This section of the post visualizes the Miracle Octad Generator (MOG) used in the generation of the Steiner octad system related to the binary (ternary) Golay codes. For more information, please see Wikipedia (WP) here and here (a 2010 post on Internet Archive for Steven H. Cullinane’s site).
For the companion Mathematica Notebook (NB) file, see here. Click the PNG image below for SVG.
MOG Visualizations and code
This section is an analysis and visualization of the Leech lattice’s related binary (ternary) Golay code. Please see here (WP) for more information.
For the companion Mathematica Notebook (NB) file, see here. Click the PNG image below for SVG.
Leech lattice binary and ternary Golay code
This section visualizes the 16D Barnes-Wall and 24D Leech Lattices. Much better visualizations from David Madore (Gro-Tsen) are here. He uses GAP and SAGE to tease out the group theoretic symmetries.
Octonion based Barnes-Wall and Leech Lattice visualizations based on my implementation of work from Geoffrey Dixon and Robert Wilson:
Octonion defined Barnes-Wall to 16D B82 projection
Octonion defined 24D Leech lattice to 8D E8 Petrie projection with all 196560 vertices in 2D and 3D, noting the similarity to the 6720 vertex rectified E8 in the same projection (above).
Octonion defined Leech lattice to 24D E83 Petrie projection with 20 increments of 1000 vertices for every 10000 out of 196560
Same as above with 50% of the inner edges filtered out
Baez’s comments regarding G2 and E6 binomial collisions
My first reaction is to question whether the “isomorphism of the irreducible representations (of the given orbit dimension)” is or is not over-constraining the binomial’s numerical pattern match.
G2 has 3 and E6 has 8 representations of orbit dimension 3003 (each shown below). I found a possible alternate path via Λ SU(4) = ΛSO(6) with the Binomial[15,5]=3003. Yet, these groups don’t have natural orbits of 3003 as E6 and G2 do. The two adjacent binomials in this 3003 triplet of collisions, namely Binomial[14,6] and Binomial[15,5], are part of the known infinite series of solutions with k=2 for binomial(n,m-1) = binomial(n-1,m) given by n = F_{2k}*F_{2k+1}; m = F_{2k-1}*F_{2k} where F_i is the i-th Fibonacci number.
Adjacent binomial collisions in the Infinite series o Fibonacci numbers
Below are the related maximal subgroups for G2 and E6. The first column shows the G2 or E6 Coxeter-Dynkin diagram. The second column is the Weyl orbit. The third column is the identified maximal subgroup elements in the 3003 rep. The fourth column shows the maximal subgroup content and the fifth column shows its Coxeter-Dynkin diagrams.
Please forgive (ignore) my use of filled instead of ringed nodes. The relevant information is the subgroup content in the third column.
The image below was created from Pane #2 Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle. It is used here to annotate the above binomial collisions which are all contained in the standard Pascal Triangle.
This #2 pane (Pascal) was created for visualizing the Pascal and Sierpinski triangles, along with the Fibonacci and Lucas numbers. In addition to allowing the change in size of the triangle, it highlights the binomial functions and allows the changing the modulus of the numbers used. The number backgrounds are colorized in the selected gradient. This pane was created with ideas from Peter S. Williams’ Mod 9 Pascal Triangle Physics http://naturalnumber.com/
Annotated Binomial Collisions of the Pascal Triangle highlighting (blue Mod8=3) dimension 3003 in G2 Binomial[14, 6], E6 Binomial[78, 2] with k=2 Binomial[15, 5] which could be Λ SO(6) = Λ SU4 (albeit not a natural orbit of those groups). The other highlights are collisions of 120 (red Mod8=8), 210 (orange Mod8=2), 1540 (yellow Mod8=4), and 7140 (magenta Mod8=4). Two partial collisions (with one factor beyond this table <128) are in white rings as Binomial[19,5]=Binomial[153,2]=11628 (Mod8=4) and Binomial[17,8]=Binomial[221,2]=24310 (Mod8=6). I also added in the OEIS A003015 listed collision of Binomial[104,39]=Binomial[103,40]=61218182743304701891431482520 as black rings Mod8=8. As is the OEIS listed k=1, this k=3 collision should more properly be assigned as part of the infinite series of adjacent Fibonacci collisions such that it is not “unique” as the other 7 that are not exclusively related to the adjacent Fibonacci series.
