Prismatorhombated tesseract

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Prismatorhombated tesseract
Schlegel half-solid runcitruncated 16-cell.png
Bowers style acronymPrit
Coxeter diagramCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, 8 small rhombicuboctahedra
Faces64 triangles, 48+96+96 squares, 64 hexagons
Vertex figureIsosceles trapezoidal pyramid, base edge lengths 1, 2, 2, 2; lateral edge lengths 2, 2, 3, 3
Measures (edge length 1)
Dichoral anglesTut–6–hip: 150°
 Sirco–4–cube: 135°
 Sirco–3–tut: 120°
Central density1
Number of pieces80
Level of complexity16
Related polytopes
DualDeltopyramidal hecatonenneacontadichoron
ConjugatePrismatoquasirhombated tesseract
Abstract properties
Euler characteristic0
Topological properties
SymmetryB4, order 384

The prismatorhombated tesseract, or prit, also commonly called the runcitruncated 16-cell, is a convex uniform polychoron that consists of 24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, and 8 small rhombicuboctahedra. 1 cube, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 small rhombicuboctahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hexadecachoron.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

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

Representations[edit | edit source]

A prismatorhombated tesseract has the following Coxeter diagrams:

  • x4o3x3x (full symmetry)
  • xxxxxx4ooxxoo3xuxxux&#xt (BC3 axial, small rhombicuboctahedron-first)
  • qo3xx3oq *b3xx&#zx (D4 symmetry)
  • Xwx xxx4oox3xux&#zx (BC3×A1 symmetry)

Semi-uniform variant[edit | edit source]

The prismatorhombated tesseract has a semi-uniform variant of the form x4o3y3z that maintains its full symmetry. This variant uses 8 small rhombicuboctahedra of form x4o3y, 16 truncated tetrahedra of form z3y3o, 24 square prisms of form z x4o, and 32 ditrigonal prisms of form x y3z as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4o3b3c), its circumradius is given by .

It has coordinates given by all permutations of:

Related polychora[edit | edit source]

The prismatorhombated tesseract is the colonel of a 3-member regiment that also includes the small cubihexadecadisoctachoron and the small rhombiprismic disoctachoron.

The great rhombicuboctahedral prism can be obtained as the central segment of the prismatorhombated tesseract in small rhombicuboctahedron-first orientation.

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]