Great prismatotetracontoctachoron

Great prismatotetracontoctachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gippic
Coxeter diagram	x3x4x3x ()
Elements
Cells	192 hexagonal prisms, 48 great rhombicuboctahedra
Faces	288+576 squares, 384 hexagons, 144 octagons
Edges	1152+1152
Vertices	1152
Vertex figure	Phyllic disphenoid, edge lengths √2 (3), √3 (2), and √2+√2 (1)
Measures (edge length 1)
Circumradius	$\sqrt{14+9\sqrt2} ≈ 5.16991$
Hypervolume	$24(62+43\sqrt2) ≈ 2947.46840$
Dichoral angles	Hip–4–hip: $\arccos\left(-\frac{2\sqrt2}{3}\right) ≈ 160.52878°$
	Girco–6–hip: 150°
	Girco–4–hip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Girco–8–girco: 135°
Central density	1
Number of pieces	240
Level of complexity	12
Related polytopes
Army	Gippic
Regiment	Gippic
Dual	Disphenoidal chiliahecatonpentaconta-dichoron
Conjugate	Great quasiprismatotetracontocta-choron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The great prismatotetracontoctachoron, or gippic, also commonly called the omnitruncated 24-cell, is a convex uniform polychoron that consists of 192 hexagonal prisms and 48 great rhombicuboctahedra. 2 hexagonal prisms and 2 great rhombicuboctahedra join at each vertex. It is the omnitruncate of the F₄ family of uniform polychora.

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

Cross-sections

Vertex coordinates

The vertices of a great prismatotetracontoctachoron of edge length 1 are given by all permutations of:

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

Representations

A great prismatotetracontoctachoron has the following Coxeter diagrams:

• x3x4x3x (full symmetry)
• xux4wxx3xxx3xwX&#zx (BC₄ symmetry)

Semi-uniform variant

The great prismatotetracontoctachoron has a semi-uniform variant of the form x3y4y3x that maintains its full symmetry. This variant uses 48 great rhombicuboctahedra of form y4y3x and 192 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 a3b4b3a), its circumradius is given by $\sqrt{2a^2+6b^2+6ab+(a^2+4b^2+4ab)\sqrt2}$.

If it has only single icositetrachoric symmetry, the variant is called a great disprismatoicositetricositetrachoron.

External links

• Bowers, Jonathan. "Category 9: Omnitruncates" (#332).

• Klitzing, Richard. "gippic".

• Quickfur. "The Omnitruncated 24-cell".

• Wikipedia Contributors. "Omnitruncated 24-cell".