Small rhombated tesseract

Small rhombated tesseract
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Srit
Coxeter diagram
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	$\sqrt{2+\sqrt2} ≈ 1.84776$
Hypervolume	$\frac{45+32\sqrt2}{3} ≈ 30.08494$
Dichoral angles	Oct–3–trip: 150°
	Sirco–4–trip: $\arccos\left(-\frac{\sqrt33}{3}\right) ≈ 125.26439°$
	Sirco–3–oct: 120°
	Sirco–4–sirco: 90°
Central density	1
Number of pieces	56
Level of complexity	9
Related polytopes
Army	Srit
Regiment	Srit
Dual	Notched enneacontahexachoron
Conjugate	Quasirhombated tesseract
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B4, 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 vertex-inscribed into a small prismatotetracontoctachoron and contains the vertices of an octagonal duoprism and the truncated cubic prism.

Gallery

• 

  B4 Coxeter plane projection

• 

  Net

• 

  Cross-sections

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  Cross-section animation

Vertex coordinates

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

• $\left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right).$

Representations

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

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 $\sqrt{a^2+b^2+ab\sqrt2}$ and its hypervolume is given by $\frac{3a^4+36a^2b^2+6b^4+(12a^3b+20ab^3)\sqrt2}{3}$.

It has coordinates given by all permutations of:

• $\left(±\frac{a+b\sqrt2}{2},\,±\frac{a+b\sqrt2}{2},\,±\frac{a}{2},\,±\frac{a}{2}\right).$

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.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#140).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "srit".

• Quickfur. "The Cantellated Tesseract".

• Wikipedia Contributors. "Cantellated tesseract".