Tag Archives: 2D

Mathematical Navigation of Ancient Sacred Texts

I’ve added some new features in the #15 pane (Gematria). This sociology pane deals with the idea of Old Testament Hebrew and New Testament Koine Greek Gematria. The histogram shows the distribution of words used in the Bible according to their gematria value in Hebrew or Greek. It also presents a clickable 3D hyperbolic tree (graph) of proper names related within verses, as well as a clickable “nearest word” graph. Each word in each book, chapter, verse are select-able by slider or clickable (as are the list of words with the same gematria values).

This combination of UI creates a powerful new way to navigate the bible. Each word in the verse is colored by their value in Hebrew (if Old Testament), Greek and English. It makes use of Wolfram’s curated LanguageData/DictionaryLookup to get a Nearest Word Graph in Hebrew and English. Note: Hebrew is properly presented from right to left.


I also decided to attempt to develop visualizations related to the alignment of sociological, theological ancient world history using the recorded sacred texts created at the start of the Bronze age. The time line runs between 4000 BCE and 100 CE and the geopolitical map centers on the cradle of civilization Sumerian (Tigris/Euphrates), and extends from the Egyptian (Nile) to China’s (Yellow), and including Indian (Indus) and Persian cultures.

The list of sacred texts aligned by creation, flood and patriarch figures (e.g. Adam, Eve, Noah, Abraham) and primary deity (e.g. Elohim) were:
1. Egyptian Book Of The Dead
2. Babylonian/Sumerian Kings List/Epic of Gilgamesh/Enuma Elish
3. Hebrew(OT)-Greek(NT) Bible
4. Chinese I Ching
5. Indian Rig Veda
6. Greek Homeric Poems (Illiad/Odyssey)
7. Persian Avesta


Given the amazing advancement of modern science in tracing the genetic lineage of human history along with radioactive dating techniques, it is untenable to interpret these sources literally with any precision. Yet, by understanding their “world view” at the time along with consistently applied “poetic license” across all the texts (e.g. Homer), one can come up with interesting theories for the veracity of their original intentions to document sociologic human history!

For example, it is interesting to note the alignment between the 10 AntiDiluvian (pre-flood) kings in both the Sumerian Kings List and the first 10 patriarchs (starting with Adam) in the Pentateuch. Both have unusually long life-times. When the extremely large (and in most cases rounded) ages of each person are reduced by factors based on the counting system in use (e.g. Sexagesimal (base 60) 10*60^2) for Sumeria and decimal 10^2 or 10^3), the reigns of kings and lifetimes of lineage align as follows:


Antediluvian Patriarchs and the Sumerian King List
Reinvestigating the Sumerian Kings List

For a modern genetic species analysis visualization, see:


See this Dymaxion Map of genetic analysis of human migration from Wikipedia Commons:

Here is an old artistic map of World History from HistoMap http://www.davidrumsey.com/. It is largely Biblical in its antidiluvian history. While it is not based on modern scientific evidence, it is an interesting mash-up of information from various sources. A more detailed map (horizontal) is here (right-click the image to download and save-as the 20MB PNG file):

Here is another vertical version from the same website:

and another histomap (linear and logarithmic) of the Earth’s Geologic/Evolutionary History (right-click/save-as for a 12MB version):

Fun with Conway's "Game of Life" in 3D -> nD and Music

I’ve integrated a few more demonstrations, see below for descriptions and pics..

Conway’s “Game of Life” in 3D -> nD


Music related




The #13 pane (Chords) will address the psychology of the mathematical beauty of music. The 2D pane presents Emmanuel Amiot and Fernand Brunschwig’s  Pythagorean Meantone And Equal Temperament Musical Scales, while the 3D pane shows Drew Lesso’s 3D Lambdoma matrix.

The #14 pane (Gematria) will address Theological Number Theory in the form of Hebrew gematria, H4, and Tori. This sociology pane deals with the idea of Old Testament Hebrew and New Testament Koine Greek Gematria. The histogram shows the distribution of words used in the Bible according to their gematria value in Hebrew or Greek. Each word in each verse is selectable and shown in the distribution. Each word in the verse is colored by their value in Hebrew (if Old Testament), Greek and English.
Note: Hebrew is properly presented from right to left.

