Tag Archives: 3D

The Rhombic Triacontahedron and E8 to H4+H4Φ folding

I was asked to clarify a particular projection of the Rhombic Triacontahedron from the 6-Cube subgroup of E8, which had I created for WikiPedia (WP) back in 2011 based on an E8 to H4+H4Φ rotation matrix I discovered in 2010.

The edge coloring on the projection above is defined by which of the 6 dimensional axis the edge aligns with.

BTW – if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at:  jgmoxness@theoryofeverthing.org. 

The following projection (basis) vectors take the 6-Cube as a subset of E8 and projects it into 3D using the following basis vectors:

Notice the last two dimensions in each vector are 0, which effectively takes the 128 8D 1/2 integer vertices of the E8 BC8 DemiCube into the 6D 6-Cube.

For the following data analysis, it is useful to reference the full list of E8 and H4+H4Φ data at the end of my blog post here.

The clarification desired was due to a conjecture that since E8 folds to H4+H4Φ (600 cells) and D6 folds to H3+H3Φ, that the 6-Cube should fold to an outer and inner rhombic triacontahedron scaled by Φ.

Below is a visualization of D6 folding into 2 concentric H3 icosadodecahedrons at a ratio of Φ using the same projection as described above. The edging displays the  480 Norm’d 6D Sqrt[2] length edges, clearly showing the outer and inner H3 polytopes.

Shown with exterior faces on edges (not a pure convex hull algorithm)…

The data below shows two sets of 30 vertices (outer on the left and inner on the right). They are sorted by the 3D Norm’d length of the vertex ray from {0,0,0}, followed by the vertex number in the E8 to H4+H4Φ list at the end of the post referenced above, along with the 6D E8 vertex that is a member of the D6 set, and the {x,y,z} projected vertex position

The last line of the data set shows that the first set of outer 30 projected vertex locations on the left is numerically equal to the second inner set after multiplication by Φ.

I did the same for the Rhombic Triacontahedron projected out of the 6-Cube.

I separated the outer and inner vertices used to make the WP projection at the top of the post. There are 64 vertices and 192 6D unit length edges forming pentagonal symmetry along specific axis (as well as hexagonal symmetries on other axis). It is clear that scaling the inner 32 vertex locations by Φ is not going to give the outer 32 vertex positions. Yet there is a similarity- they both have 12 vertices of one 3D Norm’d length and 20 of another.

Outer 32 vertices and 60 unit length Norm’d 6D edges.

The edge coloring on the projection above is defined by which of the 6 dimensional axis the edge aligns with.

Inner 32 vertices and 60 unit length Norm’d 6D edges.

The last line of the data set above shows that the outer data (left) is the same as the inner data set multiplied by a factor of Φ when the first 12 (shorter 3D Norm’d)  inner vertices (right) are multiplied by an additional factor of Φ.

This explains the lack of isomorphism, but why is it not like the D6 to H3+H3Φ and E8 to H4+H4Φ in the symmetry of Φ scaling?

The best explanation I can give involves the pattern of left vs. right 3 digits in the 6-Cube E8 vertices. This is due to the left right symmetry of the basis vectors applied to binary pattern of half integer vertices in the 6-Cube.

All vertices have a consistent pattern that differentiates the 20 longer Norm’d vertices from the 12 shorter ones. Specifically, the shorter 12 that have the inner ones needing the added factor of Φ always show that:

  • the right inner vertices all have an even number in 6 +/- elements but not always evenly split across the left and right
  • the left outer vertices all have an odd number of in 6 +/- elements  always unevenly split across the left and right

Whereas, the longer 20 that match the Φ scaling reverse the pattern where:

  • the right inner vertices all have an odd number in 6 of +/- elements  always unevenly split across the left and right
  • the left outer vertices all have an even number in 6 of +/- elements but not always evenly split across the left and right

