Runcinated tesseract

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Orthogonal projections in BC₄ Coxeter plane
Tesseract	Runcinated tesseract (Runcinated 16-cell)	16-cell
Runcitruncated tesseract (Runcicantellated 16-cell)	Runcitruncated 16-cell (Runcicantellated tesseract)	Omnitruncated tesseract (Omnitruncated 16-cell)

In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform polychoron, being a runcination (a 3rd order truncation) of the regular tesseract.

There are 4 degrees of runcinations of the tesseract including with permutations truncations and cantellations.

Runcinated tesseract

Runcinated tesseract
Schlegel diagram with 16 tetrahedra
Type	Uniform polychoron
Schläfli symbol	t_0,3{4,3,3}
Coxeter-Dynkin diagrams
Cells	80	16 3.3.3 32 3.4.4 32 4.4.4
Faces	208	64 {3} 144 {4}
Edges	192
Vertices	64
Vertex figure	Equilateral-triangular antipodium
Symmetry group	BC₄, [3,3,4], order 384
Properties	convex
Uniform index	14 15 16

The runcinated tesseract has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.

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.

Cartesian coordinates

The Cartesian coordinates of the vertices of the runcinated tesseract with edge length 2 are all permutations of:

$\left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)$

Images

orthographic projections
Coxeter plane	B₄	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	F₄	A₃
Graph
Dihedral symmetry	[12/3]	[4]

Schlegel diagrams
Wireframe	Wireframe with 16 tetrahedra.	Wireframe with 32 triangular prisms.

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.

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.