarities that easily flow into one another, from the center to the outer circle. The red line of Noble gases serves as the “spine” of total angular momentum corresponding to the degrees of 180°, 360°, 540°, 720°, 900°, 1080° and 1260°. (See my figure 6. This will be explained below along with fibrations at 720° and 1080°) There have been many circular and spiral periodic tables over the past 150 years but none have achieved the level of integration as the PTOE.



The proof of my PTOE comes from the exceptional real E(8) Lie algebra of the Lisi TOE and from Cl(8) exceptional real Clifford algebras of Tony (Frank) Smith, where there are 256 vertices that can be split into two groups of 128 vertices i.e. the E(7)(7) SU (8) and exceptional Lie sub-algebras E(6) and F(4). As with the Lisi E(8) polytope, the 128 vertices of my modified polytope each contain the fundamental particles that build a quantum mechanics, fermionic structure representing real matter itself and the elements of my periodic table. After Garrett Lisi published his Theory of Everything, physicists began the assignment of quantum field particles to each of the 248 vertices matching them to most of the known elements. (see figure 10) Here is an excellent website with a complete explanation of their particle assignments and the periodic table counterparts. J. Gregory Moxness sells his personal graphics of many of these beautiful images on his web site. (Please go to http://theoryofeverything.org/TOE/JGM/Integrated%20E8,%20Binary,%20Octonion-web.pdf)  Mr Moxness has placed the 4D Stowe-Janet-Scerri Periodic Table on the Chemogenisis Web page at, ( https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=589)


These are his remarks from that web page: “Interestingly, it has 120 elements, which is the number of vertices in the 600 Cell or the positive half of the 240 E8 roots. It is integrated into VisibLie_E8 so clicking on an element adds that particular atomic number’s E8 group vertex number to the 3rd E8 visualizer pane. The code is a revision and extension of Enrique Zeleny’s Wolfram Demonstration” If you want a closer look at this just click on the Enrique Zeleny’s link.



Figure 10 ( credits to J.Gregory Moxness, visit his excellent web site) The fano plane has the red arrow. (The Fano plane will be fully explained below)

Please note that there is an exact difference of 8 particles (vertices/elements) between the 248 of the Lisi polytope and the 256 particles of the exceptional Clifford algebras. I am using (figure 11) below, from the work of Frank (Tony) Smith, to illustrate this. These are the 8 missing chemical elements in the modern table and they now appear in my completed Periodic Table of Everything. I will show in principle, the existence of these 8 heavy, rare earth elements.




Figure 11 (Many thanks to Frank (Tony) Smith for the above)

Please notice his last sentence in the Figure 11, “…..the 8 elements of 256-dim Cl(8) that do not directly correspond elements of 248-dim E(8).” Here again we observe a deficit of 8 unitss. When he uses the term “elements” here it is referring to algebraic, geometric points (vertices) in the 8 dimension projections which I am using here as a Quantum Particle Element but extrapolating to their chemical element counterparts in my PTOE. (this has become a popular way to illustrate the periodic table as you will see below)

For close to 100 years there has been a little talked about division between physical chemistry and quantum physics mainly because the two methods do not fit together as naturally imagined. Take for instance the modern periodic table which has two rows of elements taken out of the middle and placed below the main body of the table (this happen in the 1930- 40’s). This certainly was not a natural flow to physicists who look to nature for physical explanations. (Newton’s apple tree did not have a big chunk taken from its very middle) Also there is no consensus on many of the phenomena of quantum Chemistry and the critical explanations behind them. Here are just a few: The odd/even elemental abundances, the triad rule, the octaves, the Madelung rule, the relativistic effect on elements like gold and silver, Lanthanide and Actinide contraction and there are many more anomalies. My new periodic table is a unifying table that gives many answers to both physicists and chemists.

Look again at figure 7, Garrett Lisi’s E(8) TOE polytope. Here is a link to Lisi’s particle Explorer Program that allows you to build polytopes with any number of particle assignments: http://deferentialgeometry.org/epe/ When he first presented his new understanding of the GUT and quantum physics, most scientists were very receptive but there was one problem that even Lisi had to acknowledge. His theory had to allow for something called mirror fermions, however as of that year, 2007, mirror fermions had not yet been discovered in nature. If they did exist, the experts pointed out in 2009, they would be very massive particles. To overcome the above objections Lisi postulated that just because they have not been discovered, does not mean mirror fermions do not exist. Recent tests at the LHC, large hadron collider, have shown there are indeed, massive, mirror-like particles close to the atomic weight of Rhenium.

Lisi continued his work on E(8) and in 2015 developed a new model called “Lie Group Cosmology” finding that the mirror fermion problem could be overcome by extending the E(8) to an infinite-dimensional Lie group like E8+++ E(11) which produces three generations of fermions with no mirror fermions. This may or may not be the ultimate GUT. I have chosen to use Lisi’s original model that allows for mirror fermions along with another model, the Cl(8) algebra that shows mirror fermions can exist. The reason for this is simple: The 8 un- named elements are very massive, like mirror fermions as shown by their position among the other heavy, rare earth elements in the table. This is a good start in the right direction and another helpful coincidence.




The concept of un-named elements has been a constant feature of the periodic table since its very beginning but it was only after the Lanthanoids and Actinoids were placed unnaturally beneath the table that it took a bad detour. It is clear from the beginning that the table founders intended the table to be a whole, natural unit. (See figures 1,2,4 & 5) In order for us to get back on the right path we first must uncover why the Rare Earth Elements were placed below the table. At that time i.e. 20’s & 30’s scientists said the rare earth elements needed to be placed below because they all have similar properties. But there are many other elements, not rare earths, that have similar chemical properties and yet they were not treated the same way. So why did this happen and why did a professor of biochemistry place them in their correct position in 1960? Dr Isaac Asimov was that professor and we can see in his table (Figure 4) the same 8 empty spaces that appeared in the 1917 version of Dr Mendeleev’s table. (See figure 2) When you look at Asimov’s table the answer to this puzzle is immediately apparent. The Atomic numbers that rule all positions, end at #71 Lutetium and skip 4 blank, un-named elements and then continue at #72 Hafnium, 4 spaces ahead. This would and did present an untenable problem for the 1930’s scientists who at the time were working on the Manhattan Project (Atomic Bomb). They had explained many of the facets of radioactivity but how could they now explain 8 missing elements, all their electrons, protons, neutrons and their missing spaces in the middle of the table? Everything they knew about nuclear physics, the atom, the electrons etc. etc. would have to be re-thought in order to account for these 8 massive, missing elements that did not appear on the face of the Earth. The concept of dark matter, dark energy and the Higgs Boson could not have even been imagined. However, lead scientists convinced others that the rare earths could and should be placed below the table even though the period length of the rare earths are 14 elements long and not 18 like the main body. Here is the standards table of the NIST (National Institute of Standards and Technology) Fig 12  https://www.nist.gov/blogs/taking-measure/periodic-table-its-more-just-chemistry-and-physics.

A strategically placed numbering within the table shows that the 8 empty spaces were there all the time. (Fig 12a):


Figure 12  (The official NIST periodic table. Credit: N. Hanacek/NIST)                                                                                                                                                               Fig 12a  (NIST periodic table as modified, Harry Tasset)


Notice that I have marked and numbered the 8 un-named/missing element spaces with the numbers of 1-8 in red.(they are not in the correct spaces, I am thinking inside the box) The original periodic table has a limited, specific number of spaces and when 8 spaces are typed over or made to disappear, someone will take notice and in this case it was the discerning eye of Dr. Isaac Asimov, author and Biochemist who found the error (See Figure 4).




In order for us to unwind the above error we need to return to the problem of the massive mirror fermions introduced by the opponents of the physicist, Garrett Lisi and to understand mirror fermions, we must look at CPT symmetry transformation. (CPT is, CHARGE, PARITY, TIME) Here is a quote from an article in Wikipedia:

“The implication of CPT symmetry is that a “mirror-image” of our universe — with all objects having their positions reflected through an arbitrary point (corresponding to


a parity inversion), all momenta reversed (corresponding to a time inversion) and with all matter replaced by antimatter (corresponding to a charge inversion) — would evolve under exactly our physical laws. The CPT transformation turns our universe into its “mirror image” and vice versa. CPT symmetry is recognized to be a fundamental property of physical laws.”

Some scientists believe these fermions violate CP, the combined symmetry of Charge and Parity. They wanted to eliminate the concept of mirror fermions as we can see in the following quote from George Triantaphyllou of the Institute of Theoretical Physics in Munich, Germany (Ida Noddack’s home country):

“Efforts to eliminate completely mirror fermions from nature are for some reminiscent of efforts several decades ago to identify the anti-electron with the proton, and amounts to not realizing that particles consistent with natural symmetries could actually exist independently. Such a gauge group and fermion extension, apart from fitting nicely into unification schemata, restores the left-right symmetry missing in the standard-model or in the simplest left-right symmetric models.” He goes on to say “Mirror particles are considered to be a possible component of the invisible dark matter.”


A lot of research has been taking place since the discovery of the Higgs Boson in 2011. Much of it is moving towards a Grand Unified Theory. One piece of the puzzle has been the

work to prove or disprove the Supersymmetric Models, or abbreviated to SUSSY. The following diagram is quite interesting:



Figure 14 (Credit to Wikipedia)


What caught my eye is the N=8 supergravity multiplet of (Figure 14). When the multiplet (superscript) of each even or odd part i.e. 2, 28, 70, 28 and 8, 56, 56, 8 ±1 is added it is 128, the same number of vertices in the E(7) polytope and the same number of elements of my PTOE. This is better explained by physicist Tony Smith in figure 15 :




Figure 15 (I will be showing his 8 dimensional HyperCube calculations later in appendix 1)


Many people do not realize that they live with anti-matter on a daily basis.  Anti-matter particles are around us all the time, in bananas, the air we breath and PET scans.  Bananas have a positron (an anti-matter particle) in the form of a +Potassium ion. Every day lightening strikes the Earth 4 million times, 1.4 billion strikes a year, releasing around 3 trillion Kg of Ozone and amid all that there are millions of positron antiparticles created. The last PET (Positron Emission Tomography) scan you received was done using anti-matter.


Reading the many scientific articles about mirror fermions I was amazed how often the integer 8 was repeated. Beginning with Octonions, 8 vectors, 8 dimensions, 8 multiplets, 8 mirror fermions, 8 periodicity, 8 half-spinors, E(8) etc. This is not by accident but by design. The fundamental principle of the periodic table is the Octave scale familiar to us all in music. Here is a excellent example of what I am talking about. The following table, (Figure 16), is from one of the foremost collections of all things chemical, https://www.meta- synthesis.com/webbook/35_pt/pt_database.php?PT_id=7

Also see the excellent article by John Frederick Sweeney on the musical Octaves in Appendix 2 below



Figure 16 (credit to https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=7)


This 1862 chemical chart and telluric graph show the elements arranged in Octaves.



Research is being conducted this year on the existence of Rare Earth mirror compounds which match my belief that the 8 un-named/missing elements are indeed among the rare earths. Here is an article entitled: Magnetic and noncentrosymmetric Weyl fermion semimetals in the RAlGe family of

compounds (R=rare earth) From: 10.1103/PhysRevB.97.041104 In the article the authors make the following statement:


“Taking advantage of the Weyl nodes generated by inversion-symmetry breaking in the nonmagnetic compound LaAlGe, we present a new type of magnetic Weyl semimetal in its iso-structural sister compounds CeAlGe and PrAlGe that are ferromagnetic.” (See fig 17)


(Figure 17 Thanks to APS Physics/CHORUS)


What they have discovered in 2020 is that there are rare earth compounds (i.e. containing Lanthanum) which break inversion-symmetry therefore meeting the definition of an anomalous quantum field particle with mirror-like fundamentals.


However, to be consistent, we must find other particles like the Mirror Higgs Boson that fit into our model of the missing heavy elements. David Dunsky, Lawrence J. Hall, University of California, Berkeley, California 94720, U.S.A. Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. and Keisuke Harigaya, Department of Physics, School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540, U.S.A. have submitted the following research: https://link.springer.com/article/10.1007%2FJHEP02%282020%29078 Showing Dark matter, dark radiation and gravitational waves from mirror Higgs parity. (See Fig 18)


Figure 18


Some of the other mirror particles shown here are: mirror bosons, mirror gluons, mirror photons, mirror quarks and a possible unknown dark matter particle. However, the significance in this article is the possibility that the neutron to proton ratio is altered. (See sentence 12 in Fig 18 above) Why is this important? When we look at my new periodic table, there is a deficit of 8 protons (in the 8 missing elements) and since the proton determines Atomic Number, i.e. Xe = Xenon = 54 but No = Nobelium = 102 not the 110 of my PTOE, how can that be possible? All of the above data points to the possibility that these 8 elements do exist as heavy rare earths with nucleonic and orbital mirror particles and are therefore mirror/dark matter elements themselves. Let us go further down the path. Perhaps beginning with the magic square will help. From: A magic square from Yang-Mills squared L. Borsten,_ M. J. Du_,y L. J. Hughes,z and S. Nagyx Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom

(Dated: January 8, 2015)



What is of particular interest to my research is the O column where it intersects the O row. This is the Octooctonionic projective plane with a supersymmetric of 16, 128 degrees of freedom, 248 dimensions in the SO(16) 120. These figures all come together below.


Here is a research article in 2017 that says a lot about the ratio of neutrons to protons in the early universe when baryonic matter was formed. From the State of Israel, “Mirror neutrons as dark matter in the Mirror Twin Two Higgs Doublet Model” by Hugues Beauchesne, Department of Physics, Ben-Gurion University,

Beer-Sheva 8410501, Israel. https://arxiv.org/abs/2007.00052 (see figure 19)



(Fig 19)





Dark Matter through the Higgs portal

Giorgio Arcadi, Abdelhak Djouadi, Martti Raidal

“We review scenarios in which the particles that account for the Dark Matter (DM) in the Universe interact only through their couplings with the Higgs sector of the theory, the so- called Higgs-portal models. In a first step, we use a general and model-independent approach in which the DM particles are singlets with spin 0, ½ or 1, and assume a minimal Higgs sector with the presence of only the Standard Model (SM) Higgs particle observed at the LHC. In a second step, we discuss non-minimal scenarios in which the spin-½ DM particle is accompanied by additional lepton partners and consider several possibilities like sequential, singlet-doublet and vector-like leptons. In a third step, we examine the case in which it is the Higgs sector of the theory which is enlarged either by a singlet scalar or pseudoscalar field, an additional two Higgs doublet field or by both; in this case, the matter content is also extended in several ways. Finally, we investigate the case of supersymmetric extensions of the SM with neutralino DM, focusing on the possibility that the latter couples mainly to the neutral Higgs particles of the model which then serve as the main portals for DM phenomenology. In all these scenarios, we summarize and update the present constraints and future prospects from the collider physics perspective, namely from the determination of the SM Higgs properties at the LHC and the search for its invisible decays into DM, and the search for heavier Higgs bosons and the DM companion particles at high-energy colliders. We then compare these results with the constraints and prospects obtained from the cosmological relic abundance as well as from direct and indirect DM searches in astroparticle physics experiments. The complementarily of collider and astroparticle DM searches is investigated in all the considered models.”

As you will see below that Dark Matter particle creation thru the Higgs portal plays a significant role in my PTOE theory. Note the placement of the spin -½ DM particles. (see figure 22 below)


Here is more research that is being done on “….a set of exotic nuclei, which are not found in the Periodic Table.” This research is being conducted by The Indian Lattice Gauge Theory Initiative, (ILGTI) https://phys.org/news/2019-10-deuteron-like-heavy-dibaryonsa-exotic-nuclei.html. See (figure 20 below):



Figure 20


I have not found what is meant by “….a set of exotic nuclei”, however I suspect that they are a part of the 8 missing/un-named elements I have been proposing. Another interesting finding is that the stability of these new nuclei increases when they become heavier, leading me to think that their natural existence on Earth may be possible after all, another pleasant coincidence.


I am including here another quote from Tony Smith’s E8 web book. It is his amazing ability to translate amorphous, algebraic clouds into the hard reality of the physical Universe that allowed me to see the reality behind the E8(8) Octooctonionic plane and thus my PTOE:


Below is a good description of the Rosenfeld Projective Plane




(Credit, Wikipedia)(please note the quarteroctonionic EVI E7(7) and the “Octooctonionic” EVIII E8(8) planes)



There may be lingering questions here about the exact nature of the PTOE and how it fits into the QFT. Including the perfect triads listed above, If my E(7) polytope PTOE is a representation of a stable, fully axiomized periodic table then each vertex is representative of one chemical element and the structure of my new table should provide answers as to the origin of the individual element’s electron spin angular momentum. Look again at figure 6, my PTOE and you will see that it is arranged in periods of 18 elements which all end with an inert noble gas. Look at the first 4 Noble elements: 18, 36, 54 and 72. If we let each element occupy a 10° turn in a circle then the first elemental period is 180°, the next 360°, then 540° and ending with 720°. In fact, this describes the fermion (quaternionic) and the 720° rotation of the ½ spinor of total angular momentum. This now describes a new principle that bridges the gap between Quantum Chemistry and Quantum Physics. Not only does this point to the fermion quaternionic but it describes the foundation of a new a priori principle inscribing the table’s periodicity and therefore all chemical and quantum laws, rules and principles that flow from it. The following quote from Wikipedia is a summation of how spinor rotation works into my theory……” Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications, the Clifford algebra is often the easiest to work with.[h] A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.” Now we see that the 720° rotation ends with the missing element 72 and may explain its unusual particle physics. To understand how spinors work I am including the following figure (21) from Frank (Tony) Smith’s web book:




Figure 21


There is an additional significance to the 72 number, besides representing the ½-spinor fermion quaternionic rotation, it represents the 72 vertices of the E(6) polytope. So within the E(7) is the E(6) rotation that ends at element 72 (720° and spin½) which represents fermionic dark matter particles! There are 7 rotations of 180° for a total of 1260° or 126 chemical elements. How can these additional rotations be explained, especially since the third one contains 4 more missing elements? At first I thought that there was a quantum field rebound, such as with the so-called anti-particle table before Hydrogen and in BBN but the E(7) spherical model continues without any hint of a rebound except perhaps with my element 108 which is another inert gas and another rotation of 360°. This would mean that elements Krypton 36 and Nihil 108 by virtue of their 360° rotation from 0° would have to be spin– ½ fermions, and being spin– ½ they would also represent mirror anti- fermions. (Please remember that my model uses the analogy of the 18 standard particles from which the particle assignment of the E(8) polytope of Lisi and the periodic table of Stowe-Scerri referenced in figure 10 are constructed) The particle assignments of my E(7) polytope in figure 12 would have to hold true including the force carriers represented by the spin 0 of Helium (Bosonic). (Bosons are just one of the force carriers and are the round, light green dots in the particle assignment diagram)



(Credit Wikipedia)




Whatever the QFT reason, the effect is that particles from one side may flow back and forth forming new element 72 at the E(6)SU(2) level (spin positive to negative) and the element 108 (spin negative back to positive). Another effect is the harmonic oscillation of the elements between the two portals that accounts for the Actinoids and Lanthanoids contractions. It is my belief that this oscillation carries thru the entire field of the table affecting all the elements and therefore some of their anomalous features. There are many other phenomena that this creates. You can see this better by looking at Frank Smith’s articles in Figures 11 and 15 where he shows the 8 particles in two different configurations. To visualize the idea of how the Higgs mechanism portals function and the idea of the quantum field rotations, look at the Schaltenbrand spiral with my PTOE format in figure 22 below:




Figure 22 (My Schaltenbrand spiral with the added P.T.O.E. format and Portals)


While the Schaltenbrand spiral is 100 years old this year, I found it very useful for explaining not only periodicity but also my new concept of quantum field rotations and the Higgs mechanism portal placement.