Another related fact in the folding pattern of the 6-Cube emerges when we note that since E8 contains 128 1/2 integer BC8 (the 8 Demi-Cube). In addition, we note that the 6D complement of BC8 and the 6-Cube is the same set of vertices. This means we get the same projection using all 128 vertices of the 6D trimmed BC8 as we do with the 64 of the 6-Cube, except now there are 768 6D Norm’d unit length edges rather than 192. Interestingly, that means that similar to E8 containing simultaneous copies of 4 600-Cells (a left and right pair scaled at Φ), E8 contains 4 simultaneous copies of the rhombic triacontahedron after adding the Φ scaling on the two sets of 12 inner BC8 vertices).

Wow – that is nice! Remember, you heard it here first!

If we try this idea with the complement of the integer elements of E8, namely the 112  6D trimmed D8 vertices complemented with 60 D6 vertices, the 6D trimmed 52 vertex complement contains 4 identical copies of the excluded 16 generator vertices of E8 {+/-1,0,0,0,0,0,0,0} permutations (plus 4 {0,0,0,0,0,0,0,0} vertices).

As you can see, the folding of the integer elements of E8 do not fold the same as the 1/2 integer elements, and in retrospect, we probably should not expect them to.

It is because of the symmetric 4D left/right symmetry of E8=BC8+D8 that we get the wonderful folding of it into  H4+H4Φ !

Fun with Tutte-Coxeter, Beordijk-Coxeter, E8 and H4

In reference to a G+ post by Baez (w/Greg Egan), it’s interesting to note the link to E8’s outer ring of the Petrie projection of a split real even E8, which creates a Beordijk-Coxeter helix.

Beordijk-Coxeter helix in 2D
helix2Db
Beordijk-Coxeter helix in 3D
cells6004b

The Beordijk-Coxeter helix connects the nearest 6 vertices on the outer ring. The Tutte-Coxeter graph is created in 3 (blk,grn,red) sets of edges by taking the (outer) ring and skipping (6,8,12) or counting (7,9,13) vertices. It shows there are 2 perfect pentagons and 1 pentagram (with different radii due to the difference in distance between the sets of vertices used).

e8ring8-1b

Of course, the crystallographic E8 is manifestly related to the 5 fold symmetry of the pentagon, with its integral relationship to the non-crystallographic H4 group (and its Coxeter-Dynkin diagram) through E8 to H4 folding using the Golden ratio Phi.

It is interesting to note that the skipping of 5+(1,3,7) vertices is similar to the creation of the 120 (240) vertex positions of H4 (E8) Petrie projection by adding to the 24 vertices of the 8-cell and 16-cell (which make up the self-dual 24-cell) the 96 vertices of the Snub 24-cell. This is done through 4 rotations skipping 5 vertices.

Also notice the (1,3,7) are the number of the imaginary parts of Complex, Quaternion, and Octonion numbers, also integrally related to E8.

snub-pics

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.

SriYantra-Refernce

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 *)
SriYantra1

Solution 2: Solving constraints (* 1,2,3,10,15 *)
SriYantra2

Solution 3: Solving constraints(* 1,2,4,5,10 *)
SriYantra3

Solution 4: Solving constraints (* 1,2,5,6,19 *)
SriYantra4

Solution 5: Solving constraints (* 1,2,6,14,19 *)
SriYantra5

Solution 6: Solving constraints (* 1,2,8,9,20 *)
SriYantra6

Solution 7: Optimum planar
SriYantra7

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).
SriYantra3D1

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

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
outGematria-god

Jehovah
outGematria-jehovah

Lord
outGematria-lord

Adam
outGematria-adam

Eve
outGematria-eve

Jeshua
outGematria-jeshua

Noah
outGematria-noah

Abraham
outGematria-abraham

Moses
outGematria-moses

Jesus
outGematria-jesus

Gabriel
outGematria-gabriel

Michael
outGematria-michael

Satan
outGematria-satan