Small rhombated tesseract
Small rhombated tesseract  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Srit 
Coxeter diagram  x4o3x3o () 
Elements  
Cells  16 octahedra, 32 triangular prisms, 8 small rhombicuboctahedra 
Faces  64+64 triangles, 24+96 squares 
Edges  96+192 
Vertices  96 
Vertex figure  Square wedge, edge lengths 1 (base square) and √2 (top and side edges) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Oct–3–trip: 150° 
Sirco–4–trip:  
Sirco–3–oct: 120°  
Sirco–4–sirco: 90°  
Central density  1 
Number of external pieces  56 
Level of complexity  9 
Related polytopes  
Army  Srit 
Regiment  Srit 
Dual  Notched enneacontahexachoron 
Conjugate  Quasirhombated tesseract 
Abstract & topological properties  
Flag count  3456 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Convex  Yes 
Nature  Tame 
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 vertexinscribed into a small prismatotetracontoctachoron and contains the vertices of an octagonal duoprism and the truncated cubic prism.
Gallery[edit  edit source]

B4 Coxeter plane projection

Net

Crosssection animation
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 (B_{3} axial, small rhombicuboctahedronfirst)
 oqowxxooo3xxwoqowxx3oooxxwoqo&#xt (A_{3} axial, octahedronfirst)
 qo3xx3oq *b3oo&#zx (D_{4} symmetry)
 wx xx4ox3xo&#zx (B_{3}×A_{1} symmetry)
 oxo4xxw oxo4wxx&#zxt (B_{2}×B_{2} symmetry)
Semiuniform variant[edit  edit source]
The small rhombated tesseract has a semiuniform variant of the form x4o3y3o that maintains its full symmetry. This variant uses 16 octahedra of size y, 8 semiuniform 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 7member 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.
Three small rhombated tesseracts can be blended at their octahedra to form the inverted prismatoicositetrachoron.
External links[edit  edit source]
 Bowers, Jonathan. "Category 6: Sphenoverts" (#140).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "srit".
 Quickfur. "The Cantellated Tesseract".
 Wikipedia contributors. "Cantellated tesseract".