The first thing in Fig 22 that you notice is the amazing synchronicity between the noble gases’ atomic numbers and the degrees of rotation x10, for instance Argon 18 – 180°, Krypton 36 – 360°,

Xenon 54 – 540°, Nihil** 72 – 720°, Radon 90 – 900°, Nihil** 108 – 1080° and Oganesson 126 – 1260°. Nay Sayers will quickly dismiss this because of the missing 8 protons but are they really missing? I think not. When you carefully look at the chart of heavy nuclides one can find an exact match of isotopic numbers in the series that are 4 atomic numbers ahead of each one of the missing elements (8 spaces for the heavier elements). This may be one reason why there was so much confusion, early on, about the placement of the “Ytterby” series of heavy rare earth elements. I think a case can be made for the saying that “form decides function”. This is strongly suggested by my PTOE polytope which coincidentally shares the 128 vertices of E(7) and whose elements align themselves in the 180° rotation through 1260° representing the new atomic number 126 of the element Oganesson, a noble gas. (There are of course 7 rotations matching the 7 dimensional circles of the E(7) polytope plus the two polar vertices which I assigned to Hydrogen and Helium). If you are still having

doubts about the structure of the new periodic table look at the following table’s diagram with the Schaltenbrand spiral: (Fig 23)



Fig 23

The two tables above are separated by the Schaltenbrand spiral and show the progression of angular spin rotation starting with 180° and terminating with 1260°. The left and right tables are separated by odd and even sums of their principle quantum number and the orbital quantum number of their outer electron subshells. So the left table is n + l = odd and the right is n + l = even. This clearly shows the secondary periodicity of the two subsets that naturally flow from the coincidental angular spin rotation of the groups defined by their noble gases. e.g. 720° = spin ½ and 1080° = spin -½. It is the two S7 twists at 720° and 1080° that may lead to the curious creation of matter/dark matter through the Higgs mechanism at field points # 72 and # 108. The tables also exhibit internal periodicity when you evenly split the rows. For instance, Sc-Mn and Fe-Zn repeat and internal d-shell internal periodicity.


Both left and right tables have this property. Angular momentum plays an important role in the complicated nuclear shell configurations. Here is an excellent reference from Wikipedia that caught my eye, https://en.wikipedia.org/wiki/Nuclear_shell_model because it answered a vexing question about why element 72 is the table’s first fibration:



As you can see from the above sequence of building-up nuclear shells, level 4 (the 5th shell) has a magic number of 30 and level 5 has 42. If these two shells were to combine because of a reversal (twist) there would be a total of 72. I don’t have all the answers but curiously, when I drew 33° diagonals on either side of the table in figure 22, the sum of their differences always added to 72, normally the atomic number of Hafnium. Why? (see the fig 6 table also) A solution could come from the fact that my diagonals inscribe the E(8) David’s star at the center of Lisi’s E(8) polytope. In addition, I have found a direct correspondence with the following E(6) variables who’s two dimensions add up to 72:



(Credit Wikipedia)


Since E(6) lives inside the E(7) then the above may point to a solution for my self-imposed puzzle. This could be a modern rule similar to the Madelung rule which required a deeper investigation. Moreover, how about the DNA spiral with 64 hexagrams? (64 x 2 = 128 = another amazing coincidence and something future medical science can certainly use.)





In order to explain these new discoveries we must return to the 8 dim HyperCube of Frank (Tony) Smith in figure 15 where it is split into two levels of 128 vertices each. As shown in that figure, 128 of those are assigned to mirror fermion status. For a better understanding of how he built his theory from Clifford algebras, look at the below reasoning using Hopf

fibrations and twisted S7. Please notice that the S7 has two twists in it before expanding to Spin(8). :





The subject of spinors is complicated but more complicated is the idea of the Higgs portal which has been receiving a lot of research lately and may provide some answers for the structured building mechanism of the absent 8 chemical elements referred to in figure 20. Here is an excellent article on this subject involving the Clockwork Mechanism of the Higgs portal:


Clockwork Higgs portal model for freeze-in dark matter Jinsu Kim1,* and John McDonald2,† 1 Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea 2 Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom from PHYSICAL REVIEW D 98, 023533 (2018): “The clockwork mechanism can explain interactions which are dimensionally very weak without the need for very large mass scales. We present a model in which the clockwork mechanism generates the very small Higgs portal coupling and dark matter particle mass necessary to explain cold dark matter via the freeze-in mechanism. We introduce a TeV-scale scalar clockwork sector which couples to the Standard Model via the Higgs portal. The dark matter particle is the lightest scalar of the clockwork sector. We show that the freeze in mechanism is dominated by decay of the heavy clockwork scalars to light dark matter scalars and Higgs bosons. In the model considered, we find that freeze-in dark matter is consistent with the clockwork mechanism for global charge q in the range 2

≲ q ≲ 4 when the number of massive scalars is in the range 10 ≤ N ≤ 20. The dark matter scalar mass and portal coupling are independent of q and N. For a typical TeV-scale clockwork sector, the dark matter scalar mass is predicted to be of the order of a MeV.” Here is a very simple graphic showing how the “clockwork mechanism” functions:



(note that the many references I have included here are given from the framework of BBN or Big Bang Nucleosynthesis via the newly found Higgs Mechanism which include the

supernovae etc. creation mechanisms in Tony Smith’s graphic below.)


In the following illustration by Frank (Tony) Smith it is clear that dark scalar mass, dark energy and gravity was on his mind when he created this great visual (there is a complete reconstruction and explanation of the below forms as one part of Tony’s calculations in Appendix 1):






Fig. 23 (Thanks to Frank “Tony” Smith)


I have illustrated the 8 missing chemical elements, 4 Lanthanoids and 4 Actinoids. The top 4 he has designated as scalar ghosts, spin (2,4) conformal gravity + dark energy (Lanthanoid) and the bottom 4 (Actinoid) as spin (2,4) conformal gravity + dark energy. (See appendix 1 below) If you go back to my figures 9 and 22 you will see that my placement of the missing elements coincides with his image above, remember E7(7) SU(8) lives in the E(8)/Cl8 model. The top 4 ghosts are on the E(4) circle and the bottom 4 fall along the E(6) of my PTOE of figure 9 and both marked as Higgs portals in figure 22. (under the Yang-Mills theorem these ghosts are complex scalar fields but anti-commute like fermions) My conjecture on the mechanism is as follows: The top scalar Higgs field is fed by the individual quantum field particles like the electrons, quarks and neutrinos from the surrounding Minkowski Spacetime. Quaternionic fermion particles and then elements are produced through the bottom portal. This is in fact the baryogenic process for the initial creation of matter. While there are many things about this process that need investigation, I predict that the 8 missing elements are matter/dark matter components that are residual mass to the Higgs Mechanism of mass creation/annihilation. If found on the Earth they would almost certainly be found in quantities much less than Rhenium. This also accounts for the diminishing atomic radius of the Lanthanoids towards missing element 72 (the Lanthanide Contraction) and the reverse process from element 108 (Fermium 104) backward to element 73. This backward


expansion would account for the natural cascading radioactivity of the elements below like Uranium etc. and the “attenuated” Actinide Semi-Contraction (the atomic radius contraction does not follow through the entire Actinoids unlike the Lanthanoids). A Coulomb type barrier is reached between missing element 72 and 73, likewise between 108 and 109. I am placing the above two portals at these four elements’ intersections (Hobf fibrations) based on the atomic radius contractions of the Lanthanoids and Actinoids and the two extremes between the Noble gases (72)(108) and the Halogens (73)(109). (if this is confusing, please refer back to figure 6 and 22)



Accepting the idea that the atomic numbers of the periodic table can be advanced by 8 protons from the very middle of the table may make some heads explode, so I am including another visual from Wikipedia. https://en.m.wikipedia.org/wiki/Extended_periodic_table#Electron_configurations




Looking closely at the article you will see that the predicted unbihexium’s (element 126) has the closed proton shell of a noble gas and lies in the so called “island of stability”. If this looks familiar to you, 126 is the atomic number in my PTOE noble gas Oganesson, normally 118 (118 + 8 = 126). It is also predicted that element 126 is one of the “magic” number

elements with a very long half-life. If these calculations, using Hartee-Fock-Bogoliubov method https://en.m.wikipedia.org/wiki/Hartree%E2%80%93Fock_method are true, this is very good news for the advancement of science because it confirms the position of my element 126 as a noble gas and an element that could actually be isolated. Also, please note that the author of the Wikipedia article predicts that 126 may exist as a doubly magic isotope but continues on saying this may be anomalous stability, reflecting my idea about its unusual nuclear structure. (See fig 19) Here is what we should see if the above calculations are true:




In the block on the right above we see the perfect closed electron structure of 2,8,18,36,36,18,8 = 126 as expected. This will bring up an understandable objection from the skeptic: “… where did you get the extra 8 electrons in the middle 36, 36?” Once again referring to Fig 6, we see that the atomic numbers have advanced from 120 to 128 because of my new findings.  There is no reason to believe that the electron shell structure would not follow by adding 8 electrons in the appropriate shells. Looking back at my PTOE in figure 6 we see that the atomic number 118 is now occupied by the element Darmstadtium in the transition metal group directly above element 126. As it turns out, the relativistic effects on Og 118, may account for some of its solid metallic properties instead of the gaseous state of the noble group that it is supposed to belong to and closer to those of Darmstadtium.  Once the 8 new elements are isolated,  a new naming sequence will have to be configured, perhaps one that includes both the new and the traditional names and states. Deciding between naming the new elements with their old name like Oganesson 126 and unbihexium 126, will present scientists with a whole new conundrum. We have already seen that element 126 ends in both closed proton and neutron shells so it would make sense that its electron shell structure would also close, which it does. However, as I noted previously, each of my new 7 periods of 18 elements each are closed by their respective noble gas thus theoretically originating a new rule that I am calling “The Scerri-Madelung” rule of n + ℓ, replacing the name of the old Madelung rule. This was done to acknowledge the fact that Dr. Eric Scerri has pushed for this completed, axiomized rule in so many of his articles and papers. https://www.researchgate.net/institution/University_of_California_Los_Angeles Here is an excerpt from just one such article from The Foundations of Chemistry 2010 12:69–83:




It is important here that the reader heed Professor Scerri’s admonition above, “…it is not the closing of the shells that is chemically relevant but the closing of the periods”. If you look at my table in figure 6 you will see that each of my periods ends in a noble gas as required by Eric Scerri above. By the way, my table in Fig 6 solves the dispute of group-3 by including all 6 of the elements, Sc, Y, La, Lu, Ac and Lr which should make people on both sides very happy. Also. imagine my surprise when I see that nuclear element synthesis research is now heavily focused on the last 8 elements of my new periodic table. The new research is directed towards SHE, Superheavy Elements like 126 and those in particular in and around the island of stability noted above. There is some question lately whether superheavy 126 belongs in the category of magic numbers. This can be easily put to rest by the following formula which has only one conclusion:


Given N = 184, A = 310, x is a multiple of 18 (i.e. 18-108) and Z = At # of a noble gas, Then Z= A – (N + x)


The magic numbers 126, 184 and 310 parent the repopulation of every noble gas from 18 to 108 and this simple formula makes it happen. Here are two examples:


Element 108 = 310 – (184+18) and Argon 18 = 310 – (184+108).


The formula excludes elements 2 (Helium) and element 10 (Neon). Magic doesn’t get any better but there are some who do not think 126 is magic, here is one such early 2001 article using a calculation method similar to Hartree-Fock-Bogoliubov method but comes to a different conclusion:


Shell Stabilization of Super- and Hyperheavy Nuclei Without Magic Gaps

M. Bender,1 W. Nazarewicz,2−4 P.–G. Reinhard5,6

1Gesellschaft f¨ur Schwerionenforschung, Planckstrasse 1, D–64291 Darmstadt, Germany 2Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996 3Physics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831 4Institute of Theoretical Physics, Warsaw University, ul. Ho˙za 69, PL–00681, Warsaw, Poland

5Institut f¨ur Theoretische Physik II, Universit¨at Erlangen-N¨urnberg, Staudtstrasse 7, D–91058 Erlangen, Germany 6Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, P. O. Box 2008, Oak Ridge, Tennessee 37831 (March 23, 2001)


Quantum stabilization of superheavy elements is quantified in terms of the shell-correction energy. We compute the shell correction using self-consistent nuclear models: the non-relativistic Skyrme- Hartree-Fock approach and the relativistic mean-field model, for a number of parameterizations. All the forces applied predict a broad valley of shell stabilization around Z = 120 and N = 172-184.   We also predict two broad regions of shell stabilization in hyperheavy elements with N _ 258 and N _ 308. Due to the large single-particle level density, shell corrections in the superheavy elements differ markedly from those in lighter nuclei. With increasing proton and neutron numbers, the regions of nuclei stabilized by shell effects become poorly localized in particle number, and the familiar pattern of shells separated by magic gaps is basically gone.



In order to give the reader a better idea of how this all comes together, I have made an cut-out section of the P.T.O.E. showing what these calculations mean for the future of the Periodic Table. I have limited my focus to those elements surrounding two Hopf fibrations at 72/73 and 108/109. The third fibration may be centered around element 126 and lead to the collapse of the field into a horn torus, thus repeating the whole field whether it is empty space or the middle of a neutron star. Take a look at my figure below starting with element 67 which leads into the first fibration at elements 72/73. It is at this point that a process called “simultaneous quadruple beta decay” begins:




If you look at my guide in the upper right corner you will see one of the most important findings, that the old original atomic mass has a matching isotope, 4 atomic numbers ahead. This Z+4 element, quadruple-beta process, is predicted by the addition of the first 4 missing elements at Z=72 and consists of the addition of 4 nucleons per element for 32 elements until reaching element 102 where the Z+4 matches the element’s actual new atomic mass. Starting with element 106 there is no Z+4 mass and the actual mass matches the elements’ projected mass. This is where the next 4 of the 8 new elements are added and the equation becomes Z+8. (I will explain more about the quadruple beta process below) Anyone familiar with the writings of Dr. Ida Noddack can reasonably accept the concept of a four element isotope asdvance when postulating our 8 missing elements. After her discovery of element 75, Rhenium, she wrote extensively about the use of isotopes in a new periodic system. This Nobel prize scientist wrote Her ideas about isotopes in the 1930’s and now they are receiving a lot of attention as researchers approach the discovery of the super-heavy elements. To easily verify the +4 isotope rule, go to the amazing resource of the Ptable by Michael Dayah who has created the periodic table in so many useful iterations: https://ptable.com/#Isotopes His most recent version allows the user to mouse-over the element to see which isotopes are associated with it. I was shocked when all the isotopes for that element kept scrolling to the bottom of the page. You will not be able to find such a thorough listing anywhere on the web. What an incredible addition! Looking at the above section of my P.T.O.E., we can state a new +4 isotope rule as follows:


given Z = atomic number and A = mass number, if Z ≥ 72 & < 102, one isotope of that element which is advanced to Z+4 thru quadruple beta transition has an A = to ± .5 A of that Z


The above formula applies only to my original periodic table shown in figure 3 and can be seen in the example above. This rule, in addition to my many other findings, confirms the existence of the 8 missing elements. This formula can’t be explained as either Alpha decay because, that transition is Z-2 or as Beta minus,  which is Z+1. This is not a decay chain but hot fusion nucleosynthesis. The periodic table of elements is a natural, stellar, supernovae or artificial build-up of one element to that of a heavier element. Decay chains are the opposite. In order for these 8 additional elements to have been created and virtually little trace of them left on the Earth, there must have been a very unusual cosmic event, one that could add 8 elements all at one event in time and all the way from 72 to element 110. This would seem to be impossible because of the Coulomb law barrier which limits the outcome but as strange as it seems, there may be process to create the Z+4. The Triple-alpha process shows that it is possible, even though it applies to elements at the beginning of the table. Here is an excellent reference from Wikipedia, https://en.wikipedia.org/wiki/Triple-alpha_process. Below is one way an element can gain +4 protons:


Recently theoretical physicists have tied E8(8) Supergravity to the 256 points of Tony’s Cl (16)SU(8) configuration HyperCube. (See Appendix 2 below the E7(7) light cone)


Some of the recent investigations of Yang-Mills squared and Supergravity are below..


Are all supergravity theories Yang-Mills squared?

A. Anastasiou,1, ∗ L. Borsten,2, † M. J. Duff,3, 4, ‡ A. Marrani,5, 6, § S. Nagy,7, 8, ¶ and M. Zoccali3, ∗∗

1Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden 2School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

3Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom 4Mathematical Institute, University of Oxford, Andrew Wiles Building,

Woodstock Road, Radcliffe Observatory Quarter, Oxford, OX2 6GG, United Kingdom 5Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89A, I-00184, Roma, Italy 6Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit`a di Padova,

and INFN, sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy

7Centre for Astronomy & Particle Theory, University Park, Nottingham, NG7 2RD, United Kingdom 8Center for Mathematical Analysis, Geometry and Dynamical Systems,

Department of Mathematics, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (Dated: July 12, 2017)




I am sure there is an international commission on naming elements where men of prestigious credentials sit down and agree what names to assign to which elements and that is probably a good thing. From the early days of its history the periodic table has been assigned a variety of element names that were left in the dust of advancing science. Names like Virginium, Alabamine, Nipponium, Keltium, Columbium, Cassiopium and Niton just to name a few. I believe that the best of the naming sequences have been to name the prominent scientist who actually contributed to the advancement of Physics and/or Chemistry. Einsteinium or Es is just one such name. I of course have my own wish list of names, so here goes: New element 70 – An – Astonium (after F.W. Aston), new element 71 or Jt – Janetium (after Charles Janet), new element 72 – Keltium or Kt (Hafnium’s original name), new element 73

– Mazurium or Mz (after Edward G. Mazurs), new element 106 – Heisenbergium or Hm (after Werner Heisenberg), new element 107 – Thomsonium or To (after J.J. Thomson), new element 108 – Asimovium or Av (after Isaac Asimov) and lastly, new element 109 – Hahnium or Ha (after Otto Hahn).


As for my original question about the expanding earth theory of my father, there is no doubt that Earth expansion has occurred and it is still in the process. All one has to do is type the words, “Expanding Earth Theory”, and thousands of references will follow. I have also included many of those in appendix #2. Here is just one:


Wen-Bin Shen, Rong Sun, Wei Chen, Zhenguo Zhang, Jin Li, Jiancheng Han, Hao Ding

The expanding Earth at present: evidence from temporal gravity field and space-geodetic data

https://www.annalsofgeophysics.eu/index.php/annals/article/view/4951 See Figure 21



The Earth expansion problem has attracted great interest, and the present study demonstrates that the Earth has been expanding, at least over the recent several decades. Space-geodetic data recorded at stations distributed globally were used (including global positioning system data, very-long-baseline interferometry, satellite laser ranging stations, and stations for Doppler orbitography and radiopositioning integrated by satellite), which covered a period of more than 10 years in the International Terrestrial Reference Frame 2008. A triangular network covering the surface of the Earth was thus constructed based on the spherical Delaunay approach, and average-weighted vertical variations in the Earth surface were estimated. Calculations show that the Earth is expanding at present at a rate of 0.24 ± 0.04 mm/yr. Furthermore, based on the Earth Gravitational Model 2008 and the secular variation rates of the second-degree coefficients estimated by satellite laser ranging and Earth mean-pole data, the principal inertia moments of the Earth (A, B, C) and in particular their temporal variations, were determined: the simple mean value of the three principal inertia moments (i.e., [A+B+C]/3) is gradually increasing. This clearly demonstrates that the Earth has been expanding, at least over the recent decades, and the data show that the Earth is expanding at a rate ranging from 0.17 ± 0.02 mm/yr to 0.21 ± 0.02 mm/yr, which coincides with the space geodetic evidence. Hence, based on both space geodetic observations and gravimetric data, we conclude that the Earth has been expanding at a rate of about 0.2 mm/yr over recent decades.

Fig 24


Many earth scientists now believe that there is an active fission process occurring in and around Earth’s core that could be driving the expansion. (See Figure 25)


Deep-Earth reactor: Nuclear fission, helium, and the geomagnetic field

D. F. Hollenbach and J. M. Herndon

PNAS September 25, 2001 98 (20) 11085-11090; https://doi.org/10.1073/pnas.201393998

Geomagnetic field reversals and changes in intensity are understandable from an energy standpoint as natural consequences of intermittent and/or variable nuclear fission chain reactions deep within the Earth. Moreover, deep-Earth production of helium, having 3He/4He ratios within the range observed from deep-mantle sources, is demonstrated to be a consequence of nuclear fission. Numerical simulations of a planetary-scale geo-reactor were made by using the SCALE sequence of codes. The results clearly demonstrate that such a geo-reactor (i) would function as a fast-neutron fuel breeder reactor; (ii) could, under appropriate conditions, operate over the entire period of geologic time; and (iii) would function in such a manner as to yield variable and/or intermittent output power.

“Further evidence of the georeactor comes from Hawaii and Iceland”, Hollenbach said, “where young lava basalts have been recovered that contain the helium isotopes He3 and He4. While He4 is a byproduct of the decay of natural uranium, He3 can only be produced deep within the Earth in a nuclear reaction.”

Herndon’s latest paper went even further, suggesting that the ratios of He3 to He4 indicated that the geo-reactor is reaching the end of its life — albeit in perhaps a billion years. Hollenbach said he disagrees with that conclusion. This research was performed at the Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC, for the U.S. Department of Energy under contract No. DE-AC05–00OR22725.

