Beginning to play with visualizing E8 and subgroups using hyperbolic Poincare projections and tiling – in 2D/3D/nD

Copyright 2015 J Gregory Moxness

Based on an excellent 1998 paper by C.S. Rao I’ve modeled the Sri Yantra in Mathematica and incorporated it into the VisibLie_E8 demonstrations (I will upload that soon).

This is a reference figure to verify the model is correct for concurrency (intersecting lines) and concentricity (inner triangles and Bindu point are centered). There are 4 of 18 other constraints used in the diagrams below numerically and/or symbolically solved within the Mathematica code.

Here are 7 of many possible solutions using the 20 constraint rules to produce optimal spherical and plane forms of the Sri Yantra.

Solution 1: Solving constraints (* 1,2,3,4,8 *)

Solution 2: Solving constraints (* 1,2,3,10,15 *)

Solution 3: Solving constraints(* 1,2,4,5,10 *)

Solution 4: Solving constraints (* 1,2,5,6,19 *)

Solution 5: Solving constraints (* 1,2,6,14,19 *)

Solution 6: Solving constraints (* 1,2,8,9,20 *)

3D using pyramids and new clipping and slicing functions to take off the front (and back) portions of the symmetric 3D object.

I will produce a version option with conical shapes just for fun (soon).

This is a list of the details behind each constraint function Fn

These are 3D hyperbolic tree graphs of some major figures in the Bible. The Hebrew, Greek transliterations for the first verse where they show up are also shown. Mouseover shows the names between the links and the number of times they occur (which changes the thickness of the link).

Another cool new feature I just added is a word translation when you mouse over any word. Clicking on the word or name in the graph changes the statistics to that word.

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:

See:

Antediluvian Patriarchs and the Sumerian King List

Reinvestigating the Sumerian Kings List

For a modern genetic species analysis visualization, see:

http://www.timetree.org/book.php

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):

I decided to take a quick dive into an interesting idea from Peter Williams related to how the modulus of binomial numbers are related to the Pascal Triangle. I’ve added this to the VisibLie_E8 viewer as well.

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

Conway’s “Game of Life” in 3D -> nD

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

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

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).