The #15 pane (QC) will address Quantum Computing in the context of MyToE. It uses several demonstrations… Qubits On The Poincare-Bloch Sphere,  Quantum Logic Gates Roots Exponents And Eigensystems and Quantum Fourier Transform Circuit by Rudolf Muradian

The #16 pane (AI) will address Artificial Intelligence, an nD version of Conway’s “Game of Life” and Cellular Automata in the context of MyToE. It is an integration of Daniel de Souza Carvalho’s demonstration


Boerdijk–Coxeter helix

The Boerdijk–Coxeter helix is a 4D helix (of 3D tetrahedral cells) that makes up the vertices on 4 of the concentric rings of E8 Petrie projection (or the H4 and H4φ rings of the 2 600 cells in E8).

Outer (Ring 4) of H4 in 2D with non-physics vertices of all 8 rings of E8 in the background



Outer (Ring 4) of H4 in 3D with physics vertices


Ring 3 of H4 in 3D with physics vertices


Ring 2 of H4 in 3D with physics vertices


Inner (Ring 1) of H4φ in 3D with physics vertices


Combined 4 rings of H4 in 3D with physics vertices


Outer (Ring 4) of H4φ in 3D with physics vertices

Ring 3 of H4φ in 3D with physics vertices


Ring 2 of H4φ in 3D with physics vertices


Inner (Ring 1) of H4φ in 3D with physics vertices


Combined 4 rings of H4φ in 3D with physics vertices


Combined 8 rings in 3D with physics vertices


E8's 10 24-cells (H4 and H4φ) made up from 10 8-cell Tesseracts and their dual 16-cell 4-Orthoplexes

I decided to visualize my decomposition of the 10 self-dual 24-cells of E8. These are shown in the Petrie projection of E8 that is split into H4 and H4φ using my E8 to H4 folding matrix. The interesting aspect of this is that the selection of the canonical H4 24-cell determines the other 4 24-cells that make up the 96 vertices of the H4 snub-24-cell by rotating 4 times by π/5. Scaling the vertices by φ on the Petrie projection give the other 5 24-cell+snub-24-cell vertices.

It is also interesting to note that (my modified) Lisi physics particle assignments (with particle/anti-particle pairing) create 30 nicely ordered 4-particle/anti-particle sets that (fairly consistently) distribute over type, spin, generation, and color. These are visualized in 3D with shape, size and color defined by the quantum numbers of the assigned particle. The 3D projection is the same as that of the Petrie with a 3rd Z vector added. For more detail, see my latest paper.

The last set of images (using only the “math version” round black vertices and numeric identifier from the position in the Pascal triangle/Clifford Algebra canonical lexicographic ordering) are the 2D rotations of 24-cells around face-3 of this projection (which turns out to be the orthonormal shape of the H4 600 cells).





















My latest paper on E8, the H4 folding matrix, and integration with octonions and GraviGUT physics models.

The full paper with appendix can be found on this site, or w/o appendix on viXra. (13 pages, 14 figures, 20Mb). The 22 page appendix contains the E8 algebra roots, Hasse diagram, and a complete integrated E8-particle-octonion list.


This paper will present various techniques for visualizing a split real even E8 representation in 2 and 3 dimensions using an E8 to H4 folding matrix. This matrix is shown to be useful in providing direct relationships between E8 and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects. A direct linkage between E8, the folding matrix, fundamental physics particles in an extended Standard Model GraviGUT, quaternions, and octonions is introduced, and its importance is investigated and described.

If you would like to cite this, you can use this BibTex format if you like (remove the LaTex href tag structure if you don’t use the hyperref package):

title={{The 3D Visualization of E8 using an H4 Folding Matrix}},
author={Moxness, J. G.},
month = nov,


The 3rd (Z) basis vector for Bathsheba & Wizzy’s 600 Cell

While the 3D model I used to create the 2D Van Oss projection isomorphic to E8 Petrie (and a beautiful pentagonal view), it was not the same as what was being used by Richter in his 3D “pre-Van-Oss” construction. Given my H (or x) and V (or y), the 3rd basis vector for this projections is most likely:
Z={0, -0.0801064, 0, 0.236818, 0, 0.0801064, 0, -0.236818}
which reproduces the Richter, Bathsheba and Wizzy’s 3D models. Interestingly, it produces one face (shown above) that is the same as all the orthonormal faces of 2 concentric 600 Cells (at the Golden Ratio). The 3rd unique face is:

I replaced the 3D spin movie of this on my main page with this new projection.