Fig 25



Before you dismiss the above as fake science, you should know about the Oklo natural fission site. “Oklo, Gabon, Africa is the only known location for this in the world and consists of 16 sites at which self-sustaining nuclear fission reactions are thought to have taken place approximately 1.7 billion years ago, and ran for a few hundred thousand years, averaging probably less than 100 kW of thermal power during that time.” (Wikipedia) Further research into J.M. Herndon’s papers/web pages reveal that he thinks there is an Actinoid and Lathanoid flux around the nuclear core feeding the nuclear reactor process. Because of its relevance to my theory of mass creation via the Higgs portal phenomena at the beginning and end of both of these rare earth element periods, I predict that an entirely different dark matter/dark energy process is taking place in the core. From one scientist I have the following quote, “Assuming a connection between Earth expansion and capture of dark matter in the Earth’s core, we have two possibilities for a conversion mechanism. First, dark matter directly reacts with Earth’s matter. Second, Earth matter only catalyses the change of dark matter into visible matter.” I like the later explanation because it agrees with the idea of a Higgs mechanism and/or a black hole.


Here is another such expanding Earth article:

https://www.sciencedirect.com/science/article/pii/S1674984718301678 A PDF article about deep mantle plumes and the expanding earth.

Here is a book describing the consequences of the Dirac equation of gravity on the expanding earth. “The Expanding Earth: Some Consequences of Dirac’s Gravitation Hypothesis

By Pascual Jordan”

Also, look for references to the expanding earth theory in the appendix 2 below. Whatever the cause, there is firm evidence that Earth expansion is an ongoing phenomenon.




I have proposed several theses in this paper. Firstly, I suggest that the PTOE is a periodic table in which there are more than 42 perfect triads,  showing that this table is in a primal reduced form on which known principles and rules are based, the triad principle being just one. Secondly, the above primal form shows a perfect sequence of circular rotational degrees in the vertical column of the noble gases that describes the advancement of spin angles from 0° to 1260° thus directly connecting it to the quantum nuclear physics of spin and the QFT itself. Thirdly, and most significant, I have made a reasoned argument backed by scientific references that there are 8 missing heavy rare earth elements. The form of the new table based on the Lie algebra of E(7) shows that these spaces actually exist and that the elements represented can be shown to have a residual bare mass based on the Higgs mechanism. (proof awaits further experiments at the LHC)

I started this study some 50 years ago, expecting to find a reasonable answer to the question about how the Earth’s radius could have increased to over ⅔ its original size. I had no idea it would lead me to one of the biggest mistakes in science, namely the placement of the rare earth elements below Mendeleev’s original periodic table of the elements. I have attempted to show that when this occurred it was done to hide a puzzle that could only be solved with time. It was most certainly an innocent “white lie” but had consequences for the advancement of science and therefore humanity. If scientists had been focused on the 8 missing elements they could have been lead to the same conclusions that I have reached.

Just from the simple view of the chemistry model many new compounds, pharmaceuticals and free energy could have saved lives and made advances in global warming.

I have shown examples of more than four prominent scientists, all of them outstanding in their field of chemistry, who knew that the 8 elements remained unaccounted for but could not change the course of history. Charles Janet made a discovery that would almost certainly explain many aspects of the Periodic Table’s form and function but without the power to disseminate it. I am coincidently the same age as Janet when he published his last work and the same age as Frank (Tony) Smith when his finished his life’s work. I am happy to admit that I stand on the shoulders of these six giants and the many others I have referenced here.


After years of searching, I stumbled across the last few pieces of evidence that I needed. The works of Ida Noddack who was nominated for the Nobel prize for her discovery of one of the last heavy elements, Rhenium, encouraged me to continue. Dr. Eric Scerri is another important figure that influenced me greatly. His openness to discovery and his philosophy of chemistry let me know that some questions have yet to been explained. His books are a deep dive into the structure, history and philosophy of the periodic table of the elements. His communications let me know which of his many books could help me in my journey of discovery and I want to personally thank him for that. When I discovered Isaac Asimov’s book on “The Physical Sciences” first published in 1960, I turned to his page on the periodic table and there it was, the same 8 missing element spaces. It was the work of all the above scientists including the great Dmitri Mendeleev with whose tables I began my journey that encouraged me to continue on. I stopped many times over the years in discouragement but just then, another great figure from history appeared. It was only after I found the monumental work of Garret Lisi on the exceptional Lie algebra of E8 that my final search became a reality. I then found the work of Frank (Tony) Smith who had devoted a lifetime to the Lie algebras and the Quantum Field Theory. His amazing insights into the physics behind E8 and the Clifford algebras made my task very easy. It was at this point that I saw the congruence of the 128 vertices of E7 and the 128 chemical elements of my table. What still remained a mystery was the quantum physics of the 8 missing elements, two of which are noble gases. When I saw the amazing parallel between the inert noble gas atomic numbers along the vertical length of the table and their correspondence with the spin integer degrees i.e. 18-180°, 36-360°, 54-540°, and 72-720°, I knew then that I could solve the mystery. Critical to this discovery was my fiddling around with a particularly long table that I was comparing to the twists of the DNA molecule, e.g. Hopf fibration twists of S7. I could not understand why it took so many twists to get back to the beginning of my completed table. This insight was explained when I saw the Feynberg, Dirac memorial lectures, explaining the reality of spin orientation, concluding his remarks by saying, “Fermions are quaternionic”. This statement helped to propel me forward with the PTOE.


A Bibliography: Important books, references and links either cited or referenced in my study.




“The Chemical Elements”, Helen Miles Davis 1952 “Physics and Chemistry of the Earth”, Ida Noddack, 1920 “The Building Blocks of the Universe”, Issac Asimov, 1960

“The Elements”, Jerome Myer, 1957


“The Abundance of the Elements”, Lawrence Aller, 2015


“The Periodic Table, its Story and its Significance”, Eric Scerri, 2006 “A Scientific Autobiography”, Otto Hahn, 1966


“The Birth of the Earth”, David E. Fisher, 1987 “Flash of Genius”, Alfred A. Garrett, 2012

“Isotopes”, F.W. Aston, London 1922, Edward Arnold and Co.


“The Origin of the Chemical Elements and the Oklo Phenomenon” Dr. Paul K Kuroda, 1982

“The Development of the Periodic Law”, F.P. Preston, 1983

“A New Diagram of the Periodic Table”, R.M. Deeley, 1893


“A tabular Expression of the Periodic Relations of the Elements”, Henry Bassett, 1892

“Graphic Representation of the Periodic System during 100 years”, Edward G. Mazurs, 1974

“The Expanding Earth”, Everett J. Tasset, 1982, SW Missouri University, Archived Books Section

“The helicoidal classification of the elements”, Charles Janet, 1929, Chemical News.





Here are the physics calculations from Frank (Tony) Smith that I have partially used in reference to my PTOE theory. Please understand that I have had to condense over 40 years of his research. If it seems to jump around, be patient. I apologize for including many of his papers (in the form of PDF’s) which were not dated. His below statement of S7-Spin(8) is only where the story begins when he discovers that the S7(7) can be seen as a Clifford Algebra. The twisted fibrations are Hopf :








The 248-dim Lie algebra E8 = 120-dim adjoint Spin(16) + 128-dim half-spinor Spin(16)


is the basis of the physics model of Garrett Lisi (whose root vector images are the basis for most of the root vector images here).


Spin(16) is the bivector Lie algebra of the real Clifford algebra Cl(16)




As Ramon Llull showed about 700 years ago in his Wheel A, the 16 basis vectors of Cl(16) (vertices/letters) combine to form 120 bivectors (vertex pair lines) of Cl(16) which act as the 120 generators of the Lie algebra Spin(16).


The real Clifford algebra 8-periodicity tensor product factorization


Cl(16) = Cl(8) (x) Cl(8)


gives correspondences between 248-dim E8 structure and 256-dim Cl(8) structure, which has graded structure


Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1


Taking the tensor product Cl(8) x Cl(8) to get Cl(16) produces the following 120 Cl(16) bivectors:


  • 28 Spin(8) bivectors of the first Cl(8) in the tensor product
  • 28 Spin(8) bivectors of the second Cl(8) in the tensor product
  • 64 = 8×8 tensor product of the two 8-dim 1-vectors of each the two Cl(8)s


to get the 28+28+64 = 120-dim Cl(16) bivector algebra that produces the 120-dim adjoint of the Lie algebra Spin(16).

The 248-dim Lie algebra E8 = 120-dim adjoint Spin(16) + 128-dim half-spinor Spin(16) is rank 8, and has 240 root vectors that form the vertices of an 8-dim polytope (the Witting polytope).


112 of the 240 vertices are the root vector polytope of the 120-dim rank 8 Spin(16) Lie algbra.


In terms of the 28 bivectors of the first Cl(8) factor and the 28 bivectors of the second Cl(8) factor and the 64 product-of vectors, the 112 are:


  • 24 of the 24-cell root vector polytope of the rank-4 Spin(8) of the first Cl(8) (colored magenta on the following diagram)
  • 24 of the 24-cell root vector polytope of the rank 4 Spin(8) of the second Cl(8) (colored cyan on the following diagram)
  • 64 of the 8×8 product-of-vectors (colored blue on the following diagram


Putting the 112 and 128 together gives the 240 vertices of the E8 root vector polytope




Note that in the above image some of the 240 E8(8) vertices are projected to the same point:


  • each of the 6 vertices in the center (with white dots) are points to which 3 vertices are projected, so that each of the 6 circles with a white dot represents 3 vertices;
  • each of the 24 vertices surrounded by 6 same-color nearest neighbors (with yellow dots) are points to which 2 vertices are projected, so that each of the 24 circles with a yellow dot represents 2 vertices.


Using the color-coding, the 240 root vector vertices of E8 correspond to the graded structure of the 256-dim Cl(8) Clifford algebra as follows:


Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 =


= 1 + 8 + (24+4) + (24+4+28) + (32+3+3+32) + (28+4+24) + (24+4) + 8 + 1

In the above, the black underlined 4+4 = 8 correspond to the 8 E8 Cartan subalgebra elements that are not represented by root vectors, and the black non-underlined 1+3+3+1 = 8 correspond to the 8 elements of 256-dim Cl(8) that do not directly correspond elements of 248-dim E8.


The 240 root vectors have the following physical interpretations:




The Spin(8) whose root vector diagram is the vertices of the first 24-cell, living in the Cl(8) bivectors




the 12 vertices of which form the root vector polytope of the 16-dim U(2,,2) = U(1) x SU(2,2) , where 15-dim rank 3 SU(2,2)


= Conformal Group Spin(2,4) produces Gravity by the MacDowell-Mansouri mechanism (see Rabindra Mohapatra, Unification and Supersymmetry (2nd edition, Springer-Verlag 1992), particularly section 14.6).

Since this group structure acts directly on the 8-dim Kaluza-Klein M4 x CP2, it acts on the associative part given by the associative 3-vector PSI of the dimensional reduction Quaternionic structure

(such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8 dim Octonionic vector space) which is the spatial part of the M4, so that the M4 on which it acts has signature -+++ The U(1) of U(2,2) provides the complex phase of propagators.

This gives Gravity similar to the Conformal Gravity of I. E. Segal, and U(1) propagator phase.




The Spin(8) whose root vector diagram is the vertices of the second 24-cell, living in the Cl

(8) 6-vectors



The 28 6-vectors of Cl(8) correspond to a 28-dim rank 4 Spin(8) Lie algebra after introduction of Quaternionic structure into the E8 physics model (such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8 dim Octonionic vector space ) by using the co-associative 4-vector PHI of the chosen Quaternionic structure to map any 6-vector A to a bivector A /\ PHI, and so mapping the 28 6-vectors onto 28 bivectors that form a

28-dim Lie algebra. The process is somewhat analagous to using a co-associative 4-vector PHI’ in Cl(7) to define a cross-product in 7-dim vector space for vectors a, b (see F. Reese Harvey, Spinors and Calibrations (Academic 1990)) by


a x b = *((a /\ b) /\ PSI)


A stereo view of a 24-cell (the 4th dimension color-coded red-green-blue with green in the middle)




shows that the 4-dim 24-cell has a 3-dim central polytope that is a cuboctahedron



that is the root vector polytope of 15-dim rank 3 Spin(6) = SU(4) that includes 8+1 = 9-dim SU(3)xU(1) = U(3) in the Twistor construction of 6-dim CP3 = SU(4) / U(3)

Projection into a 2-dim space for the root vectors of the rank 2 group SU(3) gives



where the 6 purple vertices form the hexagonal root vector polygon of 8-dim rank 2 SU(3) and the 6 gold vertices correspond to the 6 dimensions of the CP3 Twistor space.


Introduction of a Quaternionic CP3 Twistor space “… induces a mapping of projective spaces CP3 -> QP1 …[with]… fibres … CP1 …” (see R. O. Wells, Complex Geometry in Mathematical Physics (Les Presses de l’Universite de Montreal 1982), particularly section 2.6).


Since CP1 = SU(2) / U(1) an introduction of Quaternionic structure into the E8 physics model (such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8 dim Octonionic vector space ) gives weak force SU(2) through QP1 = Sp(2) / Sp(1)xSp(1) = Spin(5) / SU(2)xSU(2) or, equivalently, through CP3 containing CP2 = SU(3) / U(2) . Since the U(1) of U(3) = SU(3) x U(1) is Abelian, it does not correspond to a root vector vertex and therefore does not appear in the root vector diagrams. Since this group structure is produced by a co-associative 4-vector PHI, it acts on the co-associative part of 8-dim Kaluza- Klein M4 x CP2, which is the CP2 4-dim Internal Symmetry Space of signature ++++

As described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105, the U(2) = SU(2) x U(1) acts on the CP2 as little group, or local isotropy group, while the SU(3) acts globally on the CP2 = SU(3) / U(2) = SU(3) / SU(2) x U(1)

This gives SU(3) x SU(2) x U(1) of the Standard Model.




The product-of-vectors 64 = 8 x 8





With respect to the Cl(8) grading, the first 8 of the 8×8 = 64 is the vector space, and therefore is a natural 8-dim spacetime that after introduction of a preferred Quaternionic substructure such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8 dim Octonionic vector space)

becomes a 4-dim plus 4-dim Kaluza-Klein space of the form M4 x CP2 as described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105, The M4 of signature -+++ contains an associative 3-dim spatial structure, while the CP2 of signature ++++ has a co associative 4-dim structure.

So, the first 8 of the 8×8 = 64, denoted by 8_v , represents 4+4 = 8-dim M4 x CP2 Kaluza-Klein space, where the compact CP2 is small.


As to the second 8 of the 8_v x 8, it lives in the 7-vectors of the Cl(8) grading, and it should represent the 8 Dirac Gammas of the Cl(8) Clifford algebra, so denote it by 8_G so that


the 64 = 8_v x 8_G describes the Kaluza-Klein space and its connection to the Dirac Gammas.




The 128 Spin(16) half-spinors 64 + 64




The 128 is the 128-dim rank 8 symmetric space E8 / Spin(16) of type EVIII known as Rosenfeld’s octo-octonionic projective plane (OxO)P2 (see Arthur L. Besse, Einstein Manifolds (Springer 1987) and Boris Rosenfeld, Geometry of Lie Groups (Kluwer 1997)). Since it is a plane (of 2 8×8 octo-octonionic dimensions), it has structure 128 = 64 + 64 = 8×8 + 8×8. Since it is a half-spinor space (of Spin(16)) its elements are fundamentally fermionic, so

  • one of the 8 in one of the two 8×8 = 64 should correspond to the 8 first-generation fermion particles (denote it by 8_f+)
  • one of the 8 in the other of the two 8×8 = 64 should correspond to the 8 first-generation antiparticles (denote it by 8_f-)

As to the second 8 in the 8_f+ x 8 = 64 and the 8_f- x 8 = 64 it should represent the 8 Dirac Gammas of the Cl(8) Clifford algebra, so denote it by 8_G so that :


128 = 64 + 64 and


the 64 = 8_f+ x 8_G describes the 8 first-generation fermion particles ( neutrino; red, blue,

green up quarks; red, blue, green down quarks, electron ) and their connection to the Dirac Gammas

the 64 = 8_f+ x 8_G describes the 8 first-generation fermion anti-particles and their connection to the Dirac Gammas

Note that these fermions are related to the 8-dim +half-spinor and -half-spinor representations of Spin(1,7), the Lorentz group for the 8-dim space of Cl(8), so that this physics model, based on E8 and Cl(8), satisfies the Coleman-Mandula theorem because, as Steven Weinberg says at pages 382-384 of his book The Quantum Theory of Fields, Vol. III (Cambridge 2000), the important thing about Coleman-Mandula is that fermions in a unified model must “… transform according to the fundamental spinor representations of the Lorentz group … or, strictly speaking, of its covering group Spin(d-1,1).” Where d is the dimension of spacetime in the model.

Note also that the fermion particles are fundamentally all left-handed, and the fermion antiparticles are fundamentally all right-handed. The other handednesses are not different fundamental states, but arise dynamically due to special relativity transformations that can switch handedness of particles that travel at less than light-speed (i.e., that have more than zero rest mass).




Quaternionic Structure

At energies below the Planck/GUT level, the Octonionic structure of the model changes, by freezing out of a preferred Quaternionic substructure, from Real/Octonionic 8-dim spacetime to Quaternionic -+++ associative 4-dim M4 Physical Spacetime plus Quaternionic +++ co- associative 4-dim CP2 = SU(3) / SU(2) x U(1) Internal Symmetry Space.


After Quaternionic structure freezes out,


  • 64 = 8_v x 8_G x 1_Real
  • 64 = 8_f+ x 8_G x 1_Real
  • 64 = 8_f+ x 8_G x 1_Real

transform from 8×8 real matrices to 4×4 Quaternionic matrices


  • 64 = 4_v x 4_G x 4_Quaternion
  • 64 = 4_f+ x 4_G x 4_Quaternion
  • 64 = 4_f+ x 4_G x 4_Quaternion

As can be seen in this chart (from F. Reese Harvey, Spinors and Calibrations (Academic 1990))



The 16×16 = 256-dim Cl(8) = Cl(1,7) = M(16,R) = 16×16 Real Matrix Algebra is transformed into the 8x8x4 = 256-dim Cl(2,6) = M(8,Q) = 8×8 Quaternionic Matrix Algebra and the 8×8 = 64-dim Cl(6) = M(8,R) = 8×8 Real Matrix Algebra is transformed into

the 4x4x4 = 64-dim Cl(2,4) = M(4,Q) = 4×4 Quaternionic Matrix Algebra


and the 8-dim Real column vectors 8_v , 8_f+ , 8_f- become the 2-Quaternionic-dim (8-Real-dim) column vectors 2_Q_v , 2_Q_f+ , 2_Q_f and the 8-dim Real row vectors 8_G become the 2-Quaternionic-dim (8-Real-dim) row vectors 2_Q_G so that

the relationships among the 64 , 64 , 64 , and Gravity and the Standard Model coming from the D3 Lie algebras of Spin(2,4) = SU(2,2) and Spin(6) = SU(4) are maintained after introduction of Quaternionic structure.






