Great prismatodecachoron

Great prismatodecachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gippid
Coxeter diagram	x3x3x3x ()
Elements
Cells	20 hexagonal prisms, 10 truncated octahedra
Faces	30+60 squares, 20+40 hexagons
Edges	120+120
Vertices	120
Vertex figure	Phyllic disphenoid, edge lengths √2 (3) and √3 (3)
Measures (edge length 1)
Circumradius	$\sqrt5 ≈ 2.23607$
Hypervolume	$\frac{125\sqrt5}{4} ≈ 69.87712$
Dichoral angles	Hip–4–hip: $\arccos\left(-\tfrac23\right) ≈ 131.81032^\circ$
	Toe–6–hip: $\arccos\left(-\tfrac{\sqrt6}{4}\right) ≈ 127.76124^\circ$
	Toe–4–hip: $\arccos\left(-\tfrac{\sqrt6}{6}\right) ≈ 114.09484^\circ$
	Toe–6–toe: $\arccos\left(-\tfrac14\right) ≈ 104.47751^\circ$
Central density	1
Number of external pieces	30
Level of complexity	12
Related polytopes
Army	Gippid
Regiment	Gippid
Dual	Disphenoidal hecatonicosachoron
Conjugate	None
Abstract & topological properties
Flag count	2880
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

The great prismatodecachoron, or gippid, also commonly called the omnitruncated 5-cell or omnitruncated pentachoron, is a convex uniform polychoron that consists of 20 hexagonal prisms and 10 truncated octahedra. 2 hexagonal prisms and 2 truncated octahedra join at each vertex. It is the omnitruncate of the A₄ family of uniform polychora.

This polychoron can be alternated into a snub decachoron, although it cannot be made uniform.

Like the omnitruncated simplex of any dimension, this polychoron can tile 4D space. The resulting tetracomb is called the omnitruncated cyclopentachoric tetracomb.

It is the 5th-order permutohedron.

Gallery

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  Cross-sections

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  Cross-section animation

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  Net

Vertex coordinates

The vertices of a great prismatodecachoron of edge length 1 are given by the following points:

• $±\left(0,\,\tfrac{\sqrt6}{3},\,-\tfrac{\sqrt3}{3},\,±2\right),$
• $±\left(0,\,\tfrac{\sqrt6}{3},\,-\tfrac{5\sqrt3}{6},\,±\tfrac32\right),$
• $±\left(0,\,\tfrac{\sqrt6}{3},\,\tfrac{7\sqrt3}{6},\,±\tfrac12\right),$
• $±\left(0,\,\tfrac{2\sqrt6}{3},\,-\tfrac{\sqrt3}{6},\,±\tfrac32\right),$
• $±\left(0,\,\tfrac{2\sqrt6}{3},\,-\tfrac{2\sqrt3}{3},\,±1\right),$
• $±\left(0,\,\tfrac{2\sqrt6}{3},\,\tfrac{5\sqrt3}{6},\,±\tfrac12\right),$
• $±\left(±\tfrac{\sqrt{10}}{2},\,\tfrac{\sqrt6}{6},\,-\tfrac{\sqrt3}{6},\,±\tfrac32\right),$
• $±\left(±\tfrac{\sqrt{10}}{2},\,\tfrac{\sqrt6}{6},\,-\tfrac{2\sqrt3}{3},\,±1\right),$
• $±\left(±\tfrac{\sqrt{10}}{2},\,\tfrac{\sqrt6}{6},\,\tfrac{5\sqrt3}{6},\,±\tfrac12\right),$
• $±\left(±\tfrac{\sqrt{10}}{2},\,±\tfrac{\sqrt6}{2},\,0,\,±1\right),$
• $\left(±\tfrac{\sqrt{10}}{2},\,±\tfrac{\sqrt6}{2},\,±\tfrac{\sqrt3}{2},\,±\tfrac12\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,\tfrac{\sqrt6}{12},\,-\tfrac{\sqrt3}{3},\,±2\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,\tfrac{\sqrt6}{12},\,-\tfrac{5\sqrt3}{6},\,±\tfrac32\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,\tfrac{\sqrt6}{12},\,\tfrac{7\sqrt3}{6},\,±\tfrac12\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,-\tfrac{\sqrt6}{4},\,0,\,±2\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,-\tfrac{\sqrt6}{4},\,±\sqrt3,\,±1\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,-\tfrac{7\sqrt6}{12},\,-\tfrac{\sqrt3}{6},\,±\tfrac32\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,-\tfrac{7\sqrt6}{12},\,-\tfrac{2\sqrt3}{3},\,±1\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,-\tfrac{7\sqrt6}{12},\,\tfrac{5\sqrt3}{6},\,±\tfrac12\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,\tfrac{3\sqrt6}{4},\,0,\,±1\right),$
• $±\left(\tfrac{\sqrt{10}}{4},\,\tfrac{3\sqrt6}{4},\,±\tfrac{\sqrt3}{2},\,±\frac12\right).$

Much simpler coordinates can be given in five dimensions, as all permutations of:

• $\left(2\sqrt2,\,\tfrac{3\sqrt2}{2},\,\sqrt2,\,\tfrac{\sqrt2}{2},\,0\right).$

Representations

A great prismatodecachoron has the following Coxeter diagrams:

• x3x3x3x (full symmetry)
• xxxux3xxuxx3xuxxx&#xt (A₃ axial, truncated octahedron-first)

Semi-uniform variant

The great prismatodecachoron has a semi-uniform variant of the form x3y3y3x that maintains its full symmetry. This variant uses 10 great rhombitetratetrahedra of form x3y3y and 20 ditrigonal prisms of form x x3y as cells, with 2 edge lengths.

With edges of length a and b (so that it is represented by a3b3b3a), its circumradius is given by $\sqrt{a^2+2b^2+2ab}$.

If it has only single pentachoric symmetry, the variant is called a great disprismatopentapentachoron.

Related polychora

The antifrustary prismatohexacosichoron is a uniform polychoron compound composed of 60 great prismatodecachora.

External links

• Bowers, Jonathan. "Category 9: Omnitruncates" (#330).

• Bowers, Jonathan. "Pennic and Decaic Isogonals".

• Klitzing, Richard. "gippid".

• Quickfur. "The Omnitruncated 5-cell".

• Wikipedia Contributors. "Omnitruncated 5-cell".