Runcinated tesseract

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Runcinated tesseract
Schlegel diagram with 16 tetrahedra
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
Symmetry group	[3,3,4]
Schläfli symbol	t_0,3{4,3,3}
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.

1 Construction
2 Images
3 Structure
4 Projections
5 See also
6 External links

[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

Schlegel diagrams:

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.