There is a Spin(8)-type Triality among the three 64 things


  • 64 = 8_v x 8_G = 2_Q_v x 2_Q_G of Kaluza-Klein space
  • 64 = 8_f+ x 8_G = 2_Q_f+ x 2_Q_G of first-generation fermion particles


  • 64 = 8+f- x 8_G = 2_Q_f- x 2_Q_G of first-generation antiparticles The model has:


  • 16 gauge bosons for MacDowell-Mansouri Gravity plus a complex propagator phase and 12 Standard Model gauge bosons, for a total of 28 gauge bosons (which is also 28 = 8 /\ 8 the number of gauge bosons to be expected from 8 dim vector space)
  • 8 types of fermions (the second and third generations being combinatorial combinations of first-generation fermions. From the point of view of high-energy 8-dim space, in which gauge boson terms have dimension 1 in the Lagrangian and fermion terms have dimension 7/2 in the Lagrangian, the Triality gives a Subtle Supersymmetry

Total Boson Dimensionality = 28 x 1 = 28 = 8 x 7 / 2 = Total Fermion Lagrangian Dimensionality





The natural Lagrangian for the model is


Integration over 8-dim base manifold from 64




MacDowell-Mansouri term from U(2,2)




Gauge Boson term from SU(3)xSU(2)xU(1)




Fermion Particle-Antiparticle term from 64 + 64

This differs from conventional Gravity plus Standard Model in three respects:

o         1 – 8-dim base manifold

o         2 – no Higgs

o         3 – 1 generation of fermions


These differences can be reconciled as follows:


Reduction to 4-dim base manifold and Higgs:

The objective is to reduce the integral over the 8-dim Kaluza-Klein M4 x CP2 to an integral over the 4-dim M4. Since the U(2,2) acts on the M4, there is no problem with it. Since the CP2 = SU(3) / U(2) has global SU(3) action, the SU(3) can be considered as a local gauge group acting on the M4, so there is no problem with it. However, the U(2) acts on the CP2 = SU(3) / U(2) as little group, and so has local action on CP2 and then on M4, so the local action of U(2) on CP2 must be integrated out to get the desired U(2) local action directly on M4. Since the U(1) part of U(2) = U(1) x SU(2) is Abelian, its local action on CP2 and then M4 can be composed to produce a single U(1) local action on M4, wo there is no problem with it. That leaves non-Abelian SU(2) with local action on CP2 and then on M4, and the necessity

to integrate out the local CP2 action to get something acting locally directly on M4. This is done by a mechanism due to Meinhard Mayer, The Geometry of Symmetry Breaking in Gauge Theories, Acta Physica Austriaca, Suppl. XXIII (1981) 477-490 where he says:

“… We start out from … four-dimensional M [ M4 ] …[and]… R …[that is]… obtained from … G/H [ CP2 = SU

(3) / U(2) ] … the physical surviving components of A and F, which we will denote by A and F, respectively, are a one-form and two form on M [M4] with values in H [SU(2)] …the remaining components will be subjected to symmetry and gauge transformations, thus reducing the Yang-Mills action …[on M4 x CP2]… to a Yang-Mills-Ginzburg-Landau action on M [M4] … Consider the Yang-Mills action on R …

S_YM = Integral Tr ( F /\ *F )


… We can … split the curvature F into components along M [M4] (spacetime) and those along directions tangent to G/H [CP2] .


We denote the former components by F_!! and the latter by F_?? , whereas the mixed components (one along M, the other along G/H) will be denoted by F_!? … Then the integrand … becomes


Tr( F_!! F^!! + 2 F_!? F^!? + F_?? F^?? )


… The first term .. becomes the [SU(2)] Yang-Mills action for the reduced [SU(2)] Yang-Mills theory …


the middle term .. becomes, symbolically, Tr Sum D_! PHI(?) D^! PHI(?) where PHI(?) is the Lie-algebra valued 0-form corresponding to the invariance of A with respect tothe vector field ? , in the G/H [CP2] direction … the third term … involves the contraction F_?? of F with two vector fields lying along G/H [CP2] … we make use of the equation [from Mayer-Trautman, Acta Physica Austriaca, Suppl. XXIII (1981) 433-476, equation 6.18]

2 F_?? = [ PHI(?) , PHI(?) ] – PHI([?,?])




… Thus, the third term … reduces to what is essentially a Ginzburg-Landau potential in the components of PHI: Tr F_?? F^?? = (1/4) Tr ( [ PHI , PHI ] – PHI )^2

… special cases which were considered show that …[the equation immediately above]… has indeed the properties required of a Ginzburg_Landau-Higgs potential, and moreover the relative signs of the quartic and quadratic terms are correct, and only one overall normalization constant … is needed. “. (see also S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, Wiley (1963), especially section II.11)

due to the work of Meinhard Mayer,


dimensional reduction to 4-dim M4 Physcial Spacetime, with respect to the SU(2) gauge group, gives the Higgs mechanism.


As to how to combine local Lagrangians in terms of E8, note that there are 7 independent Root Vector Polytopes / Lattices of


type E8, denoted E8_1, E8_2, E8_3, E8_4, E8_5, E8_6, E8_7. Some of them have vertices in commmon, but they are all distinct.

All of the 7 independent Root Vector Polytope Lie algebras E8_i correspond to E8 Lattices consistent with Octonion Multiplication, and the the 7 Lie algebras / Lattices / Root Vector Polytopes E8_i are related to each other as the 7 Octonion imaginaries i,j,k,e,ie,je,ke , so the copies of E8 might combine according to the rules of octonion multiplication, globally arranging themselves like integral octonions.



Such a Spin Foam model might be related to the 26-dim Bosonic String model described in CERN preprint CERN-CDS-EXT

If the 128 Spin(16) half-spinors are put on integral octonion vertices, and the 120-dim adjoint Spin(16) generators on links between integral octonion vertices, a realistic Spin Foam model might be produced, related to the copies of the 27 dimensional exceptional Jordan algebra contained in each E8.

2004-031 in which fermions come from orbifolding and the 7 independent E8_i are used in constructing D8 branes.


Given the E8 / Cl(8) model and its Lagrangian, how about Physics Calculations ?




It is my opinion that recent experimental results from CDF and D0 are substantially consistent with 3 mass states of the Truth Quark, including the existence of a Tquark-Higgs state around ( M_H


= 143-160 , M_T = 130-145 ), but the consensus view is otherwise.




Force Strengths

The model Lagrangian (just looking at spacetime and gauge bosons and ignoring spinor fermions etc) is the integral over spacetime of gauge boson terms, so THE FORCE STRENGTH IS MADE UP OF TWO PARTS:

  • the relevant spacetime manifold of gauge group global action


  • the relevant symmetric space manifold of gauge group local action

Ignoring for this exposition details about the 4-dim internal symmetry space, and ignoring conformal stuff (Higgs etc), the 4 dim spacetime Lagrangian gauge boson term is the integral over spacetime as seen by gauge boson acting globally of the gauge force term of the gauge boson acting locally for the gauge bosons of each of the four forces:



Higgs and W-boson Masses


As with forces strengths, the calculations produce ratios of masses, so that only one mass need be chosen to set the mass scale. In the E8 model, the value of the fundamental mass scale vacuum expectation value v = <PHI> of the Higgs scalar field is set to be the sum of the physical masses of the weak bosons, W+, W-, and Z0, whose tree-level masses will then be shown by ratio calculations to be 80.326 GeV, 80.326 GeV, and 91.862 GeV, respectively,

and so that the electron mass will then be 0.5110 MeV.

The relationship between the Higgs mass and v is given by the Ginzburg-Landau term from the Mayer Mechanism as (1/4) Tr ( [ PHI , PHI ] – PHI )^2

or, in the notation of hep-ph/9806009 by Guang-jiong Ni

(1/4!) lambda PHI^4 – (1/2) sigma PHI^2 where the Higgs mass M_H = sqrt( 2 sigma )

Ni says: “… the invariant meaning of the constant lambda in the Lagrangian is not the coupling constant, the latter will change after quantization … The invariant meaning of lambda is nothing but the ratio of two mass scales:


lambda = 3 ( M_H / PHI )^2 which remains unchanged irrespective of the order …”.

Since <PHI>^2 = v^2, and assuming at tree-level that lambda = 1 ( a value consistent with the Higgs Tquark condensate model of Michio Hashimoto, Masaharu Tanabashi, and Koichi Yamawaki in their paper at hep-ph/0311165 ), we have, at tree-level M_H^2 / v^2 = 1 / 3

In the E8 model, the fundamental mass scale vacuum expectation value v of the Higgs scalar field is the fundamental mass parameter that is to be set to define all other masses by the mass ratio formulas of the model and


v is set to be 252.514 GeV


so that


M_H = v /sqrt(3) = 145.789 GeV



To get W-boson masses, denote the 3 SU(2) high-energy weak bosons (massless at energies higher than the electroweak unification) by W+, W-, and W0, corresponding to the massive physical weak bosons W+, W-, and Z0. The triplet { W+, W-, W0 } couples directly with the T – Tbar quark-antiquark pair, so that the total mass of the triplet { W+, W-, W0 } at the electroweak unification is equal to the total mass of a T – Tbar pair, 259.031 GeV.

The triplet { W+, W-, Z0 } couples directly with the Higgs scalar, which carries the Higgs mechanism by which the W0 becomes the physical Z0, so that the total mass of the triplet { W+, W-, Z0 } is equal to the vacuum expectation value v of the Higgs scalar field, v = 252.514 GeV. What are individual masses of members of the triplet { W+, W-, Z0 } ?

First, look at the triplet { W+, W-, W0 } which can be represented by the 3-sphere S^3. The Hopf fibration of S^3 as S^1 –> S^3 –> S^2

gives a decomposition of the W bosons into the neutral W0 corresponding to S^1 and the charged pair W+ and W- corresponding to S^2. The mass ratio of the sum of the masses of W+ and W- to the mass of W0 should be the volume ratio of the S^2 in S^3 to the S^1 in S3.

o         The unit sphere S^3 in R^4 is normalized by 1 / 2.

o         The unit sphere S^2 in R^3 is normalized by 1 / sqrt( 3 ).

o         The unit sphere S^1 in R^2 is normalized by 1 / sqrt( 2 ).

The ratio of the sum of the W+ and W- masses to the W0 mass should then be (2 / sqrt3) V(S^2) / (2 / sqrt2) V(S^1) = 1.632993

Since the total mass of the triplet { W+, W-, W0 } is 259.031 GeV, the total mass of a T – Tbar pair, and the charged weak bosons have equal mass, we have M_W+ = M_W- = 80.326 GeV and M_W0 = 98.379 GeV.

The charged W+/- neutrino-electron interchange must be symmetric with the electron-neutrino interchange, so that the absence of right-handed neutrino particles requires that the charged W+/- SU(2) weak bosons act only on left-handed electrons. Each gauge boson must act consistently on the entire Dirac fermion particle sector, so that the charged W+/- SU(2) weak bosons act only on left-handed fermion particles of all types.


The neutral W0 weak boson does not interchange Weyl neutrinos with Dirac fermions, and so is not restricted to left-handed fermions, but also has a component that acts on both types of fermions, both left-handed and right-handed, conserving parity.


However, the neutral W0 weak bosons are related to the charged W+/- weak bosons by custodial SU(2) symmetry, so that the left-handed component of the neutral W0 must be equal to the left-handed (entire) component of the charged W+/-. Since the mass of the W0 is greater than the mass of the W+/-, there remains for the W0 a component acting on both types of fermions.

Therefore the full W0 neutral weak boson interaction is proportional to (M_W+/-^2 / M_W0^2) acting on left-handed fermions and


(1 – (M_W+/-^2 / M_W0^2)) acting on both types of fermions.


If (1 – (M_W+/-2 / M_W0^2)) is defined to be sin( theta_w )^2 and denoted by K,

and if the strength of the W+/- charged weak force (and of the custodial SU(2) symmetry) is denoted by T, then the W0 neutral weak interaction can be written as W0L = T + K and W0LR = K.

Since the W0 acts as W0L with respect to the parity violating SU(2) weak force and as W0LR with respect to the parity conserving U(1) electromagnetic force of the U(1) subgroup of SU(2), the W0 mass mW0 has two components: the parity violating SU(2) part mW0L that is equal to M_W+/ the parity conserving part M_W0LR that acts like a heavy photon.

As M_W0 = 98.379 GeV = M_W0L + M_W0LR, and as M_W0L = M_W+/- = 80.326 GeV, we have M_W0LR = 18.053 GeV.


Denote by *alphaE = *e^2 the force strength of the weak parity conserving U(1) electromagnetic type force that acts through the U(1) subgroup of SU(2). The electromagnetic force strength alphaE = e^2 = 1 / 137.03608 was calculated above using the volume V(S^1) of an S^1 in R^2, normalized by 1 / sqrt( 2 ).

The *alphaE force is part of the SU(2) weak force whose strength alphaW = w^2 was calculated above using the volume V (S^2) of an S^2 \subset R^3, normalized by 1 / sqrt( 3 ).


Also, the electromagnetic force strength alphaE = e^2 was calculated above using a 4-dimensional spacetime with global structure of the 4-torus T^4 made up of four S^1 1-spheres, while the SU(2) weak force strength alphaW = w^2 was calculated above using two 2-spheres S^2 x S^2, each of which contains one 1-sphere of the *alphaE force.



o         *alphaE = alphaE ( sqrt( 2 ) / sqrt( 3) )(2 / 4) = alphaE / sqrt( 6 ),

o         *e = e / (4th root of 6) = e / 1.565 ,


and the mass mW0LR must be reduced to an effective value M_W0LReff = M_W0LR / 1.565 = 18.053/1.565 = 11.536 GeV for the *alphaE force to act like an electromagnetic force in the E8 model:


*e M_W0LR = e (1/5.65) M_W0LR = e M_Z0,


where the physical effective neutral weak boson is denoted by Z0.

Therefore, the correct E8 model values for weak boson masses and the Weinberg angle theta_w are:


M_W+ = M_W- = 80.326 GeV;

M_Z0 = 80.326 + 11.536 = 91.862 GeV;


Sin(theta_w)^2 = 1 – (M_W+/- / M_Z0)^2 = 1 – ( 6452.2663 / 8438.6270 ) = 0.235.


Radiative corrections are not taken into account here, and may change these tree-level values somewhat.




Kobayashi-Maskawa Parameters

The Kobayashi-Maskawa parameters are determined in terms of the sum of the masses of the 30 first-generation fermion particles and antiparticles, denoted by Smf1 = 7.508 GeV, and the similar sums for second-generation and third-generation fermions, denoted by Smf2 = 32.94504 GeV and Smf3 = 1,629.2675 GeV.

The reason for using sums of all fermion masses (rather than sums of quark masses only) is that all fermions are in the same spinor representation of Spin(8), and the Spin(8) representations are considered to be fundamental.

The following formulas use the above masses to calculate Kobayashi-Maskawa parameters:


o         phase angle d13 = 1 radian ( unit length on a phase circumference )

o      sin(alpha) = s12 = [me+3md+3mu]/sqrt([me^2+3md^2+3mu^2]+[mmu^2+3ms^2+3mc^2]) = 0.222198

o      sin(beta) = s13 = [me+3md+3mu]/sqrt([me^2+3md^2+3mu^2]+[mtau^2+3mb^2+3mt^2]) = 0.004608

o         sin(*gamma) = [mmu+3ms+3mc]/sqrt([mtau^2+3mb^2+3mt^2]+[mmu^2+3ms^2+3mc^2])

o         sin(gamma) = s23 = sin(*gamma) sqrt( Sigmaf2 / Sigmaf1 ) = 0.04234886

The factor sqrt( Smf2 /Smf1 ) appears in s23 because an s23 transition is to the second generation and not all the way to the first generation, so that the end product of an s23 transition has a greater available energy than s12 or s13 transitions by a factor of Smf2 / Smf1 .



E8 Geometry and Physics (undated)

( E8, the Lie algebra of an E8 Physics Model, is rank 8 and has 8+240 = 248 dimensions – Compact Version – Euclidean Signature – for clarity of exposition – much of this is from the book Einstein Manifolds (by Arthur L. Besse, Springer- Verlag 1987): Type EVIII rank 8 Symmetric Space Rosenfeld’s Elliptic Projective Plane (OxO)P2

E8 / Spin(16) = 64 + 64


The Octonionic structure of (OxO)P2 gives it a natural torsion structure * for which 64 looks like ( 8 fermion particles ) x ( 8 Dirac Gammas ) and 64 looks like ( 8 fermion antiparticles ) x ( 8 Dirac Gammas ) Type BDI(8,8) rank 8 Symmetric Space real 8-Grassmannian manifold of R16 or set of the RP7 in RP1

Spin(16) / ( Spin(8) x Spin(8) ) = 64


Spin(16) is rank 8 and has 8+112 = 120 dimensions and looks like a 64-dim Base Manifold whose curvature is determined by a 28+28=56-dim Gauge Group Spin(8) x Spin(8) The 64-dim Base Manifold looks like ( 8-dim Kaluza-Klein spacetime ) x ( 8 Dirac Gammas )


Due to the special isomorphisms Spin(6) = SU(4) and Spin(2) = U(1) and the topological equality RP1 = S1


Spin(8) / ( Spin(6) x Spin(2) ) = real 2-Grassmannian manifold of R8 or set of the RP1 in RP7 Spin(6) gives Conformal MacDowell-Mansouri Gravity

Spin(8) / U(4) = Spin(8) / SU(4) x U(1) = set of metric-compatible fibrations S1 -> RP7 -> CP3 SU(4) / SU(3)xU(1) = CP3 SU(3) gives color force U(1) gives electromagnetism

CP3 contains CP2 = SU(3) / U(1) x SU(2) and so gives SU(2) weak force



Note that in the above image some of the 240 E8(8) vertices are projected to the same point:


o         each of the 6 vertices in the center (with white dots) are points to which 3 vertices are projected, so that each of the 6 circles with a white dot represents 3 vertices;

o         each of the 24 vertices surrounded by 6 same-color nearest neighbors (with yellow dots) are points to which 2 vertices are projected, so that each of the 24 circles with a yellow dot represents 2 vertices.

Using the color-coding, the 240 root vector vertices of E8 correspond to the graded structure of the 256-dim Cl(8) Clifford algebra as follows:

Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 =


= 1 + 8 + (24+4) + (24+4+28) + (32+3+3+32) + (28+4+24) + (24+4) + 8 + 1

In the above, the black underlined 4+4 = 8 correspond to the 8 E8 Cartan subalgebra elements that are not represented by root vectors, and the black non-underlined 1+3+3+1 = 8 correspond to the 8 elements of 256-dim Cl(8) that do not directly correspond elements of 248-dim E8.





Note that in the above image some of the 240 E8 vertices are projected to the same point:

o         the first and third circles from the center, which have 12 points, to each of which 2 vertices are projected, so that each circle represents 24 vertices; and


o         the central point of each of the 6 clusters corresponding to the 27-dimensional exceptional Jordan algebra J3 (O), to each of which 3 vertices are projected, so that each cluster represents 27 vertices.

Note further that I am pretty sure of the color-identification of the points in the above figure, except for possible misplacements within the central 72-vertex E6 root vector polytope ( red 24-cell Spin(8) root vector polytope, blue 8 Spin(8) vector vertices of dual 24-cell plus 8 complex Spin(10) correspondents, and green 16 Spin(8) +/- half-spinor vertices of dual 24-cell plus 16 complex E6 correspondents). Since an important aspect of both the E8 model and the Cl(8) model is the representation of fermions by spinor- type structures, such as by 8-dimensional Spin(8) +half-spinors and 8-dimensional Spin(8) -half-spinors in the Cl(8) model and by 128-dimensional Spin(16) half-spinors in the E8 model (based on the identification of 248- dimensional E8 as the sum of the 120-dimensional Spin(16) adjoint plus a 128-dimensional Spin(16) half-spinor space) it is useful to see how spinor-type structures appear in the above-described structure of E8 by enclosing their enumeration in [ brackets ]:

Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 =

= 1 + 8 + (24+4) + (27+1+27+1) + (32+27+3+8) + (27+1+27+1) + (27+1) +8 + 1

D4 = (24+4)

D5 = D4 + 1 + 8 + 8 E6 = D5 + [32] + 1 E7 = E6 + ([16]+8+3+1) + [16]+8+3

E8 = E7 + ([16]+8+3+1) + ([16]+8+3+3) + (1+[16]+8+3+1) + ([16]+8+3+1)

256-dim Cl(8) – 248-dim E8 = 8

Note that each of the [16] above live in the 27-dimensional exceptional Jordan algebra J3(O), which is the algebra of 3×3 Hermitian Octonion matrices


E8 Root Vector Projections – Frank Dodd (Tony) Smith, Jr. – 2008


The basis of my E8 physics model, which is based on Garrett Lisi’s E8 physics model, is the figure




in which the two D4 inside E8 are shown in cyan and magenta.




Another figure I use, based on another projection of E8 root vectors into 2 dimensions, shows a nesting D4 in D5 in E6 in E7 in E8 in which the two D4 of E8 are ( showing multiplicities 3 and 2 of points to which multiple root vectors are projected




The central red 24 of the inner D4 are obviously contained in E6 in E8.


The outer magenta 6 of the outer D4 in E7 outside E6 are the two central 3 of: 126 root vectors of E7 – 72 root vectors of E6 = 54 = 2x(24+3) =

2 circular 12+12 + 2 central 3


The magenta 6 root vectors of the two central 3 correspond to the root vectors of a 7-sphere S7 ( which, although not a Lie algebra due to Octonion non-associativity, is a Malcev algebra )


The magenta 48 = 54-6 of the two E7 circular 12+12 are related to the blue 16 of 8-complex dimensional Kaluza-Klein vector spacetime D5 outside red D4 so that the magenta 48 and blue 16 combine to form a 48+16 = 64-real-dimensional = 8-octonionic-dimensional vector spacetime .

Therefore, E7 looks like E6 plus octonification of vector spacetime plus a 7-sphere S7.


The outer cyan 18 of the outer D4 in E8 outside E7 are the four central 3 plus outside 6 of: 240 root vectors of E8 – 126 root vectors of E7 = 114 = 108 + 6 = 4x(24+3) + 6 =4 circular 12+12 + 4 central 3 + outside 6


The cyan 12 root vectors of the four central 3 correspond to the root vectors of the 14-dimensional Lie algebra G2.