For a nice introduction to roots and their fractal nature, see this post from 2011 on John Baez’s Azimuth blog or his 2023 UCR page here. Some of my code here is sourced from a great repository of interesting math visualizations from many others including the author of the pywonderland GitHub of Zhao Liang.
The SVG images below can be clicked to zoom in on another tab, but here is a Mathematica Notebook with Tooltip info on the selected roots. That SVG is 8000×8000, so you may have to <cntrl>- to zoom out to see any of the interesting parts (depending on your screen resolution and browser setup).
All roots of polynomials up to 224 (~16.7 million) with +/-1 coefficients.
The roots are complex valued (Real x-axis and Imaginary y-axis) with normalized binVal(ue) defining the color from 0 to 1. The ColorRules are as follows:
A binVal{x,y} is incremented when the {x,y}={Real,Imaginary} root elements fall in that particular bin. The results are normalized by dividing each bin by the Log2@binVal{x,y} / Max@Log2@binVal. So the two x-axis binVal={+/-1,0}width/2 have a binVal=1 means the cumulative roots in every degree have the most roots at that 0/π location. The two y-axis binVal={0,+/-1}height/2 have a binVal of ~.903 that land in those +/-i π/2 bins.
The analysis below provides more detail on the Roots-of-Unity Circle (RUC) at the center of the light yellow area. These are interesting in that there are 4 quadrants each with 12 roots that all have 1/2< binVal <1 at the center of holes.
There are 6 roots that sit in holes surrounded by zeros (i.e. 2π/{12,10,8,6,5,4}) and 6 roots in holes with surround values <1/2 (i.e. 2π/{24, 14, 7, 11/2, 29/6 (75°), ~437/90 (77°)}). The only other binVal roots >1/2 are on the 1 pixel wide x-axis with 5936 of 8000 (~75%) from the x-y grid of 8000×8000 (64 million) bins which sum over each of the 224 (~16 million) roots-of-polynomials with -/+1 coefficients. The rest of the x-axis roots are 2020 bins equal to 0 and only 44 bins between 0 and 1/2).
The large table below shows 100 detail areas of “special” algorithmically selected locations from bin location Tally counts of 7732 unique binVals. About 50% of these occur in 2 to 4 (symmetric) locations with a binVal>1/2. The other 50% of the Tally counts have binVal<1/2 and tend to have 103 to 106 locations with all but 5980 (of 64 million) bins. This is made clear from the above MatrixPlots.
Each detail area shows a 400×400 ArrayPlot around the selected location and an 11×11 zoom-in of that binVal matrix with its ArrayPlot. They include the 48 RUC roots and 12 x-axis roots with binVal>1/2, and 40 off-axis roots with ~2/3 bins of zero and 1/3 with binVal<1/2. This off-axis list includes one set of 4 quadrant roots that is a proper hole (in the 4 yellow rings located just above/below the bottom/top apex in the full image or 10-14th from the bottom in the detail list below). The other 9×4 roots are what I call “pseudo-holes” that have no enclosed high central binVal.
Zoom-in of detail in the upper-right quadrant RUC root values – click SVG to create another tab and zoom in your browser.
The annotated upper-right quadrant of the SVG image has the RUC roots ringed in colors (shown in the zoom-in graphic above) with large colored rings assigned by their binVal. Annotated rings with binVal>1/2 are cyan, magenta, and red. The binVal<1/2 are yellow, orange, or blue rings. The smaller black ring is what is visible in the 11×11 MatrixPlots below. To the right is the text with the angle at which the root is located.
FYI – The color scheme on these detail zoom MatrixPlots are set to the default in order to maximize and highlight the numeric differences, so they don’t match the ColorRules in the large 8000×8000 MatrixPlot at the top of the page.
Click the image below to see this SVG and zoom further.
Detail areas of special roots with a 1000×1000 ArrayPlot and an 11×11 zoom of the Abs root values and its ArrayPlot
Stay tuned for more images. I think I will use this post to place my more beautiful works!
E8 in Petrie projection with 3D theoretically assigned physics particles
These use a better combination of specularity and opacity with both math and physics (theoretical assigned) 3D vertices. This is accomplished by actually projecting into 3D but using only X-Y basis vectors with Z={0,0,0,0,0,0,0,0} and the (virtual) camera at XYZ={0,0,∞}.
Some images use color coding for vertices that indicate overlaps (e.g. when in the spherical (math) vertices E8 hexagonal basis). The overlap number data may then be presented in the lower left. This data can be removed using SVG editors such as Inkscape.
