Small rhombated tesseract

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Small rhombated tesseract
Schlegel half-solid cantellated 8-cell.png
Bowers style acronymSrit
Coxeter diagramCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells16 octahedra, 32 triangular prisms, 8 small rhombicuboctahedra
Faces64+64 triangles, 24+96 squares
Vertex figureSquare wedge, edge lengths 1 (base square) and 2 (top and side edges) Cantellated 8-cell verf.png
Measures (edge length 1)
Dichoral anglesOct–3–trip: 150°
 Sirco–3–oct: 120°
 Sirco–4–sirco: 90°
Central density1
Number of pieces56
Level of complexity9
Related polytopes
DualNotched enneacontahexachoron
ConjugateQuasirhombated tesseract
Abstract properties
Euler characteristic0
Topological properties
SymmetryB4, order 384

The small rhombated tesseract, or srit, also commonly called the cantellated tesseract, is a convex uniform polychoron that consists of 16 regular octahedra, 32 triangular prisms, and 8 small rhombicuboctahedra. 1 octahedron, 2 triangular prisms, and 2 small rhombicuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantellating the tesseract.

The small rhombated tesseract can be vertex-inscribed into a small prismatotetracontoctachoron and contains the vertices of an octagonal duoprism and the truncated cubic prism.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

The vertices of a small rhombated tesseract of edge length 1 are given by all permutations of:

Representations[edit | edit source]

The small rhombated tesseract has the following Coxeter diagrams:

  • x4o3x3o (full symmetry)
  • xxxx4oxxo3xoox&#xt (BC3 axial, small rhombicuboctahedron-first)
  • oqowxxooo3xxwoqowxx3oooxxwoqo&#xt (A3 axial, octahedron-first)
  • qo3xx3oq *b3oo&#zx (D4 symmetry)
  • wx xx4ox3xo&#zx (BC3×A1 symmetry)
  • oxo4xxw oxo4wxx&#zxt (BC2×BC2 symmetry)

Semi-uniform variant[edit | edit source]

The small rhombated tesseract has a semi-uniform variant of the form x4o3y3o that maintains its full symmetry. This variant uses 16 octahedra of size y, 8 semi-uniform small rhombicuboctahedra of form x4o3y, and 32 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by two small rhombicuboctahedra) and b (of octahedra), its circumradius is given by and its hypervolume is given by .

It has coordinates given by all permutations of:

Related polychora[edit | edit source]

The small rhombated tesseract is the colonel of a 7-member regiment. Its other members include the retrosphenoverted tesseractitesseractihexadecachoron, small rhombic disoctachoron, small pseudorhombic prismatotesseract, grand rhombic prismatotesseract, prismatotesseractintercepted tesseractihexadecachoron, and prismatointercepted prismatotesseractihexadecachoron.

The small rhombated tesseract can be seen as a truncated cubic prism with the bases augmented by small rhombicuboctahedron atop truncated cube segmentochora. The octagonal prisms of the central prism will combine with the square cupolas of the segmentochoral caps to produce small rhombicuboctahedral cells.

Another cap of the small rhombated tesseract is the square pucofastegium. In fact, 8 of these caps, in 2 sets of 4, can be chopped off to give an inscribed octagonal duoprism, with the octagonal prisms formed alternatingly either from middle segments of small rhombicuboctahedra or bases of the removed caps.

The small rhombicuboctahedra of the small rhombated tesseract can be augmented by octahedron atop small rhombicuboctahedron segmentochora. If all eight are augmented, the result is the small prismatotetracontoctachoron.

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

External links[edit | edit source]