The outside 6 root vectors correspond to the root vectors of a 7-sphere S7 ( which, although not a Lie algebra due to Octonion non-associativity, is a Malcev algebra )

The cyan 96 = 108-12 of the four E8 circular 12+12 are related to the green 32 of 16-complex dimensional full-spinor E6 fermion first-generation particles and antiparticles so that the cyan 96 and green 32 combine to form 96+32 = 128-real-dimensional = 16-octonionic-dimensional representation space for full-spinor fermion first-generation particles and antiparticles.

Therefore, E8 looks like E7 plus octonification of representation space for full-spinor fermion first- generation particles and antiparticles plus G2 plus a 7-sphere S7.


The outer magenta-cyan 24 of the outer D4 of the nesting D4 in D5 in E6 in E7 in E8 is made up of:


7-sphere from E7 plus G2 and 7-sphere from E8 The outer magenta-cyan 24




look like the outer red-green-blue 24




in my basic E8 physics model figure



in which the second D4 is represented by its magenta 24




so the physics interpretations of the two projections are related by interchanging, in the basic figure, its outer red-green-blue 24 and its magenta 24.


The basic figure, so interchanged, and with cyan changed to bright yellow and magenta changed to dark yellow. looks like




from the view of the 240 of E8 as 8 circles of 30.


To get more feel for the 8 circles of 30, consider the comment by rntsai on N-category Cafe that mentioned Kostant’s “… decomposition of e8 into 31 cartan’s …” and said: “… It’s … related to e8/(d4

+d4) decomposition :


e8/(d4+d4)= (28,1)+(1,28) + (8v,8v) + (8S+,8S+) + (8S-,8S-)

The last 3 terms can be seen as 24 8-dim spaces …The other 7 cartans are inside d4+d4 … there are probably several ways to identify [them]…”.

Another way is to decompose d4 into a 14-dim G2 plus two 7 spheres S7 + S7, getting d4 = 14 + 7 + 7

14- dim rank-2 G2 has 7 = 14/2 Cartans and G2 can be seen as the sum of two 7-dimensional representations. If each 7 is represented by the 7 imaginary octonions { i,j,k,e,ie,je,ke } then the 7 Cartans of G2 are the 7 pairs (one from each of the 7 in G2):


i i

  1. j
  2. k

e e

ie ie je je ke ke

Note that to make an Abelian Cartan, the pairs must match, because only matching pairs close to form an Abelian Cartan (this can be seen by looking at the octonion products).

  1. dim rank-4 d4 has 28/4 = 7 Cartans and d4 looks like G2 plus S7 plus S7 and since G2 decomposes into two 7 representations d4 decomposes into 7 + 7 + 7 + 7 ( where the first two 7 are from G2 and the other two come from the two S7


and the 7 Cartans of d4 are ( in terms of octonion imaginaries )


i i i i j j j j k k k k e e e e

ie ie ie ie je je je je ke ke ke ke


Again, note that all elements of the quadruples must match to get Abelian Cartan structure.

When you look at d4 + d4 to get 8-element Cartans of E8, all 8 elements must again match up to get Abelian Cartan structure, so the 7 Cartans of E8 that come from d4 + d4 look like


i i i i i i i i j j j j j j j j k k k k k k k k e e e e e e e e

ie ie ie ie ie ie ie ie je je je je je je je je ke ke ke ke ke ke ke ke


Of course, this octonion structure is also reflected in the “… 24 8-dim spaces …” described as “… (8v,8v) + (8S+,8S+) + (8S-,8S-) …” so that all 31 of the 8-dim Cartans of E8 have nice octonionic structure.

Also note that when you make a 240-element E8 root vector diagram of 8 circles each with 30 vertices, 8 of the 248 E8 generators are missing, so that you must leave out one of the 31 Cartan 8-element sets. Seeing E8 in terms of E8 = 120 + 128 = d4 + d4 + 8×8 + 8×8 + 8×8 it is most natural to see the Cartan as being one of the Cartan sets of 8 coming from the d4 + d4, but you could see the E8 from other points of view by using other Cartan sets of 8 to determine which of the 248 were the 8 omitted from the root vector diagram where the E8/octonionic { 1,i,j,k,e,ie,je,ke } represent 8-dim spacetime (8v) or 8 fermion fundamental first-generation particles (8S+) or 8 fermion fundamental first-generation antiparticles (8S-) and the g1 g8 are Dirac gammas of 8-dimensional Kaluza-Klein spacetime.

Those Dirac gammas, although they have intrinsic Clifford algebra structure, can be regarded with respect to E8/octonionic structure as only indicating physical Dirac gamma component structure of the E8/octonionic { 1,i,j,k,e,ie,je,ke } so that they are consistent with each of the rows


1g1 1g2 1g3 1g4 1g5 1g6 1g7 1g8 ig1 ig2 ig3 ig4 ig5 ig6 ig7 ig8 jg1 jg2 jg3 jg4 jg5 jg6 jg7 jg8 kg1 kg2 kg3 kg4 kg5 kg6 kg7 kg8 eg1 eg2 eg3 eg4 eg5 eg6 eg7 eg8 ieg1 ieg2 ieg3 ieg4 ieg5 ieg6 ieg7 ieg8 jeg1 jeg2 jeg3 jeg4 jeg5 jeg6 jeg7 jeg8 keg1 keg2 keg3 keg4 keg5 keg6 keg7 keg8

being able to represent an 8-element E8 Cartan subalgebra, no matter which of the three representations 8v, 8S+, or 8S- (which are related to each other by triality) is used.


D4 and D4* and Higgs

Frank Dodd (Tony) Smith, Jr. – March 2008


Consider the two D4 in the E8 physics model based on E8 / Spin(16) that I like to use, and denote them D4 and D4* to distinguish between them.



When transformed from the 8-circle projection to the basic projection of my E8 physics model, D4 and D4* look like




The basic figure of my E8 physics model




has, for the D4 and D4*, cyan intead of bright yellow and magenta instead of dark yellow, so that in the basic figure the D4 and D4* look like




28-dim D4 ( with 24 root vectors ) gives Gravity from its 15+1 = 16-dimensional D3xU(1). The 12-dimensional symmetric space D4 / D3xU(1) corresponds to the Lie spheres in R8.


28- dim D4* ( with 24 root vectors ) gives the Standard Model SU(3) and SU(2) and U(1) from its 15+1 = 16 dimensional A3xU(1) = U(4).


The 12-dimensional symmetric space D4* / U(4) corresponds to the set of complex structures in R8.

Since D3 = A3 and D3xU(1) = U(4), the Lie spheres in R8 looks like the set of complex structures in R8, so from when I refer to the “set of complex structures in R8” I am referring to both of those things.


Since D4 describes Gravity acting on 4-dimensional M4 physical spacetime, the 12-dimensional set of complex structures in R8 of the D4 symmetric space correspond to the ways that M4 can be fit inside the prior-to-dimensional reduction 8-dimensional spacetime.


Since D4* describes the Standard Model SU(3) and SU(2) and U(1) acting on 4-dimensional CP2 , the 12-dimensional set of complex structures in R8 of the D4* symmetric space correspond to the ways that CP2 can be fit inside the prior- to-dimensional-reduction 8-dimensional spacetime.

After dimensional reduction, the uniform R8 is transfomed into a 4+4-dimensional M4xCP2 Kaluza-Klein space, and consistency with the structure of the M4xCP2 Kaluza-Klein space is a restriction on the 12+12 = 24 degrees of freedom of the D4 and D4* symmetric spaces, and the geometry of that dimensional reduction gives, by the Mayer Mechanism, the Higgs scalar, which is 2-complex dimensional or 4-real dimensional ( see, for example, Introduction to Gauge Field Theory, by Bailin and Love (rev ed IOP 1993 at pages 235, 238)).


Since the 12+12 = 24 degrees of freedom of the D4 and D4* symmetric spaces produce the 4 degrees of freedom of the Higgs scalar,


the remaining 24-4 = 20 degrees of freedom do not correspond to physics in our M4xCP2 low-energy Kaluza-Klein realm, but to phenomena in the high-energy realm of prior-to-dimensional-reduction 8-dimensional spacetime. Having seen how the Higgs etc comes from the 28-16 = 12-real-dimensional symmetric spaces Spin(8) / U(4) of D4 and D4* consider the physical interpretation of the 16-real dimensional U(4) subgroup of Spin(8) in D4* that produces the Standard Model.


E6 in E8, PSL(2,11) and E8(p)




E6 in E8

Note that in the below images some of the 240 E8(8) vertices are projected to the same point, so that when counting root vectors keep in mind:

o         each of the vertices in the center with white dots are points to which 3 vertices are projected, so that each of the 6 circles with a white dot represents 3 vertices;

o         each of the vertices surrounded by 6 same-color nearest neighbors with yellow dots are points to which 2 vertices are projected, so that each of the 24 circles with a yellow dot represents 2 vertices.


The right figure in the image below shows the 240 root vectors of 248-dimensional E8:




The left figure in the image above shows the 72 root vectors of 78-dimensional E6 which is made up of:

o         28-dimensional D4 (24 cyan root vectors)

o         ( 8+8 ) complex D4 vectors ( 8+8 blue root vectors )

o         1 Cartan subalgebra element for complexification of D4 vectors

o         ( 8+8 ) complex D4 +half-spinors( 8+8 red root vectors )

o         ( 8+8 ) complex D4 -half-spinors ( 8+8 green root vectors )

o         1 Cartan subalgebra element for complexification of D4 spinors

Given a basis {1,i} of the complex numbers, the 3 sets of 8+8 in E6 can each be regarded as representing 8 complex elements of the form


8 x 1 + 8 x I

so that the representation spaces of 8-dimensional Kaluza-Klein spacetime and the 8 fundamental first-generation fermion particles and the 8 fundamental first-generation fermion antiparticles can be seen as complex as is useful for calculation of particle masses and force strength constants using an approach motivated by that of Armand Wyler.



To see how to expand from E6 to E8, consider that E8 has octonionic structure, evidenced by the fact that E8 / Spin(16)= (OxO)P2 = Rosenfeld’s ocoto-octonionic projective plane., so that E6 must be “Octonified” as follows: First, consider the D4 part of E6, which is not explicitly complexified, so it must be extended to operate on the octonions of E8. Ignoring signature subtleties, E6 has one D4 = Spin(8), whose action must be extended to octonion space. Consider the full spinor representation of Spin(8). According to F. Reese Harvey in his book “Spinors and Calibrations” (Academic 1990 at page 287): “… Spin(8) acts transitively on S7 x S7 …”, where each of the two S7 are the unit sphere in each of the 8-dimensional half-spinor representation spaces of Spin(8). So, to expand to E8, each of the S7 must be Octonified. This is done by introducing an octonion product among the points of each S7. Unlike S3 with a quaternion product that closes to form a Lie group, S7 under an octonion product does not close, but expands to form a 28-dimensional Spin(8) that can be seen as an S7, another S7, and a 14-dimensional G2. Since each of the two S7 expands to a Spin(8): Expanding E6 to E8 goes from the one D4 in E6 to 2 D4 in E8. The 24 root vectors of the second D4 are the 24 magenta root vectors in the central figure of the above image.

o         Second, consider each of the 3 ( blue, red, and green in the E6 left figure of the above image ) sets of 8+8 root vectors in E6 with complex form 8 x 1 + 8 x e ( for complex basis here denoted {1,e} )

To Octonify them they must be expanded from complex with basis {1,e} to octonion with basis {1,i,j,k,e,ie,je, ke} by adding 6 more root vectors ( {i,j,k} added corresponding to {1} and {ie,je,ke} added corresponding to {e} ) such that the each of the 3 sets of 8+8 = 16 can, when expanded to E8, each be regarded a representing 8+8+8+8+8+8+8+8 = 64 octonionic elements of the form

8 x 1 + 8 x i + 8 x j + 8 x k + 8 x e + 8 x ie + 8 x je + 8 x ke

by adding 6 new sets of 8 root vectors for each of the vector blue, +half-spinor red, and -half-spinor green as shown in the central figure of the above image, for a total of 3 x 6×8 = 3×48 = 6 x 24 =144 of the root vectors in the central figure of the above


image. ( Note that, since the complex structure of E6 remains implicitly in the structure of E8, it is still available for use by Armand Wyler-type approaches ( such as I use in my model ) for calculation of force strengths, particle masses, etc. ) So, to expand the 72 root vectors of 78-dimensional E6 to the 240 root vectors of 248 dimensional E8, add the 72 root vectors of the left figure of the above image to the 24+144 = 168 root vectors of the central figure of the above image to get the 72 + 168 = 240 root vectors of the right figure of the above image.

( Note that 168 is the order of PSL(2,7) = PSL(3,2) and is related to the Klein Quartic. )


Those three images are shown on larger scale in the three images immediately below:













According to “The Classification of the Finite Simple Groups” (AMS Mathematical Surveys and Monographs, Vol. 40, No. 1, 1994) by Gorenstein, Lyons, and Solomon ( in the following I change their notation from prime number q to prime number p ): “… It is our purpose … to prove the following theorem:

CLASSIFICATION THEOREM. Every finite simple group is


  • cyclic of prime order,


  • an alternating group,


  • a finite simple group of Lie type,


  • or one of the twenty-six sporadic finite groups.


… the bulk of the set of finite simple groups consists of finite analogues of Lie groups … called finite simple groups of Lie type, and naturally form 16 infinite families … In 1968, Steinberg gave a uniform construction and characterization of all the finite groups of Lie type as groups of fixed points of endomorphisms of linear algebraic groups over the algebraic closure of a finite field …

The finite simple groups are listed …[including]… Group … E8(p) …[ for prime p ]..Order … p^120 (p^2 – 1 ) (p^8 – 1 ) (p^12 – 1 ) (p^14 – 1 ) (p^18 – 1 ) (p^20 – 1 ) (p^24 – 1 ) (p^30 – 1 ) …”. To get a feel for E8(p), ignore the -1 part of the Order formula for E8(q) and see that the order of E8(q) is roughly (somewhat less than)

p^120 p^(2+8+12+14+18+20+24+30) = p^(120+128) = p^248


Note that 248-dim E8 = 120-dim adjoint of Spin(16) + 128-dim half-spinor of Spin(16) and that p^248 is the set of maps from 248 top and that the exponents are one greater than each of the primes 1, 7, 11, 13, 17, 19, 23, and 29, but not similarly related to the primes to 2, 3, or 5 and that

o         E8(2) = the number of ways to assign the 2 elements + and 1 (as in + and – electric charge of the U(2) electroweak gauge group) to each of the 248 basis elements of E8

o         E8(3) = the number of ways to assign the 3 = 2+1 = 4-1 elements + and 1 (as in r, g and b color charge of the SU

(3) color force gauge group) to each of the 248 basis elements of E8

o   E8(5) = the number of ways to assign the 5 = 6-1 = 4+1 elements x, y, z, t and m (as in spatial x, y and z , and time t and scale/mass m of the Spin(2,3) anti-deSitter group of MacDowell-Mansouri gravity) to each of the 248 basis elements of E8l

  • E8(7) = the number of ways to assign the 7 = 6+1 = 8-1 Imaginary Octonion basis elements (as in spatial/ internal symmetry part of 8-dim Kaluza-Klein spacetime and tree-level-massive first generation fermion particles and antiparticles and in 7 of the 8 Dirac gammas of E8 physics) to each of the 248 basis elements of E8
  • E8(11) = the number of ways to assign 11 = 12-1 elements (as in the 11 generators of charge-carrying SU(3) and SU(2) of the 12 generators of the Standard Model SU(3)xSU(2)xU(1) in E8 physics) to each of the 248 basis elements of E8
  • E8(13) = the number of ways to assign 13 = 12+1 elements (as in 12 root vectors of Conformal Spin(2,4) = SU (2,2) of MacDowell-Mansouri gravity in E8 physics) to each of the 248 basis elements of E8
  • E8(17) = the number of ways to assign 17 = 16+1 elements as in the 16-dim vector representation of Spin(16) and the 16-dim full spinor representation of Spin(8) and 16-dim pairs of octoniions representing second- generation fermions and in the complexification of 8-dim Kaluza-Klein spacetime and 8-dim representation spaces of first-generation particles and antiparticles) to each of the 248 basis elements of E8
  • E8(19) = the number of ways to assign 19 = 18+1 elements (as in the 18 root vectors of 21-dimensional rank 3 Spin(7)) to each of the 248 basis elements of E8
  • E8(23) = the number of ways to assign 23 = 24-1 elements (as in 24-dim triples of octonions representing third- generation fermions and 24 full octonionic dimensions of the 27-dim Jordan algebra J(3,O)) to each of the 248 basis elements of E8
  • E8(29) = the number of ways to assign 29 = 28+1 elements (as in 28-dim d4 for MacDowell-Mansouri gravity and 28-dim d4 for the Standard Model in E8 physics) to each of the 248 basis elements of E8
  • E8(113) = the number of ways to assign 113 = 112+1 elements (as in the 112 root vectors of 120-dim Spin(16)) to each of the 248 basis elements of E8


  • E8(127) = the number of ways to assign 127 = 128-1 elements (as in 64+64 = 128-dim half-spinors of Spin(16) representing first-generation fermion particles and antiparticles, and the related Dirac Gammas) to each of the 248 basis elements of E8
  • E8(257) = the number of ways to assign 257 = 256+1 elements (as in 256-dim Cl(8)) to each of the 248 basis elements of E8


  • E8(65537) = the number of ways to assign 65,537 = 65,536+1 elements (as in 65,536-dim Cl(16)) to each of the 248 basis elements of E8


Physics of F4, E6, and E8




Here is how I see Physics Models based on F4, E6, and E8: F4


My root vector decomposition (using only one so(8) or D4) is one of the things that causes Garrett Lisi to say that I have “… a lot of really weird ideas which …[ he, Garrett ]… can’t endorse …”. So, from a conservative point of view, that you must use group or Lie algebra decompositions (not even considering a somewhat unconventional Weyl group factor approach, for which the f4 approach also will not work) , f4 will not work because one copy of D4 so(8) is not big enough for gravity and the SM. Also, f4 has another problem for my approach: f4 has basically real structures, while I use complex-bounded-domain geometry ideas based on ideas of Armand Wyler to calculate force strengths and particle masses.


So, although f4 gives you a nice natural idea of how to build a Lagrangian as


  • integral over vector base manifold


  • of curvature gauge boson term from adjoint so(8)


  • and spinor fermion terms from half-spinors of so(8) f4 has two problems:


  • 1 – no complex bounded domain structure for Wyler stuff (a problem forme)


  • 2 – only one D4 (no problem for me, but a problem for more conventional folks). So, look at bigger exceptional Lie algebra:



e6 is nice, and has complex structure for my Armand Wyler-based calculation of force strengths and particle masses, so e6 solves problem 1 with f4 and I can and have constructed an e6 model, but e6 still has only one D4, so e6 is still problematic from the conventional view, as e6 does not solve the conventional problem 2 with f4. So, do what Garrett Lisi did, and go to the largest exceptional Lie algebra, e8:



If you look at e8 in terms of E8(8) EVIII = Spin(16) + half-spinor of Spin(16) you see two copies of D4 inside the Spin(16) (Jacques Distler mentioned that) which are enough to describe gravity and the SM. I think that Garrett’s use of e8 is brilliant, and have written up a paper about e8 (and a lot of other stuff). There is a link to a pdf version, and there is a misprint on page 2 where I said EVII instead of EVIII, and probably there are more misprints, but as I said in the paper”… Any errors in this paper are not Garrett Lisi’s fault. “.


