Representations of A4
There is an obvious problem to visualize four dimensional polytopes. However, it is convenient to display projections on a two dimensional plane, the so-called Coxeter plane, where a huge amount of symmetry of the polytope is preserved. To make these projections more illustrative, first the three dimensional analouges are shown. The projections of the representations of A3 reappear as subgraphs in the projections of the representations of A4.
Tetrahedron
Octahedron
CUBOCTAHEDRON
Truncated Tetrahedron
TRuncated Octahedron
Now the projections of the four-dimensional polytopes are shown in the usual way. Additionally, it is stated what kind of three-dimensional boundaries are contained and how many of them meet at one single vertex.
A120
- 120 vertices
- 30 3d-boundaries
- 150 2d-faces
- 240 edges
The 30 boundaries split into:
- 20 hexagonal prims
- 10 truncated octahedra
The 150 faces split into:
- 60 hexagons
- 90 squares
On one vertex, there meet
- two prims and
- two truncated octahedra
A60A
- 60 vertices
- 30 3d-boundaries
- 120 2d-faces
- 150 edges
The 30 3d-boundaries split into:
- 10 triangular prisms
- 10 hexagonal prisms
- 5 truncated tetrahedra
- 5 cuboctahedra
The 120 2d-faces split into:
- 40 triangles
- 60 squares
- 20 hexagons
On one vertex, there meet
- one triangual prism
- two hexagonal prisms
- one cuboctahedron
- one truncated tetrahedron
A60b
- 60 vertices
- 20 3d-boundaries
- 80 2d-faces
- 120 edges
The 20 boundaries split into:
- 10 triangular prisms
- 5 truncated tetrahedra
- 5 truncated octahedra
The 120 2d-faces split into:
- 20 triangles
- 30 squares
- 30 hexagons
On one vertex, there meet
- two truncated octahedron
- one truncated tetrahedron
- one triangular prism
A30
- 30 vertices
- 20 3d-boundaries
- 80 2d-faces
- 90 edges
The 20 3d-boundaries split into:
- 5 octahedra
- 5 cuboctahedra
- 10 triangular prisms
The 80 2d-faces split into:
- 50 triangles
- 30 squares
On one vertex, there meet
- two triangular prisms
- two cuboctahedra
- one octahdron
A30b
- 30 vertices
- 10 3d-boundaries
- 40 2d-faces
- 60 edges
The 10 boundaries are truncated tetrahedra
The 30 faces split into:
- 20 triangles
- 20 hexagons
On one vertex, there meet four truncated tetrahedra
A20a
- 20 vertices
- 30 3d-boundaries
- 70 2d-faces
- 60 edges
The 30 boundaries split into:
- 10 tetrahedra
- 20 triangular prisms
The 150 faces split into:
- 40 triangles
- 30 squares
On one vertex, there meet
- two tetrahedra and
- six prisms
A20b
- 20 vertices
- 10 3d-boundaries
- 30 2d-faces
- 40 edges
The 10 boundaries split into:
- 5 tetrahedra
- 5 truncated tetrahedra
The 30 2d-faces split into:
- 20 triangles
- 10 hexagons
On one vertex, there meet
- one tetrahedron
- three truncated tetrahedra
A10
- 10 vertices
- 10 3d-boundaries
- 30 2d-faces
- 30 edges
The 10 boundaries split into:
- 5 tetrahedra
- 5 octahedra
The 30 faces are triangles
On one vertex, there meet
- two tetrahedra
- three octahedra
A5
- 5 vertices
- 5 3d-boundaries
- 10 2d-faces
- 10 edges
The 5 boundaries tetrahedra
The 10 faces are triangles
On one vertex, there meet four tetrahedra
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