Some images also show the 3D technical axes (basis) projection vectors as well as the vertex, edge, and other data. Also, bear in mind that the PNG images shown below (so they are visible in Google Images after being crawled) tend to show pixelation artifacts and can present with black background vs. white. The SVG presentation (obtained by clicking the embedded image link) will be correct and the transparent background can be modified easily (e.g. with Inkscape).
Please note, these are provided under Creative Commons CC BY-SA 4.0 Attribution-ShareAlike 4.0 International license, so if you use them please cite me at this site (or me on the Wikipedia (WP) article or WP commons source):
Below is the well known PR photo for E8 in Petrie projection with basis vector axes and data presented (click the image for the downloadable SVG):
E8 hexagonal (aka. as the “E6/D4 Mox [6]” basis in Wikipedia) projection
The E8 hexagonal with the 3D assigned theoretical physics vertices:
E8 with the 3D assigned theoretical physics vertices in hexagonal projection
The E8 hexagonal projection with 6720 edges:
E8 with edges in hexagonal projection
The Rectified E8 in hexagonal projection with 6720 vertices:
Rectified E8 with edges in hexagonal projection
E8142 in hexagonal projection with 17280 vertices:
E8142 in hexagonal projection
E8142 in Petrie projection with 17280 vertices:
E8142 in Petrie projection
The periodic table with orbitals in 2D:
The periodic table with orbitals in 2D
The periodic table with orbitals in 3D:
The periodic table with orbitals in 3D
Kneser neighborhood graph of Niemeier lattices with associated Mathematica Notebook here:
Kneser neighborhood graph of Niemeier lattices with associated Mathematica Notebook here.
The E8 subgroup tree:
E8 Subgroup Tree with associated Mathematica Notebook here.
Riemann Zeta function zeros:
This is a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points along the critical line ζ(1/2+it) for real values of running from 0 to 50. The consecutive zeros have 50 red plot points between each with zeros identified by magenta concentric rings (scaled to show the relative distance between their values of t).
All roots of polynomials up to 224 (~16.7 million) with +/-1 coefficients.
Bott Periodicity Clock of Cl(9,0) as it relates to the Hopf Fibration of CP4
This is not the typical E8 Gram matrix representation as the Cartan matrix using only the simple roots (shown at the end of this post).
It uses all 240 E8 vertices (+120/-120 roots) of the E8 Split Real Even (SRE) vertices to produce the Gram matrix (with vertices sorted in simple numeric order):
Full E8 Split Real Even (SRE) vertex Gram matrix in vertex numeric sort order
The same 240 E8 vertices (+120/-120 roots) below are sorted by the 9th row of the binary structure Pascal Triangle, excluding the 2nd & 2nd to last entries of -/+ 8 (i.e. {-1,-8,-28,-56,70=(-35/35),56,28,8,1}, where the 112 integer D8 are -56/56 and the 128 half-integer BC8 are {-1,-28,-35,35,28,1}).
Full E8 Split Real Even (SRE) vertex E8 Gram matrix in Pascal Triangle order
As above, but with H4+H4φ 8D Gram matrix created using the dot product of E8 against the 8×8 E8 to H4 folding matrix U:
H4+H4φ 8D Gram matrix
Now with only the H4φ 4D Gram matrix created as 120 H4 4D vertices:
H4φ 4D Gram matrix
Compare this to the Gram matrix using the root weights of E8 (in Pascal triangle order vs. sorted into root and/or weight order):
E8 Gram Weights (in Pascal Triangle order)
Then compare to the Cartan matrix produced as the Gram matrix of only the 8 E8 simple roots:
This post presents my independent analysis of Jenny Lorraine Nielsen’s (JLN’s) Topological Unified Field Theory (TUFT). It symbolically implements her equations related to the fundamental constants, General Relativity (GR) cosmological parameters, and Standard Model (SM) particle masses and magnetic gyrometric ratios and anomalous moments. It then calculates the values and compares them against the PDG CODATA from Wolfram’s curated data sets.
