Runcinated tesseract
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| Runcinated tesseract | |
|---|---|
 Schlegel diagram with 16 tetrahedra |
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| Type | Uniform polychoron |
| Cells | 16 3.3.3  32 3.4.4  32 4.4.4  |
| Faces | 64 {3} 144 {4} |
| Edges | 192 |
| Vertices | 64 |
| Vertex configuration | Equilateral-triangular antipodium |
| Schläfli symbol | t0,3{4,3,3} |
| Coxeter-Dynkin diagrams |     |
| Symmetry group | [3,3,4] |
| Properties | convex |
In geometry, the runcinated tesseract or runcinated 16-cell is a 4-dimensional convex uniform polytope (or polychoron) made of 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
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[edit] Construction
The runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra (vertex figures), cubes (face prisms), and triangular prisms (edge figures). The same process applied to a 16-cell also yields the same figure.
[edit] Images
 Wireframe |
 Wireframe with 16 tetrahedra. |
 Wireframe with 32 triangular prisms. |
[edit] Structure
Eight of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces.
[edit] Projections
The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a (small) rhombicuboctahedral envelope. The images of its cells are laid out within this envelope as follows:
- The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope.
- Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron. These are the images of 12 of the cubical cells (each pair of cubes share an image).
- The 18 square faces of the envelope are the images of the other cubical cells.
- The 12 wedge-shaped volumes connecting the edges of the central cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms (a pair of cells per image).
- The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms.
- Finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra (again, a pair of cells per image).
This layout of cells in projection is analogous to the layout of the faces of the (small) rhombicuboctahedron under projection to 2 dimensions. The rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron.
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