# Small rhombated tesseract

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Small rhombated tesseract
Rank4
TypeUniform
Notation
Bowers style acronymSrit
Coxeter diagramx4o3x3o ()
Elements
Cells16 octahedra, 32 triangular prisms, 8 small rhombicuboctahedra
Faces64+64 triangles, 24+96 squares
Edges96+192
Vertices96
Vertex figureSquare wedge, edge lengths 1 (base square) and 2 (top and side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2+{\sqrt {2}}}}\approx 1.84776}$
Hypervolume${\displaystyle {\frac {45+32{\sqrt {2}}}{3}}\approx 30.08494}$
Dichoral anglesOct–3–trip: 150°
Sirco–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Sirco–3–oct: 120°
Sirco–4–sirco: 90°
Central density1
Number of external pieces56
Level of complexity9
Related polytopes
ArmySrit
RegimentSrit
DualNotched enneacontahexachoron
ConjugateQuasirhombated tesseract
Abstract & topological properties
Flag count3456
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
Flag orbits9
ConvexYes
NatureTame

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.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

The small rhombated tesseract has the following Coxeter diagrams:

• x4o3x3o () (full symmetry)
• xxxx4oxxo3xoox&#xt (B3 axial, small rhombicuboctahedron-first)
• oqowxxooo3xxwoqowxx3oooxxwoqo&#xt (A3 axial, octahedron-first)
• qo3xx3oq *b3oo&#zx (D4 symmetry)
• wx xx4ox3xo&#zx (B3×A1 symmetry)
• oxo4xxw oxo4wxx&#zxt (B2×B2 symmetry)

## Semi-uniform variant

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 ${\displaystyle {\sqrt {a^{2}+b^{2}+ab{\sqrt {2}}}}}$ and its hypervolume is given by ${\displaystyle {\frac {3a^{4}+36a^{2}b^{2}+6b^{4}+(12a^{3}b+20ab^{3}){\sqrt {2}}}{3}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a}{2}},\,\pm {\frac {a}{2}}\right)}$.

## Related polychora

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.

Three small rhombated tesseracts can be blended at their octahedra to form the inverted prismatoicositetrachoron.