Cantellated tesseract

Four cantellations

tesseract
Cantellated tesseract
Cantellated 16-cell
(Rectified 24-cell)

16-cell
Cantitruncated tesseract
Cantitruncated 16-cell
(Truncated 24-cell)
Orthogonal projections in A4 Coxeter plane

In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular tesseract.

There are four degrees of cantellations of the tesseract including with permutations truncations. Two are also derived from the 24-cell family.

Cantellated tesseract

Cantellated tesseract

Schlegel diagram
Centered on rhombicuboctahedron
octahedral cells shown
Type Uniform 4-polytope
Schläfli symbol rr{4,3,3}
Coxeter diagram
Cells 56 8 3.4.4.4
16 3.3.3.3
32 3.4.4
Faces 248 128 {3}
120 {4}
Edges 288
Vertices 96
Vertex figure
Square wedge
Symmetry group B4, [3,3,4], order 384
Properties convex
Uniform index 13 14 15
Net

The cantellated tesseract, bicantellated 16-cell, or small rhombated tesseract is a convex uniform 4-polytope or 4-dimensional polytope bounded by 56 cells: 8 small rhombicuboctahedra, 16 octahedra, and 32 triangular prisms.

Construction

In the process of cantellation, a polytope's 2-faces are effectively shrunk. The rhombicuboctahedron can be called a cantellated cube, since if its six faces are shrunk in their respective planes, each vertex will separate into the three vertices of the rhombicuboctahedron's triangles, and each edge will separate into two of the opposite edges of the rhombicuboctahedrons twelve non-axial squares.

When the same process is applied to the tesseract, each of the eight cubes becomes a rhombicuboctahedron in the described way. In addition however, since each cube's edge was previously shared with two other cubes, the separating edges form the three parallel edges of a triangular prism—32 triangular prisms, since there were 32 edges. Further, since each vertex was previously shared with three other cubes, the vertex would split into 12 rather than three new vertices. However, since some of the shrunken faces continues to be shared, certain pairs of these 12 potential vertices are identical to each other, and therefore only 6 new vertices are created from each original vertex (hence the cantellated tesseract's 96 vertices compared to the tesseract's 16). These six new vertices form the vertices of an octahedron—16 octahedra, since the tesseract had 16 vertices.

Cartesian coordinates

The Cartesian coordinates of the vertices of a cantellated tesseract with edge length 2 is given by all permutations of:

\left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)

Structure

The 8 small rhombicuboctahedral cells are joined to each other via their axial square faces. Their non-axial square faces, which correspond with the edges of a cube, are connected to the triangular prisms. The triangular faces of the small rhombicuboctahedra and the triangular prisms are connected to the 16 octahedra.

Its structure can be imagined by means of the tesseract itself: the rhombicuboctahedra are analogous to the tesseract's cells, the triangular prisms are analogous to the tesseract's edges, and the octahedra are analogous to the tesseract's vertices.

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]

Wireframe
16 octahedra shown.
32 triangular prisms shown.

Projections

The following is the layout of the cantellated tesseract's cells under the parallel projection into 3-dimensional space, small rhombicuboctahedron first:

This layout of cells in projection is analogous to the layout of faces in the projection of the truncated cube into 2 dimensions. Hence, the cantellated tesseract may be thought of as an analogue of the truncated cube in 4 dimensions. (It is not the only possible analogue; another close candidate is the truncated tesseract.)

Another uniform 4-polytope with a similar layout of cells is the runcitruncated 16-cell.

Cantitruncated tesseract

Cantitruncated tesseract

Schlegel diagram centered on truncated cuboctahedron cell with octagonal faces hidden.
Type Uniform 4-polytope
Schläfli symbol tr{4,3,3}
Coxeter diagrams
Cells 56 8 4.6.8
16 3.6.6
32 3.4.4
Faces 248 64 {3}, 96 {4}
64 {6}, 24 {8}
Edges 384
Vertices 192
Vertex figure
Sphenoid
Symmetry group B4, [3,3,4], order 384
Properties convex
Uniform index 17 18 19
Net

In geometry, the cantitruncated tesseract or great rhombated tesseract is a uniform 4-polytope (or uniform 4-dimensional polytope) that is bounded by 56 cells: 8 truncated cuboctahedra, 16 truncated tetrahedra, and 32 triangular prisms.

Construction

The cantitruncated tesseract is constructed by the cantitruncation of the tesseract. Cantitruncation is often thought of as rectification followed by truncation. However, the result of this construction would be a polytope which, while its structure would be very similar to that given by cantitruncation, not all of its faces would be uniform.

Alternatively, a uniform cantitruncated tesseract may be constructed by placing 8 uniform truncated cuboctahedra in the hyperplanes of a tesseract's cells, displaced along the coordinate axes such that their octagonal faces coincide. For an edge length of 2, this construction gives the Cartesian coordinates of its vertices as all permutations of:

\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right)

Structure

The 8 truncated cuboctahedra are joined to each other via their octagonal faces, in an arrangement corresponding to the 8 cubical cells of the tesseract. They are joined to the 16 truncated tetrahedra via their hexagonal faces, and their square faces are joined to the square faces of the 32 triangular prisms. The triangular faces of the triangular prisms are joined to the truncated tetrahedra.

The truncated tetrahedra correspond with the tesseract's vertices, and the triangular prisms correspond with the tesseract's edges.

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]

A stereographic projection of the cantellated tesseract, as a tiling on a 3-sphere, with its 64 blue triangles, 96 green squares and 64 red hexagonal faces (the octagonal faces are not drawn).

Projections

In the truncated cuboctahedron first parallel projection into 3 dimensions, the cells of the cantitruncated tesseract are laid out as follows:

This layout of cells in projection is similar to that of the cantellated tesseract.

Alternative names

Related uniform polytopes

References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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