My view of e8 differs in some details from Garretts:


o         I don’t use triality for fermion generations, since my second and third generations are composites of the first, as described above in talking about f4




There are D4+D4+64 = 24+24+64 = 112 root vectors of Spin(16) : 24 yellow points for one D4 in the Spin(16) in E8

24 purple points for the other D4 in the Spin(16) in E8



64 blue points for the 8 vectors times 8 Dirac gammas in the Spin(16) in E8 There are 64+64 = 128 root vectors of a half-spinor of Spin(16) : 64 red points for the 8 first-generation fermion particles times 8 Dirac gammas

64 green points for the 8 first-generation fermion antiparticles time 8 Dirac gammas


if you put in some indices on the fields so that there are many fields of each type, then the first three terms of the equation give just the so-called standard model …


In the same Workshop proceedings I said (at pages 377-379, 381-383):


“……………. The 16-dimensional spinor representation of Spin(8) reduces to two irreducible 8-dimensional half-spinor representations that can correspond to the 8 fundamental fermion lepton and quark first-generation particle and to their 8 antiparticles …Numerical values for force strengths and ratios of particle masses to the electron mass are given. … Armand Wyler …(1971), C. R. Acad. Sci. Paris A272, 186 wrote a paper in which he purported to calculate the fine structure constant to be a = 1 / 137.03608 … from the volumes of homogeneous symmetric spaces. … Joseph Wolf (1965), J. Math. Mech. 14, 1033 wrote a paper in which he classified the 4-dimensional Riemannian symmetric spaces with quaterniuonic structure. There are just 4 equivalence classes, with the following representatives:


o         T4 = U(1)^4


o         S2 x S2 = SU(2) / U(1) x SU(2) / U(1)


o         CP2 = SU(3) / S(U(2) x U(1))


o         S4 = Spin(5) / Spin(4)


… Final Force Strength Calculation .


o         fine structure constant for electromagnetism = 1 / 137.03608


o         weak Fermi constant times proton mass squared = 1.03 x 10^(-5)


o         color force constant (at about 10^(-13) cm.) = 0.6286


o         gravitational constant times proton mass squared = 3.4 – 8.8 x 10^(-39).




o         the electron mass …[ is assumed to be given at its experimentally observed value ]…


o         electron-neutrino mass = 0 … [ Note that this is only a tree-level value. ] …


o         down quark constituent mass = 312.8 Mev …


o         up qaurk constituent mass = 312.8 Mev …


o         muon mass = 104.8 Mev …


o         muon-neutrino mass = 0 … [ Note that this is only a tree-level value. ] …


o         strange quark constituent mass = 523 Mev …


o         charm quark constituent mass = 1.99 Gev …


o         tauon mass = 1.88 Gev …


o         tauon-neutrino mass = 0 … [ Note that this is only a tree-level value. ]…


o         beauty quark constituent mass = 5.63 Gev …


o         truth quark constituent mass = 130 Gev …


CERN has announced that the truth quark mass is about 45 Gev (Rubbia … (1984), talk at A.P.S. D.P.F. annual meeting at Santa Fe … but I think that the phenomena observed by CERN at 45 Gev are weak force phenomena that are poorly explained … As of the summer of 1985, CERN has been uable to confirm its identification of the truth quark in the 45 Gev events, as the UA1 experimenters have found a lot of events clustering about the charged … W mass and the UA2 experimenters have not found anything convincing. (Miller … (1985), Nature 317, 110 … I think that the clustering of UA1 events near the charged … W mass indicates that the events observed are … weak force phenomena. “.


Since I have been critical of CERN for its error in truth quark obersvations, I should state that my paper in that 1985 Workshop also contained errors, the most conspicuous of which may have been my statement that “. there should be three generations ofweak bosons”.


Together …[ those ]……………. elements form a basis for a representation of the Lie algebra so( dimO + dimO ). I’ll call this External_OO and call it the external subalgebra. To this collection we now add the spinors of O(x)O(2) , namely the elements of (O(x)O)^2 , without yet specifying a commutator product on this linear space. I’ll denote this Spinor_OO ….The total resulting linear space will be denoted MS_OO , MS for magic square …Let e_La and e’_Lb be distinct and mustually commuting bases for the hypercomplex octonions. External_OO is spanned by W , e_La A , e’_La B ( 1-vector basis for …[ the Clifford algebra Cl(0,15) ]. ) and e_Lab B , e’_Lab A , e_La e’_Lb W , e_Lab E , e’_Lab E ( 2-vectors )…[ with dimension 1 + 7 + 7 + 21 + 21 + 21 + 21 + 21 = 1+14 +105 = 120 ]…External_OO = so(16) . Spinor_OO is 128-dimensional, and … because OL = OR …[ there is no Internal_OO ] That’s 120 + 128 = 248 elements altogether, and we make the identification:


MS_OO = LE8 . …


Getting LE8 from O(x)J3(O) …[ where J3(O) is the 27-dimensional exceptional Jordan algebra ]……………………… is slightly trickier. In this case there are two distinct copies of O commuting with each other ( denote them O1 and O2 ) …


We begin .. with the 28 so(8) generators …[ that ]… are elements of LF4… and the 3 so(3) generators … [that]………………… Account for 3 of the 52 dimensions of LF4 … Together …[ they ]… account… for 3 + 28 = 31 of the 52-dimensional LF4. …in …[ this ] O1(x)J3(O2) case we expand so(3) to LF4 , the Lie algebra of the sutomorphism of J3(O1) . That gives us 28 elements from so(8 , and 52 elements from LF4 ( which contains another distinct so(8) ). Of the 52 generators of theis new LF4 , 28 are diagonal. and 24 are off-diagonal. Commutators of the 28 diagonal generators ( the so(8) of O1 ) with the so(8) of O2 yield

nothing new, but each of the 24 off-diagonal generators gives rise to a 7-dimensional space of new generators. That yields, 28 + 52 + 168 = 248

generators all together, and the set closes here on LE8….. “. Note that 168 is the order of PSL(2,7) = SL(3,2) which can be thought of as the group of linear fractional transformations of the vertices of a heptagon and is so related to octonion multiplication rules, and that SL(2,7) of order 336 double covers the Klein Quartic







Some History of my Physics Model

In the 1960s-early 1970s Armand Wyler wrote a calculation of the fine structure constant using geometry of bounded complex domains. It was publicized briefly (almost as much as Garrett Lisi’s E8 model is publicized now) but Wyler never showed convincing physical motivation for his interpretation of the math structures, and it was severely ridiculed and ignored (with sad personal consequences for Wyler) Also in the 1960s, Joseph Wolf classified 4-dim spaces with quatenionic structure:


  • (I) Euclidean 4-space [ the 4-torus T4];


  • (II) SU(2) / S(U(1)xU(1)) x SU(2) / S(U(1)xU(1)), … [ S2 x S2 ]…;


  • (III) SU(3) / S(U(2)xU(1)), … [ CP2 ] …; and


  • (IV) Sp(2) / Sp(1)xSp(1) … [ = Spin(5) / Spin(4) = S4 ] …, and the noncompact duals of II, III, and IV


  • and I noticed that they corresponded to


o         U(1) electromagnetism,


o         SU(2) weak force,


o         SU(3) color force, and


o         Sp(2) MacDowell-Mansouri gravity


So I thought that it might possibly be useful to apply Wyler’s approach to the geometries of those 4-dim quaternionic structures. It was only in the 1980s that I was able to cut back on the time devoted to my law practice to try to learn enough math/physics to try to work out the application of Wyler’s stuff to Wolf’s classification, and I did so by spending a lot of time at Georgia Tech auditing seminars etc of David Finkelstein (who was tolerant enough to allow me to do so). I had learned some Lie group / Lie algebra math while an undergrad at Princeton (1959-63), but I did not know Clifford algebras very well until studying under David Finkelstein. Then (early 1980s) N=8 supergravity was popular, so I looked at SO(8) and its cover Spin(8), and noticed that:


  • Adjoint Spin(8) had 28 gauge bosons enough to do MacDowell-Mansouri gravity plus the Standard Model, but not if they were included as conventional subgroups;


  • Vector Spin(8) looked like 8-dim spacetime;


  • +half-spinor Spin(8) looked like 8 left-handed first-generation fermions;


  • -half-spinor Spin(8) looked like 8 right-handed first-generation fermions.



To break the 8-dim spacetime into a 4-dim physical spacetime plus a 4-dim internal symmetry space I used the geometric methods that had been developed by Meinhard Mayer (working with Andrzej Trautman) around1981. A consequence of that dimensional reduction was second and third generations of fermions as composites (pairs and triples) of states corresponding to the first-generation fermions. When I played with the Wyler-type geometry stuff, I got particle masses that looked roughly realistic, and a (then) prediction- calculation of the Tquark mass as around 130 GeV (tree-level, so give or take 10% or so).


When in 1984 CERN announced at APS DPF Santa Fe that they had seen the Tquark at 45 GeV, I gave a talk there (not nearly as well- attended as Carlo Rubbia’s) saying that CERN was wrong and the Tquark was more massive (I will not here go into subsequent history of Dalitz, Goldstein, Sliwa, CDF, etc except to say that I still feel that experimental data supports the Tquark having a low-mass state around 130-145 GeV, and that the politics related to my position may have something to do with my current outcast status with the USA physics establishment.) Since Spin(8) is bivector Clifford algebra of the real Clifford algebra Cl(8), and since real Clifford algebra 8-periodicity means that any very large real Clifford algebra can be factored into tensor products of Cl(8), it can be a building block of a nice big algebraic QFT (a real version of the complex hyperfinite II1 von Neumann factor). Since the Adjoint, Vector, and two half- Spinor reps of Spin(8) combine to form the exceptional Lie algebra F4, I tried to use it as a unifying Lie algebra, but I eventually saw that the real structure of F4 was incompatible with the complex bounded domain structures of the Wyler approach, so I went to E6, which is roughly a complexification of F4, and used E6 to construct a substantially realistic version of 26-dim bosonic string theory (fermions coming from orbifolding). Since by then I was blacklisted by the Cornell arXiv, I put that up on the CERN website as CERN-CDS-EXT-2004-031 As of then, the major conventional objection to my model was how I got 16 generators for a MacDowell-Mansouri gravity U(2,2) and 12 generators for the Standard Model from the 28 generators of Spin(8) (I used root vector patterns, because they do not consistently fit as subgroups and subalgebras). Now, Garrett Lisi’s E8 model has two copies of the D4 Spin(8) Lie algebra, so I can use it to be more conventional and get MacDowell- Mansouri gravity from one D4 and the Standard Model from the other one.

We thus consider the decomposition of the adjoint representation of E8 into representations of the maximal subgroup Spin(16)/Z2. The adjoint decomposes into the adjoint 120 and a chiral spinor 128. …

Our convention for chirality is GAMMA_(a1. a16) PHI = + e_(a1. a16) PHI . The e8 algebra becomes

( 2.1 )


[ T^(ab) , T^(cd) ] = 2 delta^([a)_([c) T^(b])_(d]) ,

[ T^(ab) , PHI^(alpha) ] = (1/4) ( GAMMA^(ab) PHI )^(alpha) ,

[ PHI^(alpha) , PHI^(alpha) ] = (1/8) ( GAMMA_(ab) )^(alpha beta) T^(ab) ,

… The coefficients in the first and second commutators are related by the so(16) algebra. The normalisation of the last commutator is free, but is fixed by the choice for the quadratic invariant, which for the case above is X2 = (1/2) T_(ab) T^(ab) + PHI_(alpha) PHI^(alpha) .

Spinor and vector indices are raised and lowered with delta . Equation (2.1) describes the compact real form, E8(-248) .


It seems that … the S7 or … non-trivial central extensions …[ of S7 ]… or Schwinger terms ghosts do not come in an S7 representation. This is also confirmed by an attempt to construct a representation (other than scalar) for imaginary octonions, which turns out to be impossible. …A part of the structure of S7 we have treated only fragmentarily is representation theory. It is not immediately clear even how to define a representation. We have quite strong feelings, though, that the spinorial representations and the adjoint, as described in this paper, in some sense are the only ones allowed, and that the spinor representation is the only one to which a variable freely can be assigned.”


B-mode Octonionic Inflation of E8 Physics



Frank Dodd (Tony) Smith, Jr. – 2014

BICEP2 in arXiv 1403.3985 said:

“… Inflation predicts … a primordial background of … gravitational waves …[that]… would have imprinted a unique signature upon the CMB. Gravitational waves induce local quadrupole anisotropies in the radiation field within the last-scattering surface, inducing polarization in the scattered light … This polarization pattern will include a “curl” or … inflationary gravitational wave (IGW) B-mode … component

at degree angular scales that cannot be generated primordially by density perturbations. The amplitude of this signal depends upon the tensor-to-scalar ratio … r = 0.20


+0.07 -0.05 … which itself is a function of the energy scale of inflation. “.


In E8 Physics, Inflation is due to Non-Unitarity of Octonion Quantum Processes that occur in 8-dim SpaceTime before freezing out of a preferred Quaternionic Frame ends Inflation and begins Ordinary Evolution in (4+4)-dim M4 x CP2 Kaluza-Klein.

The unit sphere in the Euclidean version of 8-dim SpaceTime ( see viXra 1311.0088 for Schwinger’s “unitary trick” to allow use of Euclidean SpaceTime ) is the 7-sphere S7.


( for E8 Physics overview see viXra 1312.0036 and 1310.0182 )


Curl-type B-modes (tensor) are Octonionic Quantum Processes on the surface of SpaceTime S7 which is a 7-dim NonAssociative Moufang Loop Malcev Algebra.


( for Malcev Algebras see Appendix I ) ( image below from Sky and Telescope )


B-modes look like Spirals on the Surface of S7


Divergence-type E modes (scalar and tensor) are Octonionic Quantum Processes from SpaceTime S7


plus a spinor-type S7 representing Dirac Fermions living in SpaceTime

plus a 14-dim G2 Octonionic Derivation Algebra connecting the two S7 spheres all of which is a 28-dim D4 Lie Algebra Spin(8).



E-modes look like Fermion Pair Creation either



off or on tensor the surface of S7


If you look at Llull’s Art (especially his Quaternary Phase) you see that it is equivalent to E8 Physics ( see viXra 1403.0178 ) with

the Clifford Algebra Cl(16) containing E8 giving the Local Lagrangian of a Region that is equivalent to a ” snapshot” of the Deutsch “multiverse”.

The completion of the union of all tensor products of all Cl(16) E8 Local Lagrangian Regions


Then emergently self-assembles into a structure = Deutsch multiverse forming a generalized hyperfinite II1 von Neumann factor AQFT

( Algebraic Quantum Field Theory ).


Stephen L. Adler in his book Quaternionic Quantum Mechanics and Quantum Fields (1995) said at pages 50-52, 561:

“… If the multiplication is associative, as in the complex and quaternionic cases, we can

remove parentheses in … Schroedinger equation dynamics … to conclude that … the inner product < f(t) | g(t) > … is invariant … this proof fails in the octonionic case, and hence one cannot follow the standard procedure to get a unitary dynamics. [so there is a]… failure of unitarity in octonionic quantum mechanics “.

The NonAssociativity and Non-Unitarity of octonions accounts for particle creation without the need for tapping the energy of a conventional inflaton field.

Inflation begins in Octonionic E8 Physics with a Quantum Fluctuation initially containing only one Cl(16) E8 Local Lagrangian Region


The Fermion Representation Space for a Cl(16) E8 Local Lagrangian Region

is E8 / D8 = the 64+64 = 128-dim +half-spinor space 64s++ + 64s+- of Cl(16) 64s++ = 8 components of 8 Fermion Particles 64s+- = 8 components of 8 Fermion Antiparticles

By 8-Periodicity of Real Clifford Algebras Cl(16) = tensor product Cl(8) x Cl(8) where the two copies of Cl(8) can be denoted by Cl(8)G and Cl(8)SM

( in E8 Physics Cl(8)G gives Gravity with Dark Energy and Cl(8)SM gives the Standard Model )

Cl(8)G and Cl(8)SM each have 8-dim half-spinor spaces 8Gs+ 8Gs- and 8SMs+ 8SMs- 8Gs+ and 8SMs+ representing 8 Fermion Particles 8Gs- and 8SMs- representing 8 Fermion Antiparticles


so that

64s++ = 8Gs+ x 8SMs+ for First Generation Particles of E8 Physics 64s+- = 8Gs+ x 8SMs- for First Generation AntiParticles of E8 Physics 64s-+ = 8Gs- x 8SMs+ for AntiGeneration Particles ( NOT in E8 Physics )



64s– = 8Gs- x 8SMs- for AntiGeneration AntiParticles ( NOT in E8 Physics )

+/- half-spinor of Cl(8)G determines +/- half-spinor of Cl(16) and Generation or AntiGeneration ( only +half-spinor Generation is in E8 )

+/- half-spinor of Cl(8)SM determines Particle or AntiParticle


E8 Physics has Representation space for 8 Fermion Particles + 8 Fermion Antiparticles on the original Cl(16) E8 Local Lagrangian Region that is 64s++ + 8 of 64s+- =




where a Fermion Representation slot _ of the 8+8 = 16 slots can be filled


by Real Fermion Particles or Real Fermion Antiparticles

IF the Quantum Fluctuation( QF ) has enough Energy to produce them as Real and IF the Cl(16) E8 Local Lagrangian Region has an Effective Path from its QF Energy

to that Particular slot. ( see Appendix III for Geoffrey Dixon’s ideas and Effective Path of QF Energy )


Since E8 contains only the 128 +half-spinors and none of the 128 -half-spinors of Cl(16) the only Effective Path of QF Energy to E8 Fermion Representation slots goes to the only Fermion Particle slots that are also of type + that is, to the 8 Fermion Particle Representation slots




Next, consider the first Unfolding step of Octonionic Inflation.It is based on all 16 = 8 Fermion Particle slots + 8 Fermion Antiparticle Representation slots whether or not they have been filled by QF Energy.


7 of the 8 Fermion Particle slots correspond to the 7 Imaginary Octonions and therefore to the 7 Independent E8 Integral Domain Lattices and therefore to 7 New Cl(16) E8 Local Lagrangian Regions.

The 8th Fermion Particle slot corresponds to the 1 Real Octonion and

therefore to the 8th E8 Integral Domain Lattice ( not independent – see Kirmse’s mistake ) and therefore to the 8th New Cl(16) E8 Local Lagrangian Region.


Similarly, the 8 Fermion Antiparticle slots Unfold into 8 more New New Cl(16) E8 Local Lagrangian Regions, so that one Unfolding Step is a 16-fold multiplication of Cl(16) E8 Local Lagrangian Regions:



If the QF Energy is sufficient, the Fermion Particle content after the first Unfolding is




so it is clear that the Octonionic Inflation Unfolding Process creates Fermion Particles with no Antiparticles,

thus explaining the dominance of Matter over AntiMatter in Our Universe

Each Unfolding has duration of the Planck Time Tplanck

and none of the components of the Unfolding Process Components are simultaneous, so that the total duration of N Unfoldings is 2^N Tplanck.


  1. S. M. Coxeter in his paper Regular and Semi-Regular Polyotpes III (Math. Z. 200, 3-45, 1988) about the 240 units of an E8 Integral Domain said: “… “. the 16 + 16 + 16 octaves


±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke, (±1±ie±je±ke)/2, (±e±i±j±k)/2,

and the 192 others derived from the last two expressions by cyclically permuting the 7 symbols [ i,j,k,e,ie,je,ke ] in the peculiar order

e, i, j, ie, ke, k, je

… It seems somewhat paradoxical … that the cyclic permutation

( e, i, j, ie, ke, k, je ),

which preserves the integral domain (and the finite projective [Fano] plane …) is not an automorphism of the whole ring of octaves; it transforms the associative triad ijk into the anti-associative triad j ie je. On the other hand, the permutation

( e ie je i k ke j ),

which IS an automorphism of the whole ring of octaves (and of the finite [Fano] plane …) transforms this particular integral domain into another one of R. H. Bruck’s cyclic of seven such domains. “.


240 Root Vectors of 248-dim E8 have physical interpretations in E8 Physics:



64 Red = 8 components of 8 Fermion Particles

64 Green = 8 components of 8 Fermion Antiparticles

64 Blue = 8 position x 8 momentum of 8-dim SpaceTime 28 Yellow include 12 of the 16 generators of U(2,2) for Conformal Gravity, Dark Energy, and Propagator Phase


28 Orange include 8 of the 12 generators of Standard Model SU(3)xSU(2)xU(1)


( the other 4 come from 4 of the 248 – 240 = 8 Cartan Subalgebra elements of E8 )


( the other 4 come from 4 of the 248 – 240 = 8 Cartan Subalgebra elements of E8 )


Appendix III ( 4 pages ): Effective Path of QF Energy and Dixon CxHxO

Another ( probably equivalent ) way to see that the Effective Path of QF Energy goes entirely to creation of Real Fermion Particles is to follow the work

of Geoffrey Dixon who said ( 2012.09.20 ) “… based on the algebra T := CxHxO , an interpretation is developed that implies the existence of a matter universe, and an anti-matter universe …”.


Represent both 64-dim 64s++ and 64-dim 64s+- as tensor product C x H x O = T where C = Complex Numbers, H = Quaternions, and O = Octonions so that T+T = 128-dim +half-spinor space 64s++ + 64s+- of Cl(16) which is in E8


Dixon says that T+T corresponds to a 1,9-spacetime and

that there are 2 ways to reduce 1,9-spacetime to our physical 1,3-spacetime with

one way producing a matter universe and the other producing an antimatter universe.


My view is that those 2 ways correspond to 2 copies of T+T which represent 128-dim +half-spinor space 64s++ + 64s+- of Cl(16) and in E8 =

= Dixon’s Matter Universe


And 128-dim -half-spinor space 64s-+ + 64s– of Cl(16) and not in E8 =

= Dixon’s AntiMatter Universe



E8 Physics uses only the 128 +half-spinors and none of the 128 -half-spinors of Cl(16) so, using Geoffrey Dixon’s ideas, E8 Physics Octonionic Inflation


produces Real Fermion Particles and a Matter Universe

and the

Effective Path of Quantum Fluctuation Energy to Creation of Real Fermion Particles

consistently with Dixon’s reduction of 1,9-spacetime to 1,3-spacetime.

Here are some details about Geoffrey Dixon’s ideas:


Geoffrey Dixon in his book Division Algebras, Lattices, Physics, Windmill Tilting (2011) said: ( in this quote I use T+T instead of Geoffrey Dixon’s notation T2 )

“… T inherits noncommutativity form H and O, and nonassociativity from O. From the combination of H and O it also loses alternativity …

TL uses only HL , and TA uses HA , which includes both HL , HR , and their combined actions.

T … is a Pauli spinor doublet for a 1,9-spacetime

in exactly the same way P is a Pauli spinor doublet for 1,3-spacetime


In the Pauli algebra case, we got Dirac spinors by doubling P to P+P , and the associated Clifford algebra is PL(2) = C x Cl(1,3)

To produce Dirac spinors we do for T what we did for P: we double up and use T+T as our spinor space, with the associated Clifford algebra TL(2) = C x Cl(1,9)

Note that in both these cases, if we absorb the C into the Clifford algebra, we expand the dimensionality of the associated spacetime. This is sometimes done, but not here. …







Geoffrey Dixon in his paper Matter Universe: A Mathematical Solution said:

( in this quote I use T+T instead of Geoffrey Dixon’s notation T2 )

“… The algebra T := C x H x O is 2 x 4 x 8 = 64-dimensional. It is noncommutative, nonassociative, and nonalternative.

In this model … the foundation is the 128-dimensional … space T+T (the doubling of T in the spinor space is modeled on the notion that a Dirac spinor is a double Pauli spinor). …

the Dirac algebra … PL := C x H. is

the complexification of the Clifford algebra of 1,3-spacetime

T+T is acted upon by the complexification of the Clifford algebra of 1,9-spacetime, represented by TL(2) , where TL is the algebra of left actions of T on itself, which in the octonion case, due to nonassociativity, requires the nesting of actions.

In the T-theory … the quarks are associated with the octonion units ep , p=1,. ,6.

The extra six space dimensions. also rest on these units


An elegant representation of the Clifford algebra Cl(1,9) represented in TL(2)

that is aligned with the choice of the octonion unit e7 arises from the following set of ten anti-commuting 1-vectors:





E8 Root Vectors from 8D to 3D




Frank Dodd (Tony) Smith, Jr.


This paper is an elementary-level attempt at discussing 8D E8 Physics based on the 240 Root Vectors of an E8 lattice

and how it compares with physics models based on 4D and 3D structures such as Glotzer Dimer packings in 3D, Elser-Sloane Quasicrystals in 4D, and various 3D Quasicrystals based on slices of 600-cells and

a natural progression from 600-cell to Superposition of 8 E8 Lattices.


The 128 Fermionic E8 Root Vectors are also consistent with Geoffrey Dixon’s fundamental tensor T^2 where T = RxCxHxO

= real x complex x quaternion x octonion.



the gauge bosons of Gravity+Dark Energy are in M4 (horizontal axis) and their ghosts are in CP2 (vertical axis) so both axes must be used and Standard Model similarly requires both axes to be used.


Now, look at the 240 E8 Root Vectors in the circle-ball projection:


My E8 Physics model Physical Interpretation of the 240 E8 Root Vectors

which break down into two sets of 120 each with H4 symmetry that correspond to the M4 gravity and CP2 standard model sectors of M4 x CP2 Kaluza-Klein is:


64 blue = Spacetime

64 green and cyan = Fermion Particles

64 red and magenta = Fermion AntiParticles

24 yellow = D4g Root Vectors = 12 Root Vectors of SU(2,2) Conformal Gravity

+ 12 Ghosts of Standard Model SU(3)xSU(2)xU(1) 24 orange = D4sm Root Vectors = 8 Root Vectors of Standard Model SU(3)xSU(2)xU(1)

+ 16 Ghosts of U(2,2) of Conformal Gravity


Here they are shown in the circle-ball 2-dim projection with 8 circles of 30 vertices each:










Here is how the 240 break down into 120 + 120 of H4grav and H4stdmod


Here are 128 Fermionic Root Vectors with the 8 components for the electron dimer that break into two (M4 and CP2) tetrahedra with 4 vertices shown connected by white lines. If you combine the dimers for the green, red, and blue up quarks with the electron dimer as shown in purple boxes then you get 4 dimers with maximum packing density


If you then take all 4 Fermion Quadrants


then you get the tetrahedral N = 32 for 16 dimers that represent E8 / D8 = (OxO)P2

= all 16 fermions x 8 components = 128 Fermionic E8 Root Vectors



The 128 Fermionic E8 Root Vectors are also consistent with Geoffrey Dixon’s fundamental tensor T^2 where T = RxCxHxO

= real x complex x quaternion x octonion.



The 240 of E8 = ( 128 spinor fermionic E8 / D8 ) + 112 of D8




The Spinor Fermion part = E8 / D8 contains 128 vertices = 64 binars = 16 dimers =

= 32 tetrahedra so it has tetrahedral N = 32



Since D8 / D4xD4 = 64-dim (OxO)P2

the 112 of D8 = ( 8×8 = 64 spacetime ) + (24+24 = 48 D4xD4 )




The Spacetime part = D8 / D4xD4 contains 64 vertices = 32 binars = 8 dimers =

= 16 tetrahedra so it has tetrahedral N = 16




and the total Spinors + Spacetime has 192 vertices = 96 binars = 24 dimers =

= 48 tetrahedra so it has tetrahedral N = 48


The Gauge Boson + Ghosts part = D4xD4 contains 48 vertices = 24 binars = 6 dimers

= 12 tetrahedra so it has tetrahedral N = 12



and the total Spinors + Spacetime + Gauge Bosons + Ghosts has 240 vertices =

= 120 binars = 30 dimers = 60 tetrahedra so the total E8 tetrahedral N = 60


Then, consider the 61,440 = 16x16x240 vertices of the second shell of Barnes-Wall /\16 rescaled for Unit Radius constructed from triples of E8 Lattices using Dixon’s XY- product with X and Y outside the E8i, E8j, E8e and their /\16 Lattices (E8i x E8j) x E8e with 16x16x240 = 61,440 vertices ( x , y , z ) (E8j x E8e) x E8i with 16x16x240 = 61,440 vertices ( x , y , z ) (E8e x E8i) x E8j with 16x16x240 = 61,440 vertices ( x , y , z )

The total inner vertices = 3 x ( 240 + 3840 + 61,440 ) = 196,560 correspond to the inner-shell vertices of the 24-dim Leech Lattic


One Cell of E8 26-dimensional Bosonic String Theory with structure J(3,O)o

with Strings being physically interpreted as World-Lines

and massless spin-2 states are interpreted as carriers of Bohm Quantum Potential can be described by taking the quotient of its

24-dimensional O+, O-, Ov subspace modulo the 24-dimensional Leech lattice.



3D Rhombic Triacontahedron — Jitterbug –> 3D Truncated Octahedron which fills 3D space with each node corresponding to

3 Elementary Sets of Cellular Automata (CA) each of which corresponds to an E8 Lattice so that


the 3 Sets of CA represent a 24D Leech Lattice


underlying the structure of the 26D String Theory of E8 Physics AQFT based on Strings as World-Lines and


massless spin-2 states as carriers of Bohm Quantum Potential



Start with the 4D H4 QC QuasiLattice whose origin-neighbor vertices form a 600-cell

From 600-cell to Superposition of 8 E8 Lattices


with unit Radius. It has 120 vertices whose physical interpretations are

32 blue = 4D M4 Minkowski part of 8D M4xCP2 Kaluza-Klein Spacetime 32 green and cyan = 4 Minkowski components of 8 Fermion Particles

32 red and magenta = 4 Minkowski components of 8 Fermion AntiParticles 24 yellow = D4g Root Vectors = 12 Root Vectors of SU(2,2) Conformal Gravity

+ 12 Ghosts of Standard Model SU(3)xSU(2)xU(1) arranged, with respect to a circle-sphere projection to 2D, in 4 circles of 30 vertices each.

Boyle and Steinhardt in arXiv 1608.08220 , arXiv 1608.08215 ,arXiv 1604.06426 say: “… there is a unique reflection QL … quasilattice … /\ … in 4D … Every vector … in /\ can … be written as an integer combination of the 120 H4 roots

… /\ must contain all the golden integers times each H4 root, and all integer linear combinations of such vectors … it is unique … H4 root QL … correspond[ing] to the … icosians…

the scaling group of the QL must be a subgroup of the scaling group of its 1D sublattice

… H4 contain[s] Z(sqrt(5)) … scaling factor is the “golden ratio” … ( 1 + sqrt(5) ) / 2 …”.


Therefore, the second shell of the 4D H4 QC QuasiLattice is also a 600-cell whose expanded Radius is the “golden ratio” … ( 1 + sqrt(5) ) / 2 = 1.61 arranged in 4 circles of 30 vertices each with physical interpretations


32 blue = 4D CP2 Internal Symmetry part of 8D M4xCP2 Kaluza-Klein Spacetime 32 green and cyan = 4 CP2 components of 8 Fermion Particles 32 red and magenta = 4 CP2 components of 8 Fermion AntiParticles

24 orange = D4sm Root Vectors = 8 Root Vectors of Standard Model SU(3)xSU(2)xU(1)

+ 16 Ghosts of U(2,2) of Conformal Gravity

In other words, the first two shells of the 4D H4 QC QuasiLattice

form the 240 Root Vector first shell of an 8D E8 Latice with 8D norm 2×1 = 2 shown here in the circle-ball 2-dim projection with 8 circles of 30 vertices each



and in a square-cube 2-dim projection with the same physical interpretation color-coding


The second shell of the 8D E8 Latice, with 8D norm 2×2 = 4, has 2160 vertices: 8 pairs of 128-vertex Cl(16) half-spinors for 2048 vertices and 112 vertices corresponding to Root Vectors of the D8 subalgebra of E8


The 112-vertex D8 combines with a left-handed 128-vertex Cl(16) half-spinor, representing E8 Physics Fermion Particles, which are left-handed, to form a 240-vertex configuration like the E8 Root Vector Gosset Polytope



Since there are 8 pairs (left-handed and right-handed) of 128-vertex Cl(16) half-spinors in the second E8 shell, if you require the 112-vertex D8 to combine with theleft-handed half-spinor to form a 240-vertex E8 Root Vector Polytope for an E8 Physics model with realistic left-handed half- spinors representing Fermion Particles, then

there are 8 ways you can produce an E8 Lattice for E8 Physics.

7 of the 8 ways produce algebraically distinct independent Integral Domains, corresponding to the 7 imaginary Octonions i, j, k, E, I, J, K The 8th way is not an algebraically independent Integral Domain and it corresponds to the real Octonion 1.




In E8 Physics, 8-dim Spacetime (described by the D8 / D4xD4 part of E8)


is a Superposition of Spacetimes of each of those 8 E8 sets of 240 Root Vectors.

If you consider each of those 8 E8 sets of 240 Root Vectors as a first shell of a second-order E8 Lattice each having a second shell of 2160 Vertices then

you get 8 x 2160 = 17280 Vertices and

if you add 240 Vertices of a first-shell set of E8 Root Vectors then you get the 17280 + 240 = 17520 Vertices of the 4th shell of an E8 Lattice


as described by the E8 theta series

Wikipedia: “… the number of E8 lattice vectors of norm 2n is 240 times the sum of the cubes of the divisors of n. The first few terms of this [theta] series are given by (sequence A004009 in … OEIS) … 240 2160 6720 17520 30240 60480 …”.




8D E8 Shell 4 shows explicitly the 240 Root Vectors of E8 Physics


I (Harry Tasset) have added the below sections from Wikipaedia and Tony for a better understanding of the PTOE mechanism:










Appendix 2, other sources:

End Tony’s Calculations


From Wikipedia:








Jan. 2008





The following web PDF file contains a 2020 update on the new supergravities research. I have inserted their magic pyramid image. (see below) Please note the E7(7) and the E8(8) Supergravities form part of the “spine” of the pyramid.


The Mile High Magic Pyramid∗ A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy, and M. Zoccali




Octonion Song By John Frederick Sweeney




Note by Author: I discovered this amazing article by John Frederick Sweeney when I was trying to find the meaning behind the “Fano” plane. The following article is only an abbreviated section. It is well worth your attention if you want to learn more about the history of how music and quantum theory merge. You will recognize the below Fano image as the same one that is currently used by nuclear physicists to construct the periodic table of particles.


By John Frederick Sweeney


Chanyal, Bisht, Li and Negi have done some ground – breaking work on Octonions, with some results reproduced here for the reader’s convenience. The author recommends that interested readers consult the original sources online for a complete understanding. Suffice it here that this work shows some directions for further research.


Recall in Vedic Physics that there exist 18 types of Quarks based on the color scheme, and that the remaining 12 levels are logarithmically derived from the known six, in addition to Anti – Quarks, which makes for a total of 36 Quarks


Note the structure of 1 first line, followed by 8 lines, each with 8+8 = 16 Sanskrit syllables left of the | line and 8 Sanskrit syllables right of the | line,for a total of 24 Sanskrit syllables per line. Note that the three sets of eight syllables correspond to




the 8 first generation fermion particles, the 8 first generation fermion antiparticles, and an 8-dimensional spacetime in the D4-D5-E6-E7-E8 VoDou Physics model, and all 24 form the vertices of a 24-cell.




This paper has shown the isomorphic relations between the Octonions, the Fano plane and the Pythagorean music system, for the purpose of explaining that Octonions produce vibrations, equivalent to musical sounds, called Chhandra in Sanskrit, which play an important role in the E8 + E8 Heteroric String Theory. Much of this has been shown by S.M. Philipps in Article 15 of his website, but Philipps fails to explain the “why” of this phenomenon.

The motivation for this paper came as an effort to reveal the secret scientific codes contained within the Rig Veda, the oldest book known to humanity, and as Smith argues, probably represents the first writing down of the oral Ifa system from Africa. Srinivasan confirms this notion, stating that Sanskrit was created as a scientific language expressly for the purpose of committing the Brahman oral knowledge, which had been transmitted from father to son within a strict social caste, in order to maintain the knowledge in pristine state, to a written form, which might survive the global catastrophes which occurred some 13,000 years ago.


Vedic Science contains a doctrine which explains how the Milky Way galaxy reaches two nodal points every 13,000 years, and when that occurs, drastic changes take place on Earth. The people of the Vedas, who then lived in polar Siberia, committed this knowledge to writing in the event that the Brahma class was destroyed during the Earth Changes period. The knowledge from that advanced civilization is superior to our own, which has just recently come into Quaternions (1850), Bott Periodicity (1960), Octonions (1995) and Sedenions (2000). Most mathematicians and physicists continue to disregard many of these concepts today – there still exists a tremendous amount of resistance in these fields.


That knowledge included nuclear secrets, as described in this paper. The Veda people did not want this crucial information to fall into the hands of idiots. We have seen what the US has done at Hiroshima, Nagasaki, with nuclear testing on Pacific atolls, and then with Nixon and others threatening nuclear war in Korea, Vietnam and Iraq. In the latter case, the US had an idiot with an IQ of 70 who could not read the New York Times, sitting in the Oval Office with his finger on the nuclear button, which he and his handlers threatened to push on numerous occasions, out of specious motivations based on greed.

Thus, the Vedic people had the foresight to understand that idiots would emerge who might destroy Earth, much as we have recently experienced. Explicitly for this reason, they encoded the advanced scientific information within the Rig Veda and other documents of Vedic Literature, to keep this knowledge out of the hands of humans who had not reached a comparatively high level of development, such as George W. Bush. Perhaps this explains the deep skepticism of Christopher Minkowski. The Rig Veda authors intended


the encoded information only for humans who had matured enough not to abuse nuclear power.


Minkowski’s criticism of Nilakantha falls down on two implications: first, if Nilakantha proved capable of decoding Magic Squares in the Rig Veda, that implies that someone encoded them there for a purpose. In a similar way, if the Katapadya system was used to encode information in the Vedas, that implies that the authors intended this be so, and that they understood the Katapadya system as well as later authors. Wikipedia and other Indologists mistakenly date the Katapadya system to much later than 11,000 BC, when the Vedas were written.


The passage quoted by Smith indicates deeper meanings below the surface. The passage mentions Prakrithi, as related to intelligence. In fact, the Prakrithi is a nuclear structure related to the Leptons in modern nuclear physics, as a forthcoming paper will demonstrate.


The explanation by Srinivasan describes other methods for encoding information in the Vedas, and so supports the idea that Nilakantha worked well within a long – established tradition. Minkowski portrays Nilakantha as engaged in some self – indulgent intellectual parlour game that holds no relation to anyone else but his own selfish intellectual curiosity. In fact, Nilakantha was engaged in a deep effort to decode the Rig Veda, in a tradition that had been long established by his day, but which Minkowski apparently knows nothing about.


Finally, K. C. Sharma details the actual nuclear process which produces the Octonion Song. While somewhat allegorical, enough details emerge to allow contemporary physicists to follow up and uncover the physical mechanism which creates Octonionic music. For this reason, this paper includes work by Weng and others, to suggest the features of Octonions, Sedenions and Trigintaduonions which might produce such vibrations. In fact, and upcoming paper by this author will give Sharma’s Rig Veda explanation for wavelengths in Vedic Nuclear Particle Physics.


The passage by S.M. Philpps relates that the Octonions bear an intimate relationship with the Pythagorean School. This author has written a paper about the Egyptian connectionto this science, in terms of the Exceptional Lie Algebra G2 and other structures in theseries:


A2 – G2 – D4 – F4 – E8

Quaternions to Octonions to Sedenions and Trigintaduonions

While the Pythagoreans were supposedly Greek, they obviously learned fromAncient Egypt, and this paper establishes an additional connection between the ancient Vedic people and the people of Ancient Egypt. A forthcoming paper from this author will provide specific details about where and how the Octonic songs emanate in Vedic Nuclear Physics, with the explanation derived primarily from Book I of the Rig Veda



Compounding Fields and Their Quantum Equations in Trigintaduonion Space, by Zihua Weng RIGVEDA, AN INTRODUCTION by G. Srinivasan, 2009.


Vedic Particle Physics, K.C. Sharma, 2008.


Nilakantha and Rig Vedic Magic Squares, Christopher Minkowski. The author may be contacted at jaq2013 at outlook dot com






So let us dedicate ourselves to what the Greeks wrote so long ago: to tame the savageness of man and to make gentle the life of this world.


Some men see things as they are and ask, why?


I see things that never have been, and ask, why not?


Robert Francis Kennedy



Another important paper by Sweeney http://citeseerx.ist.psu.edu/viewdoc/download?doi=


Another interesting book and paper on the expanding Earth: https://www.earth-prints.org/bitstream/2122/8658/6/INDEX%20fullpage.pdf

The above book has an up-to-date assessment on the expanding earth theory. Below is just a snipit of the amazing research that is taking place. The Stephen Hurrell section caught my eye.

G. Scalera, E. Boschi and S. Cwojdzin´ski (eds.), 2012

THE EARTH EXPANSION EVIDENCE – A Challenge for Geology, Geophysics and Astronomy Selected Contributions to the Interdisciplinary Workshop of the 37th International School of Geophysics EMFCSC, Erice (4-9 October 2011)


Ancient Life’s Gravity and its Implications for the Expanding Earth


Stephen Hurrell


11 Farmers Heath, Great Sutton, Ellesmere Port, Cheshire, CH66 2GX, United Kingdom – Phone: +44 (0) 7753 587469 (dinox@btinternet.com)


Abstract. Galileo Galilei emphasised in the 17th century how scale effects impose an upper limit on the size of life. It is now understood that scale effects are a limiting factor for the size of life. A study of scale effects reveals that the relative scale of life would vary in different gravities with the result that the relative scale of land life is inversely proportional to the strength of gravity. This implies that a reduced gravity would explain the increased scale of ancient life such as the largest dinosaurs. In this paper, various methods such as dynamic similarity, leg bone strength, ligament strength and blood pressure are used to estimate values of ancient gravity assuming a Reduced Gravity Earth. These results indicate that gravity was less on the ancient Earth and has slowly increased up to its present-day value. The estimates of the Earth’s ancient reduced gravity indicated by ancient life are also compared with estimates of gravity for Constant Mass and Increasing Mass Expanding Earth models based on geological data. These comparisons show that the Reduced Gravity Earth model agrees more closely with an Increasing Mass Expanding Earth model.


