Cantellated tesseract

Four cantellations
Orthogonal projections in A₄ Coxeter plane
tesseract	Cantellated tesseract	Cantellated 16-cell (Rectified 24-cell)
16-cell	Cantitruncated tesseract	Cantitruncated 16-cell (Truncated 24-cell)

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} [math]\displaystyle{ r\begin{Bmatrix} 4 \\ 3 , 3 \end{Bmatrix} }[/math]
Coxeter diagram
Cells	56	8 3.4.4.4 16 3.3.3.3 20px 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:

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right) }[/math]

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:

The projection envelope is a truncated cube.
The nearest and farthest small rhombicuboctahedral cells from the 4D viewpoint project to the volume of the same shape inscribed in the projection envelope.
The axial squares of this central small rhombicuboctahedron touches the centers of the 6 octagons of the envelope. The octagons are the image of the other 6 small rhombicuboctahedral cells.
The 12 wedge-shaped volumes connecting the non-axial square faces of the central small rhombicuboctahedron to the neighbouring octagons are the images of 24 of the triangular prisms.
The remaining 8 triangular prisms project onto the triangular faces of the envelope.
Between the triangular faces of the envelope and the triangular faces of the central small rhombicuboctahedron are 8 octahedral volumes, which are the images of the 16 octahedral cells.

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} [math]\displaystyle{ t\begin{Bmatrix} 4 \\ 3 , 3 \end{Bmatrix} }[/math]
Coxeter diagrams
Cells	56	8 4.6.8 16 3.6.6 20px 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.