Prismatorhombated icositetrachoron

Prismatorhombated icositetrachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Prico
Coxeter diagram	x3x4o3x ()
Elements
Cells	96 triangular prisms, 96 hexagonal prisms, 24 truncated octahedra, 24 small rhombicuboctahedra
Faces	192 triangles, 144+288+288 squares, 192 hexagons
Edges	288+576+576
Vertices	576
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths 1, √2, √2, √2; lateral edge lengths √3, √3, √2, √2
Measures (edge length 1)
Circumradius	$\sqrt{8+3\sqrt2} ≈ 3.49895$
Hypervolume	$305+224\sqrt2 ≈ 621.78384$
Dichoral angles	Hip–4–trip: $\arccos\left(-\frac{2\sqrt2}{3}\right) ≈ 160.52878°$
	Toe–6–hip: 150°
	Sirco–3–trip: 150°
	Sirco–4–hip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Sirco–4–toe: 135°
Central density	1
Number of pieces	240
Level of complexity	16
Related polytopes
Army	Prico
Regiment	Prico
Dual	Deltopyramidal pentacosiheptacontahexachoron
Conjugate	Prismatoquasirhombated icositetrachoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	F4, order 1152
Convex	Yes
Nature	Tame

The prismatorhombated icositetrachoron, or prico, also commonly called the runcitruncated 24-cell, is a convex uniform polychoron that consists of 96 triangular prisms, 96 hexagonal prisms, 24 truncated octahedra, and 24 small rhombicuboctahedra. 1 triangular prism, 2 hexagonal prisms, 1 truncated octahedron, and 1 small rhombicuboctahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the icositetrachoron.

The truncated octahedra can be alternated into icosahedra to create the scaliform prismatorhombisnub icositetrachoron.

Cross-sections

Vertex coordinates

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

• $\left(±\frac{2+3\sqrt2}{2},\,±\sqrt2,\,±\frac{\sqrt2}{2},\,0\right),$
• $\left(±\frac{1+3\sqrt2}{2},\,±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$

The second set of vertices is identical to the vertices of an inscribed great disprismatotesseractihexadecachoron.

The runcitruncation of the dual icositetrachoron has vertices given by all permutations of:

• $\left(±\frac{5+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right),$
• $\left(±\frac{3+\sqrt2}{2},\,±\frac{3+\sqrt2}{2},\,±\frac32,\,±\frac12\right),$
• $\left(±\frac{4+\sqrt2}{2},\,±\frac{2+\sqrt2}{2},\,±1,\,±1\right).$

Representations

A prismatorhombated icositetrachoron has the following Coxeter diagrams:

• x3x4o3x (full symmetry)
• s3s4x3x (as runcicantic snub icositetrachoron)
• ox4xx3xx3wx&#zx (BC4 symmetry)
• wxx3xxx3xwx *b3xxw&#zx (D4 symmetry)
• xux4qoo3xxx3oqQ&#zx (BC4 symmetry, dual ico)

Semi-uniform variant

The prismatorhombated icositetrachoron has a semi-uniform variant of the form x3y4o3z that maintains its full symmetry. This variant uses 24 truncated octahedra of form o4y3x, 24 small rhombicuboctahedra of form y4o3z, 96 triangular prisms of form x z3o, and 96 ditrigonal prisms of form z x3y as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a3b4o3c), its circumradius is given by $\sqrt{a^2+3b^2+c^2+3ab+(ac+2bc)\sqrt2}$.

Variations

Besides the semi-uniform variant, another isogonal variant known as the runcicantic snub icositetrachoron also exists, where the small rhombicuboctahedra have pyritohedral symmetry, the truncated octahedra have tetrahedral symmetry, and the hexagonal prisms have pyramidal symmetry only.

Related polychora

The prismatorhombated icositetrachoron is the colonel of a 3-member regiment that also includes the small prismatotrisicositetrachoron and small rhombiprismic disicositetrachoron.

The segmentochoron truncated octahedron atop great rhombicuboctahedron can be obtained as a cap of the prismatorhombated icositetrachoron. If eight of these caps are removed the result is the great disprismatotesseractihexadecachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#361).

• Klitzing, Richard. "prico".

• Quickfur. "The Runcitruncated 24-cell".

• Wikipedia Contributors. "Runcitruncated 24-cell".