The structure of this analysis is shown (below) in PNG (click for SVG version) or see here for the full PDF of the analysis or here for the Mathematica Notebook (NB) version:
Outline of my Mathematica analysis of JLN’s TUFT predictive theoretical parameters
E8 Hopf Fibration
The Aizawa Chaotic Attractor
This is all also related to the Clifford algebra of Cl(9,0) with the Bott Periodicity clock showing the related algebras (click for SVG):
Bott Periodicity Clock of Cl(9,0) as it relates to the Hopf Fibration of CP4
The Bott periodicity clock uses a Mod 8 clock face with second hand mnemonics taken from the I-Ching with the real Clifford algebra of signature (p,q) denoted as Clp,q(R)=Cl(p,q). An animated (GIF) example is on the Bott Periodicity Wikipedia page.
If you are working with topological field theoretic Hopf fibrations, chaotic attractors, and/or E8 and any of its extensive set of maximal subgroups, polytopes, lattices and codes or its folded (rotated) H4 family of polytopes (e.g. 600 and 120 cells in their vertex, cell, face, and edge orientations), VisibLie_E8 can group theoretically integrate all of these concepts visually!
Click on the PNG image to get the SVG version.
E8 Hopf Fibration
H4 600-cell Hopf Fibration
The H4 cell-first {5,3,3} 120-cell Hopf Fibration (with envelope of a Chamfered Dodecahedron)
The Aizawa Chaotic Attractor
This is all also related to the Clifford algebra of Cl(9,0) with the Bott Periodicity clock showing the related algebras (click for SVG):
Bott Periodicity Clock of Cl(9,0) as it relates to the Hopf Fibration of CP4
The Bott periodicity clock uses a Mod 8 clock face with second hand mnemonics taken from the I-Ching with the real Clifford algebra of signature (p,q) denoted as Clp,q(R)=Cl(p,q). An animated (GIF) example is on the Bott Periodicity Wikipedia page.
A few more Hopf geometries:
The envelope of the vertex first 600-cell aka. PentakisIcosidodecahedron
The envelope of the cell first 120-cell aka. Chamfered Dodecahedron
If anyone is using my E8 to H4 folding matrix in their research, this post provides a bit of context from the original form (2012-2014) as it was improved to the current form. Drop me an email if you are using my work or have questions, or find errors, etc.
I am working with a few who are using my work on cryptography, quantum computing, and computation optimization, but they have yet to post or publish results.
The image below is also available as a Mathematica notebook (NB) or PDF. Click the PNG image for the SVG version.
The first part reviews the current Hermitian, centrosymmetric, traceless (Tr=0), and volume preserving (Det = 1) form of the folding matrix (U).
The last part reviews the traceless Hermitian unitary matrix (uU) of determinant 1 (i.e. A7=SU(8)) with characteristic polynomial |coefficients| of the 5th row of the Pascal triangle (1,4,6,4,1}. This matrix has the same characteristic polynomial as that of the 3-Qubit Hadamard matrix. Both are related to QM Hadamard, CNOT and SWAP gates and Pauli matrices from which it, cmU, and U are constructed.
Below that is another matrix analysis of the 6 8×8 Dirac matrices (i.e. DiracMatrix[#, Dimension -> 6]&/@Range@6) with the Identity and +/- Traceless Identity matrices prepended. These have similar properties to uU with the addition of being (skew)Hamiltonian matrices.
Comparing E8 to H4 folding matrix versions
Matrix Analysis of Identity and Dirac Matrices
Now for good measure, we show the matrix characteristics of the 2×2 Pauli, 3×3 Gell-Mann, and 4×4 Dirac matrices:
Matrix Analysis of 2×2 Pauli, 3×3 Gell-Mann, and 4×4 Dirac Matrices
And finally, we analyze the 8×8 Dimension 6 n=4 Dirac Matrices:
My latest ToE paper here. This paper presents compelling E8 group-theoretic properties that are found in the set of 240 O and N H-bond acceptors which support Watson-Crick codon:anticodon pairing. Empirically correct consequences are deduced from the group-theoretic model.
This paper is related to another recent post here.
Canonical Watson-Crick RNA Codon Acceptor Sites
Here is the WikiMedia version of the Canonical Watson-Crick Acceptor Sites (Figure 11).
Stainless & Glass Public Art Sculpture in Tucson By Ray King.
It’s shape is similar to, but not exactly, a Pentakis Icosidodecahedron, which is 3D the orthographic shadow of the 4D of H4 (and H4φ) 600-cell (s) obtained by folding the 8D object (E8) to 4D. E8 is also known as the 421 polytope.
I thought it was interesting that the 2D color shadow on the patio that looks like my E8 diagram called a E8 Petrie Projection:
Dedicated to the pursuit of beauty and Truth in Nature!