Key words. Ancient gravity – Reduced gravity Earth – Scale effects – Expanding Earth


1. Introduction


Galileo Galilei (1638) was probably the first scientist to point out that larger an- imals need relatively thicker bones than smaller animals. He noted that the bones of very large animals must be scaled out of proportion in order to support the weight of the animal. This is because when any object increases in size its volume (l3) in- creases quicker than its area (l2), and its area increases quicker than its length (l). For example, a simple box which was dou- bled in length would be four times the area and eight times the volume of the original box. The leg stress in a large animal is pro- portionally more than a geometrically sim- ilar small animal because the weight of the large animal has increased quicker than its strength. This is commonly known as the scale effect.


To overcome this shortfall in strength with increased size, the legs of real large- scale animals generally tend to be propor- tionally thicker. Take for comparison the thigh bones of a deer, a rhinoceros and an elephant. As animals increase in scale the relative thickness of their legs is greater. The deer has the most slender legs, the rhinoceros relatively thicker ones, while the elephant’s legs are thicker still to sup- port its massive bulk. The elephant is near the upper size limit for land-based life.

The same basic principles can be seen in land-based animals, plants and flying birds. The largest insects have reached the upper size limit for creatures without bones. Mammals have reached the largest size for animals with bones and a com- plex four-chambered heart. Reptiles have reached the largest size for animals with bones and simple hearts. The largest plants


have reached the upper limit in size and the largest birds have reached the upper lift- ing capacity of their wings. For every form of living creature there is an upper limit to how large it can be.

The scale effect means that gravity limits the scale of present-day land-based life. This has been well understood for many years by specialists in the field such as Thompson (1917), Schmidt-Nielson (1984) and others. The scale effect limit presents a difficult problem for ancient gi- gantic animals like dinosaurs. Over the years, many different solutions to the prob- lem of their large scale have been sug- gested. Until at least the 1980s it was widely thought that large sauropods lived in water so the buoyancy effect permit- ted them to grow large (Schmidt-Nielson 1984), but this idea is now considered in- correct. Bakker (1986) was the main cham- pion for the evidence that these large ani- mals lived on land and his interpretation is widely accepted today. Hokkanen (1985) calculated a theoretical upper mass limit same gravity, the stress in the larger ani- mal’s legs would be double the stress in the smaller animal’s legs. This variation can be compensated for by adjusting the strength of gravity: if gravity was one half as strong for the larger animal, it would be four times as heavy. Both the small and large animals would have the same leg stress because of the difference in gravity. They would be dynamically similar despite their dif- ference in size because of the variation in gravity.


Ancient Dragonflies – Dragonflies similar to modern forms were present in the Carboniferous, dating from about 300 million years ago. These dragonflies were usually large and occasionally gigantic in size. The Muséum national d’Histoire naturelle in Paris contains the only two known examples of the famous giant drag- onfly, Meganeura monyi. With a wingspan of about 75 cm, it is still claimed by some



Fig. 3. A life-size reconstruction (72 cm wingspan) of Meganeuropsis permiana by Werner Kraus for the University Museum of Clausthal-Zellerfeld. Ⓧc Werner Kraus 2003.


authorities to be the largest known insect species to ever fly. This wingspan of 75 cm is gigantic compared to that of 19 cm for one of the largest modern species of drag- onfly, the Giant Hawaiian Darner dragon- fly, Anax strenuus.

Applying formula (2) to the ancient and modern forms of dragonflies gives a value for gravity 300 million years ago:


gravity300 = 1/3.95 = 0.25g


Gravity 300 million years ago was 25% of today’s gravity from a simple dy- namic similarity comparison. There are some fundamental assumptions used with this comparison; both the ancient dragon- fly fossils and the largest modern species of dragonfly are assumed to have reached the largest size possible for a dragonfly in their respective gravities, and both the an- cient and modern dragonfly are assumed to be dynamically similar and have followed similar lifestyles.

How accurate are these assumptions? The ancient dragonfly which is commonly accredited as being the largest has sev- eral rivals which are very close to the fa- mous giant dragonfly, Meganeura monyi. Examples of these are Meganeuropsis americana and Meganeuropsis permiana, as shown in Fig.3, from the Lower Permian fauna of Elmo. Given the fact that these are both very close in size to Meganeura monyi it would seem likely that this is about as large as these ancient dragonflies grew, even if there is some disagreement about which was the largest.

A similar argument applies to the largest present-day dragonfly. Although the Giant Hawaiian Darner dragonfly is the largest recorded size of dragonfly there are other species approaching this size: the Giant Petaltail dragonfly Petalura ingentis-sima has a wingspan of approximately 16 cm, for example. It would seem that we can safely assume that the sizes of the largest ancient and modern dragonflies are suffi- ciently accurate to calculate gravity 300 million years ago.

Is there any other way to check the results? Since there are still dragonflies around today there is an interesting method of doing this. Experiments performed by Marden (1987) loaded dragonflies with weights to measure the maximum amount that a range of dragonflies could lift. The largest dragonfly that Marden exper- imented with was Anax junius, which is commonly known as the Green Darner Dragonfly. The five individuals measured had an average mass of 0.9752 grams and an average maximum lifting force of 2.58 grams with an average wingspan of 10 cm. Comparing these dragonflies to the ancient dragonfly Meganeura monyi would give a scaling factor of 75/10 = 7.5. Using the scale effect to calculate the weight and lift- ing force of the ancient dragonfly assum- ing it was dynamically similar to the mod- ern dragonfly gives:



Ws = (W1/3 × s)3 = (0.97521/3 × 7.5)3


force was exactly the same as the dragon- fly’s weight. This gives a maximum possible force of gravity 300 million years ago as 0.35g. Even this seems beyond reason- able limits since it is difficult to imagine a dragonfly that didn’t have any power re- serves. It doesn’t seem a realistic proposal especially if we consider that dragonflies are predators that need to capture small insects to survive and the female dragon- fly must also mate in flight and then lay its eggs in water – a sudden gust of wind would drown our large dragonfly. It proba- bly means that these calculations represent an absolute size limit that could not be ex- ceeded and was unlikely to be reached in practice.



Fig. 5. V-shaped neck vertebra probably held the neck ligament used to keep Diplodocus’s neck erect and this enables the ligament’s size to be estimated.


Diplodocus is now mostly depicted with a stiff, relatively useless long neck that it couldn’t lift to reach the higher plants. Many museums around the world and TV series like Walking with Dinosaurs show Diplodocus like this even though some paleontologists disputed this view (Bakker 1986).


2.  Blood pressure of Brachiosaurus

Because of its long neck the giraffe has the maximum hydrostatic blood pressure of any animal alive today. This high blood pressure seems to be about the maximum possible since the giraffe needs to use ex- treme measures to maintain it. Because the central blood pressure is high the heart’s muscle has to be strong and a giraffe’s heart can weigh up to 10 kg and mea- sure about 60 cm long. The heart of an adult giraffe is about 2% of its body weight whereas in people it’s only about half a percent. Giraffes have arterial blood pres- sures of 25 kPa at the bases of their necks whilst standing. By extrapolation, the pres- sure in the heart must exceed 30 kPa which is about double the normal pressure in a mammal.

Brachiosaurus lived in the Late Jurassic to Early Cretaceous, about 145 million years ago. It is generally recon- structed with its neck sloping steeply up, in a giraffe-like posture so the brain of Brachiosaurus was about 7.9 metres above its heart as shown in Fig.6. Calculations assuming our present gravity reveal that the total pressure difference between the brain and the heart would be 8590 kPa.

These problems of high blood pres- sure would not exist on a Reduced Gravity Earth because blood pressure is lower in a reduced gravity. Blood pressure is propor- tional to blood mass, gravity and height, so it is possible to estimate ancient gravity by comparing the blood

pressure in ancient life with the blood pressure in modern life. The hydrostatic pressure difference be- tween the blood in the brain and the heart can mostly defined as the hydrostatic head in metres. The hydrostatic pressure at the base of the Brachiosaurus’s neck 145 million years ago can be calculated by: Hydrostatic Pressure =

= blood density × gravity145 × height.

In a reduced gravity the hydrostatic pressure would be reduced because the weight of the column of blood would be less and this would allow a



Fig. 6. The position of the head above the heart determines the blood pressure, or the hydrostatic head, at the heart for a giraffe and Brachiosaurus.


Brachiosaurus’s neck to become much longer than today’s giraffe. Blood is an incompressible fluid whose density would not vary in a different gravity so it seems safe to assume that the density of dinosaur blood was the same as giraffe blood. A large giraffe about 5.5 metres tall would hold its head 2.8 metres above its heart so the hydrostatic head in its heart would be 2.8 metres.




Fig. 8. A typical Expanding Earth reconstruction based on geological data.



Fig. 9. Earth’s changing gravity over time based on geological reconstructions of an Expanding Earth.


3. Implications for the expanding Earth

The magnetic recordings on the ocean floor have been mapped to give a detailed account of the age of the Earth’s ocean floor. By removing the ocean floor that is known to be younger than a particu- lar age, it is possible to reconstruct an- cient Expanding Earth globes by rejoin- ing the remaining ocean floors. A number of reconstructions have been produced by Hilgenberg (1933), Vogel (2003), Hurrell

(1994, 2011), Luchert (2003), Maxlow (2005) and many others. Fig.8 shows a typ- ical Expanding Earth reconstruction.

The estimates of ancient Earth’s re- duced gravity, indicated by the larger rela- tive scale of ancient life, can be compared with estimates of gravity for Constant Mass and Increasing Mass Expanding Earth models. The force of the Earth’s gravity is:

F = G × M1 × M2/R2, (5)

where M1 and M2 are the masses of the two mutually attracting bodies, R is the dis- tance separating them and G is Universal Constant of Gravity and the calculated force F is effectively the force of gravity.

For a Constant Mass Expanding Earth

ancient gravity would be about four times the present value which does not agree with the results from ancient life. For an Increasing Mass Expanding Earth gravity would gradually increase over time as the Earth grew in diameter and mass so this agrees with the gravity results from ancient life.

This is a simplistic method of calculating the force of gravity since it assumes that the density of the ancient Earth is exactly the same as the present Earth. It is much more probable that as the an- cient Earth grew larger in size and mass it would become denser as its core became more compact due to the increasing surface gravity (Hurrell 1994, 2003). This density increase can be estimated by plotting the known variation of gravity against the ra- dius of other known celestial bodies, and a graph of changing gravity on the ancient Earth taking account of density variations in the Earth’s core and mantle based on other celestial bodies is shown in Fig.9.

The Reduced Gravity Earth model agrees most closely with an Increasing Mass Expanding Earth model rather than a Constant Mass Expanding Earth model. Estimates of ancient life’s gravity indicate that Earth Expansion is due to mass in- crease.




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Author’s Biographical Notes: Stephen Hurrell lives near Liverpool in the UK where he has worked in mechanical engineering design positions for various companies. It was his role as a mechanical design engineer at the UK’s Electricity Research Centre that first oered him his insight into how scale eects were pertinent to the biomechanical problems of the dinosaurs’ large size. These thoughts about dinosaurs as engineering structures, and the influence of scale eects, fostered the development of the Reduced Gravity Earth theory and its implications for the Expanding Earth.


As you can see from Stephen Hurrell’s list of references, the expanding earth theory has gained a lot of supporters.


Below is another excellent article by H Kragh discussing some of the

most brilliant minds in mathematics and physics adding their input as to the expanding Earth.


Expanding Earth and declining gravity: a chapter in the recent history of geophysics


H. Kragh

Niels Bohr Archive, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark

Correspondence to: H. Kragh (helge.kragh@nbi.ku.dk)

Received: 4 February 2015 – Accepted: 16 April 2015 – Published: 5 May 2015


Here is just one section of this excellent source:


Claiming that there was “a considerable body of evidence”supporting a decreasing G, Hoyle considered the effects on the motion of the Moon, solar luminosity, the temperature of Earth in the past, and the motion of the continents. All

of these subjects had been covered in detail previously, but now Hoyle reconsidered them from the point of view of the Hoyle–Narlikar gravitation theory (Hoyle, 1972). For the rate of increase of Earth’s radius he calculated a value between 6 and 10 km per 108 years (0.1mmyr1), suggesting that the expansion combined with mantle convection explained how the continental plates were set in horizontal motion. Hoyle’s somewhat amateurish excursion into geophysics

made even less of an impact than Jordan’s longer and more serious excursion. Hugh Owen, a palaeontologist and cartographer at the British Museum, was among the few who found Hoyle’s theory promising. A late advocate of the expanding

Earth, he suggested that Earth’s diameter was approximately half its current value 700 million years ago (Owen,

1976, 1984; Fig. 4). Hoyle’s old friend and collaborator, the Cambridge astronomer Raymond Lyttleton, did not believe in continental drift. In a critical reply to the Hoyle–Narlikar hypothesis, he and his collaborator John Finch recalculated the change of Earth’s radius on the assumption of G1=t and Ramsay’s hypothesis of the inner Earth. They found a

rate of change in radius of dR=dt D 2.3 km per 108 yr, or less than a quarter of the value reported by Hoyle and Narlikar. “It is clearly impossible,” they concluded, “that a decreasing G could alone cause expansion on such a scale that a fissure between Africa and South America would yawn some 5000 km in width” (Lyttleton and Finch, 1977).


Hist. Geo Space Sci., 6, 45–55, 2015 www.hist-geo-space-sci.net/6/45/2015/ doi:10.5194/hgss-6-45-2015

© Author(s) 2015. CC Attribution 3.0 License.




http://www.tony5m17h.net/d4d5e6hist.html (Tony)

http://www.tony5m17h.net/TShome.html (Tony) http://www.tony5m17h.net/higgsmass.html#masscales (Tony) http://www.tony5m17h.net/simph.html (Tony)

http://www.tony5m17h.net/E8GLTSCl8xtnd.html (Tony’s E-8)

http://www.valdostamuseum.com/hamsmith/Sets2Quarks4.html#ultraVfinite (Tony) http://www.valdostamuseum.com/hamsmith/allspaces.html#correl (Tony) http://www.valdostamuseum.com/hamsmith/GLA.html#e6gla (Tony) http://www.tony5m17h.net/GLe8Cl8.html (Tony)

http://www.tony5m17h.net/ (Tony) http://valdostamuseum.com/hamsmith/NDalg.html#purespinor (Tony) http://valdostamuseum.com/hamsmith/clfpq.html#SPIN

http://citeseerx.ist.psu.edu/viewdoc/download?doi= (John Frederick Sweeney) http://theoryofeverything.org/TOE/JGM/Integrated%20E8,%20Binary,%20Octonion-web.pdf (Moxness) http://theoryofeverything.org/theToE/2015/03/23/e8-particle-assignment-triality-symmetries/ (Moxness) http://theoryofeverything.org/TOE/JGM/Fano.pdf (Moxness)

https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=589 (chemogenisis) http://theoryofeverything.org/theToE/2013/04/26/created-a-new-4d-stowe-janet-scerri-periodic-table/ (Moxness) http://theoryofeverything.org/TOE/JGM/ToE_Demonstration.cdf (Moxness) https://fgmarcelis.wordpress.com/witting-polytope/ (Marcelis polytopes)

http://theoryofeverything.org/TOE/JGM/Integrated%20E8,%20Binary,%20Octonion-web.pdf (Moxness, Scerri example)

https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=1076 (Asimov table) https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=1126 (my table)

http://theoryofeverything.org/theToE/2015/03/23/e8-particle-assignment-triality-symmetries/ (particle assignment 256)

https://ncatlab.org/nlab/show/Spin%287%29%2FG2+is+the+7-sphere https://ncatlab.org/nlab/show/7-sphere https://ncatlab.org/nlab/show/exotic+7-sphere https://ncatlab.org/nlab/show/Hopf+construction https://en.wikipedia.org/wiki/Seven-dimensional_space https://en.wikipedia.org/wiki/7-cube https://en.m.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghost https://en.m.wikipedia.org/wiki/Representation_theory_of_SU(2) https://en.m.wikipedia.org/wiki/Lorentz_group https://en.m.wikipedia.org/wiki/Spin-%C2%BD https://en.m.wikipedia.org/wiki/Relativistic_quantum_mechanics https://en.m.wikipedia.org/wiki/Spinor https://en.m.wikipedia.org/wiki/Creation_and_annihilation_operators https://en.m.wikipedia.org/wiki/Azimuthal_quantum_number

https://en.m.wikipedia.org/wiki/Extended_periodic_table#Electron_configurations see Unbipentium https://en.m.wikipedia.org/wiki/Nuclear_shell_model see magic numbers of nucleons https://en.m.wikipedia.org/wiki/3D_rotation_group SO3

https://en.m.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem Gauge ghosts are spinless fermions, but they include states of negative norm. https://en.m.wikipedia.org/wiki/Spin_(physics)

https://www.researchgate.net/profile/Eric_Scerri/publication/225802509_Explaining_the_periodic_table_and_the_role_of_chemical_triads/links/00b7d5230770c7ff3c000000.pdf Dr Scerri explains triads


https://www.youtube.com/watch?v=foYIHXkHH5w&t=701s (A Tony Smith video recording explaining everything, poor quality)


https://youtu.be/LzpbDttfh9g (animated rotation of the E(8)


https://www.youtube.com/watch?v=WAIw123DIcI (Garrett Lisi explains his particle explorer) https://www.youtube.com/watch?v=dK0qkkjimfo (Garrett Lisi explains the Higgs Boson) https://edu.rsc.org/feature/the-trouble-with-the-aufbau-principle/2000133.article (Problems with the Aufbau) https://phys.org/news/2016-06-knowns-unknowns-dark.html (great explanation of dark matter)

https://web.archive.org/web/20060313032546/http://www.chemcases.com/cisplat/cisplat06.htm (explanation of the Lanthanide contraction) https://www.dinox.org/sugbooks.html (list of expanding earth books)

http://nuclearplanet.com/Herndon’s%20Nuclear%20Georeactor.html (Herndon’s georeactor) https://en.wikipedia.org/wiki/Axion

https://en.wikipedia.org/wiki/Cosmic_ray_spallation#/media/File:Nucleosynthesis_periodic_table.svg (explanation of Nucleosunthesis) http://www.horntorus.com/illustration/unrolling_line_1to2.html (horn torus animation)

https://www.thoughtco.com/what-is-the-higgs-field-2699354 (explanation of the Higgs field) https://www.thoughtco.com/what-is-quantum-gravity-2699360 (the graviton) https://arxiv.org/pdf/0803.2398v1.pdf (Lorentz invariance violation) https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Example:_3D_isotropic_harmonic_oscillator https://en.wikipedia.org/wiki/Zero-point_energy

https://en.wikipedia.org/wiki/Table_of_nuclides#Tables (Table of Nuclides) https://phys.org/news/2017-03-portal-unveil-dark-sector-universe.html (Axion portal) https://mathworld.wolfram.com/StarofDavidTheorem.html (Star of David theory) https://en.wikipedia.org/wiki/Adjoint_representation https://en.wikipedia.org/wiki/List_of_particles

https://www.researchgate.net/publication/270394417_The_expanding_Earth_a_sound_idea_for_the_new_millennium (Scalera expanding earth) http://www.eearthk.com/ (extensive database on the expanding earth)

https://arxiv.org/ftp/astro-ph/papers/0408/0408539.pdf extensive J. Marvin Herndon’s earth theory) https://arxiv.org/ftp/arxiv/papers/1903/1903.12574.pdf Iron slurry in core) https://arxiv.org/pdf/1904.11837.pdf (Eigenmodes in liquid cores) http://nuclearplanet.com/pnas%202003.pdf Herndon’s helium eveidence) https://www.nature.com/articles/srep37740 (link to earth core fusion PDF)

https://www.sciencedirect.com/science/article/pii/S1674984715000518 (Geodetic measurement of expanding earth) https://www.annalsofgeophysics.eu/index.php/annals/article/view/4951 (Expanding earth) https://www.annalsofgeophysics.eu/index.php/annals/article/view/3057/3100 (changing radius cartography) http://www.frontier-knowledge.com/earth/ (Bill Erickson’s web)

http://www.jamesmaxlow.com/ (Dr. James Maxlow’s expanding earth) https://www.lenr-canr.org/acrobat/JonesSEgeofusiona.pdf (Geo-fusion) https://www.dinox.org/comparept-et.html (Paleographic globes)


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