All posts by jgmoxness

E8 Projected to the Concentric Hulls of H4+H4 Phi

The 8 concentric projected 3D hulls from the vertices of the 4_21 polytope (using the split real even E8 roots) and a projection basis from 3 of the 4 rows of the E8 to the H4+H4 Phi folding matrix produce from inner to outer (sorted by vertex Norm):

  • 4 Points at the origin
  • 2 Icosahedron
  • 2 Dodecaheron
  • 4 Icosahedron
  • 1 Icosadodecahedron
  • 2 Dodecaheron
  • 2 Icosahedron
  • 1 Icosadodecahedron

Projection Basis:

Looking at it as an orthonormal 3D projection of 2 600-cells (from the fully folded E8 to H4+H4 Phi), which is the same (as it should be), but rotated.

Here is just one 600-cell (interior).

Here is E7 doing the same procedure:

And again for E6

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

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. 

Perspective Enhanced HyperCube Projections to 3D

These objects are projected using an {x,y,z} orthonormal basis. For n={5,6,7} I add a small delta (.2) to {x,y,z} basis vectors in columns {n-2,n-1,n} in order to keep vertices from overlapping. There is an enhanced perspective applied to vertex location as well.

4Cube


4Cube 3D StereoLithography (STL) model for 3D Printing

5Cube


5Cube 3D StereoLithography (STL) model for 3D Printing

6Cube


6Cube 3D StereoLithography (STL) model for 3D Printing

7Cube


7Cube 3D StereoLithography (STL) model for 3D Printing

Rotating 6D D6 to 3D Pentagon Centered H3 for 2D Decagon Symmetry


Interestingly, these 2D and 3D projections includes all 240 vertices and 6720 edges of E8 with the same 2D and 3D projection. Only the vertex and edge overlap are different.

The D6 to H3 projection based on 3 rows of the E8 to H4+H4 Phi folding matrix.

The rotation off of the D6 to H3 projection based on the E8 to H4+H4 Phi folding matrix is

Using this rotated basis, we now show rectified D6 (Cantellated 6-Orthoplex t0,2{3,3,3,3,3,4})

Bi-rectified D6

6-Orthoplex

Rectified 6-Orthoplex (D6)

Bi-Rectified 6-Orthoplex

Tri-Rectified 6-Orthoplex

6 Demicubes Projected via H4 Folding Matrix

Taking the 32 vertex 6 Demicube with an even number of -1 elements and projecting with 3 of the 4 rows of the H4 folding matrix gives an dodecahedron hull with 12 vertices and 30 edges on the hull out of 240 edges of 6D length Sqrt[2]).

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. 

While taking the 32 vertex 6 Demicube with odd number of -1 elements and projecting with 3 of the 4 rows of the H4 folding matrix gives the icosahedron with 20 vertices and 30 edges on the hull out of 240 edges of 6D length Sqrt[2]).

Rhombic Triacontahedron Animations

Best Viewed in HD

3D Stereoscopic

Red-Cyan Stereoscopic

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. 

More Symmetries of E8 folding, including 5-Cube and 4-Cube (Tesseract)

The same 3 (projection basis vectors that produce the H3 Icosadodecahedron from D6 and the Rhombic Triacontahedron from the 6-Cube (or 2 sets from the 128 1/2 Integer BC8 vertices of E8) form lower dimensional objects within E8.

For more information on the above symmetries, see this post.

The 3D object identification has been confirmed by …

32 vertex 5-Cube with 80 5D Norm’d Unit Length Edges
Projection Basis Vectors {x,y,z}:

Projected Vertex Data:

3D Rhombic 20-Hedron Outer Hull of 22 Vertices


The edge coloring on these projections are defined by which of the 6 dimensional axis the edge aligns with.

10 Interrior Vertices with 10 Edges

All Vertices in 3D

2D faces

16 vertex 4-Cube with 32 4D Norm’d Unit Length Edges
Projection Basis Vectors {x,y,z}:

Projected Vertex Data:

3D Rhombic Dodecahedron with 2 Interior Vertices

2D faces

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.