There are three truncations, including a bitruncation, and a tritruncation, which creates the truncated 16-cell.
| bgcolor=#e7dcc3 colspan=3 | Truncated tesseract | ||
|---|---|---|---|
| align=center colspan=3 | Schlegel diagram (tetrahedron cells visible) | ||
| Type | Uniform 4-polytope | ||
| Schläfli symbol | t | ||
| Coxeter diagrams | |||
| Cells | 24 | ||
| Faces | 88 | 64 24 | |
| Edges | 128 | ||
| Vertices | 64 | ||
| Vertex figure | v | ||
| Dual | Tetrakis 16-cell | ||
| Symmetry group | B4, [4,3,3], order 384 | ||
| Properties | convex | ||
| Uniform index | 12 13 14 | ||
The truncated tesseract may be constructed by truncating the vertices of the tesseract at
1/(\sqrt{2}+2)
The Cartesian coordinates of the vertices of a truncated tesseract having edge length 2 is given by all permutations of:
\left(\pm1, \pm(1+\sqrt{2}), \pm(1+\sqrt{2}), \pm(1+\sqrt{2})\right)
In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows:
The truncated tesseract, is third in a sequence of truncated hypercubes:
| bgcolor=#e7dcc3 colspan=3 | Bitruncated tesseract | ||
|---|---|---|---|
Two Schlegel diagrams, centered on truncated tetrahedral or truncated octahedral cells, with alternate cell types hidden. | |||
| Type | Uniform 4-polytope | ||
| Schläfli symbol | 2t 2t h2,3 | ||
| Coxeter diagrams | = | ||
| Cells | 24 | ||
| Faces | 120 | 32 24 64 | |
| Edges | 192 | ||
| Vertices | 96 | ||
| Vertex figure | Digonal disphenoid | ||
| Symmetry group | B4, [3,3,4], order 384 D4, [3<sup>1,1,1</sup>], order 192 | ||
| Properties | convex, vertex-transitive | ||
| Uniform index | 15 16 17 | ||
A tesseract is bitruncated by truncating its cells beyond their midpoints, turning the eight cubes into eight truncated octahedra. These still share their square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other.
The Cartesian coordinates of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of:
\left(0, \pm\sqrt{2}, \pm2\sqrt{2}, \pm2\sqrt{2}\right)
The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces.
The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a truncated cubical envelope. Two of the truncated octahedral cells project onto a truncated octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled by 8 flattened truncated tetrahedra, each of which is the image of a pair of truncated tetrahedral cells.
The bitruncated tesseract is second in a sequence of bitruncated hypercubes:
| bgcolor=#e7dcc3 colspan=3 | Truncated 16-cell Cantic tesseract | ||
|---|---|---|---|
Schlegel diagram (octahedron cells visible) | |||
| Type | Uniform 4-polytope | ||
| Schläfli symbol | t t h2 | ||
| Coxeter diagrams | = | ||
| Cells | 24 | ||
| Faces | 96 | 64 32 | |
| Edges | 120 | ||
| Vertices | 48 | ||
| Vertex figure | square pyramid | ||
| Dual | Hexakis tesseract | ||
| Coxeter groups | B4 [3,3,4], order 384 D4 [3<sup>1,1,1</sup>], order 192 | ||
| Properties | convex | ||
| Uniform index | 16 17 18 | ||
It is related to, but not to be confused with, the 24-cell, which is a regular 4-polytope bounded by 24 regular octahedra.
The truncated 16-cell may be constructed from the 16-cell by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures).
(Truncating a 16-cell at 1/2 of the edge length results in the 24-cell, which has a greater degree of symmetry because the truncated cells become identical with the vertex figures.)
The Cartesian coordinates of the vertices of a truncated 16-cell having edge length √2 are given by all permutations, and sign combinations of
(0,0,1,2)
An alternate construction begins with a demitesseract with vertex coordinates (±3,±3,±3,±3), having an even number of each sign, and truncates it to obtain the permutations of
(1,1,3,3), with an even number of each sign.
The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces.
The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:
This layout of cells in projection is analogous to the layout of faces in the projection of the truncated octahedron into 2-dimensional space. Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.
The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:
A truncated 16-cell, as a cantic 4-cube, is related to the dimensional family of cantic n-cubes: