# E8 in E6 Petrie Projection

An article (interview) with John Baez used an E8 projection which I introduced to Wikipedia in Feb of 2010 here. Technically, it is E8 projected to the E6 Coxeter plane.

The projection uses X Y basis vectors of:
X = {-Sqrt[3] + 1, 0, 1, 1, 0, 0, 0, 0};
Y = {0, Sqrt[3] – 1, -1, 1, 0, 0, 0, 0};

Resulting in vertex overlaps of:
24 Yellow with 1 overlap
24 Dark Blue each with 8 overlaps (192 vertices)
1 Light Blue with 24 overlaps (24 vertices)

After doing this for a few example symmetries, Tom took my idea of projecting higher dimensional objects to the 2D (and 3D) symmetries of lower dimensional subgroups – and ran with it in 2D – producing a ton of visualizations across WP. 🙂

It was one of those that was subsequently used that article from the 4_21 E8 WP page.

Here is a representation of E6 in the E6 Coxeter plane:

Resulting in vertex overlaps of:
24 Yellow with 1 overlap
24 Orange each with 2 overlaps (48 vertices)

# 2D Fourier Transforms on E8 Vertex Projections

Interesting stuff as it relates to QuasiCrystals and non-Crystallographic analysis.
Original source demonstration from John Holland modified and integrated into my VisibLie_E8 viewer…Cool!

# QuasiCrystals and Geometry

I found a 1995 book (PDF online) “QuasiCrystals and Geometry” by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below:

12-gon Rectified E8

Diffraction

Overlay

18-gon Rectified E8

Diffraction

Overlay

30-gon Rectified E8

Diffraction

Overlay

and another 12-gon rectified E8 pattern overlay from M. & N. Koca’s in “12-fold Symmetric Quasicrystallography from affine E6, B6, and F4”:

# My E8 F4 projection and so called "Sacred Geometry"

Please note, this is about the GEOMETRY used in what has come to be called “Sacred Geometry”. Upon further investigation, along with the Kabbalah’s “Tree of Life”, it seems even the Jewish “Star of David” (Solomon) is medieval and not sourced in authentic ancient Abrahamic theology.

As it stands, the so-called “sacred” origin of these, including the pentagon that shows up in the icosahedral/dodecahedral Platonic solids due to the Sqrt(5) Golden Ratio (Phi) relationship within my E8 to H4 folding, is medieval and mostly pagan. I am merely pointing out that E8 contains the geometry that many find “interesting” for whatever reason, and am not promoting the underlying cults that use them.

Video (Best in HD):

My E8 F4 projection w/blue triality edges and physics particle assignments:

My E8 F4 projection with 6720 edge lines:

My E8 F4 projection rectified:

Star Of David Overlay:

Metatron’s Cube Overlay:

Flower of Life Overlay:

Kabbalah Tree of Life Overlay:

Rectified E8 and F4 Triality (Original Work)

WikiMedia Commons source images:
Metatron’s Cube (User:Barfly2001)
Flower of Life (User:Life_of_Riley)
Tree of Life (User:AnonMoos)

# A new Lisi Arxiv paper along with a new E8 projection

For Lisi’s latest ideas on a “Lie Group Cosmology (LGC)” ToE, see http://arxiv.org/abs/1506.08073.
Be warned, it is over 40 heavy pages.

He also posted a new projection on FB.

He uses the (H)orizontal and (V)ertical projection vectors of:
H = {.12, -.02, .02, .08, .33, -.49, -.49, .63};
V = {-.04, .03, .13, .05, .20, .83, -.45, .19};

Here it is rendered with H and V reordered to match in VisibLie_E8 tool:
H/V order is {2, 3, 4, 7, 1, 6, 5, 8};
Note: the particle assignments in my VisibLie_E8 tool (as in the Interactive visualizations on this website) are not the same and have different triality rotation matrices.

The pic below has a slightly altered projection that creates a bit more hexagonal symmetry and integer projections (without deviating more than .05 off his coordinates):
Note: vertex colors indicate the number of overlapping vertices.
This uses H/V projection vectors (reverse ordered back to his coordinate system).
Symbolic:
H={3 Sqrt[3]/40, 0, 0, 3 Sqrt[3]/40, 1/3, 1/2, 1/2, -2/3};
V={-3/40, 0, 3/20, 3/40, 3/20, -4/5, 1/2, -3/20}};
Numeric:
H={.13, 0., 0.,.13,.33,.5, .5,-.66`};
V={-.075, 0., .15, .075, .15,-.8, .5,-.15};

Here is Lisi’s version with my coordinates:

Here is a version of the full hexagonal (Star of David) triality representation using coordinates I found with the same process used above. Particle color, shape and size are based on the assigned particle quantum numbers (spin, color, generation). There are 86 trialities indicated by blue triangles.
Projection vectors (in my “physics” coordinate system) are:
H={2-4/Sqrt[3],0,0,-Sqrt[(2/3)]+Sqrt[2],0,0,Sqrt[2],0};
V={0,-2+4/Sqrt[3],Sqrt[2/3]-Sqrt[2],0,0,0,0,-Sqrt[2]};

See also a rectified version of this, where the particle overlaps show the inner beauty of E8!!

The underlying quantum symmetry patterns within this coordinate system are:

# Hyperbolic E8 Visualizations

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

# 7 Perfect Sri Yantras in 2D and 3D

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 *)

Solution 7: Optimum planar

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

# The latest VisibLie_E8

It now has 3D clipping (and slicing) plane control sliders on the the left side controls within a new multiplexed slider control. I also added the 2D I-Ching and 2D/3D Sanskrit Sri Yantra visualizations.

# Working on visualizing the I-Ching Trigrams and Hexagrams

Notice that the binary can be associated with Clifford algebras

# Ever wonder what the Bible's social network looks like?

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.

God

Jehovah

Lord

Eve

Jeshua

Noah

Abraham

Moses

Jesus

Gabriel

Michael

Satan