Platonic solids in 2D orthogonal projection
I made some images showing each figure in a Flower of Life (geometry) pattern of circles, and made cube vertices green, octahedron vertices blue, and intermediate vertices purple.
More information Stellated octahedron{3,3} + {3,3} +, Cube and octahedron{4,3} + {3,4} + ...
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Here's the vertices projected onto Metatron's cube, in dual polyhedron configurations (red and green edges), sharing the same outer hexagon vertices.
3D presentation
Metatron's cube in 3-dimensions
As spheres |
As tangent cubic spheres |
Here's another representation, visualized as an actual cube, viewed on the diagonal, with 8 corners of the cube as transparent blue spheres, and 6 transparent red spheres on the centers of the 6 faces. The projection center has 2 cube corners, so there are (8+6) 14 spheres in 3D space of this cube. A second one has larger spheres/circles which might be harder to see.
So the 3D presentation the connection to the cube and octahedron are apparent. The cube vertices are the 8 blue spheres, and the octahedron vertices are the cube face centers. You can also see a relation to the face-centered cubic lattice cell.
Metatron's Cube
13 vertices |
birectified 5-simplex
A5 Coxeter plane
(doubled vertices in orange) |
birectified 5-simplex
A4 Coxeter plane
20 vertices visible |
 |
The 5-dimensional birectified 5-simplex seen in projection of the A5 Coxeter plane has its vertices match this geometry exactly, although it is missing some of the edges.
It has 20 vertices rather than 13, but the center and first ring have double-vertices, coinciding within the projection.
It has a very symmetric Coxeter diagram: 
. It has 90 edges, 120 triangular faces, made of 30 tetrahedra, and 30 octahedra.
It also represents the intersection of 2 5-simplices (like the 3D tetrahedron as a 3-simplex), in dual configurations (
+ 
), like the hexagon center of a hexagram in 2D, and the octahedron () center of a tetrahedron star in 3D (
+ 
).
Also interestingly the birectified 5-simplex has a 10-fold symmetry 2D projection in the A4 Coxeter plane.
I wonder if that means anything helpful to the Metatron cube? I've not seen any evidence anyone else has noticed this similarity, but I'll look around.
Incidentally, the 5-dimensional birectified 5-simplex is the vertex figure for the 6-dimensional 122 polytope, with symmetric Coxeter diagram: 
. It also has a hexagonal projection, although more of a seed of life, containing total 72 vertices, but overlapping with 2,3,6 multipliers (red, orange, yellow), into 19 vertices. It also represents the root vectors of the simple Lie group E6. It is most symmetrically seen in the E6 Coxeter plane with order 12 symmetry.
122
A5 coxeter plane |
A4 coxeter plane |
E6 coxeter plane |
