Great disprismatotesseractihexadecachoron

Great disprismatotesseracti-hexadecachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gidpith
Coxeter diagram
Elements
Cells	32 hexagonal prisms, 24 octagonal prisms, 16 truncated octahedra, 8 great rhombicuboctahedra
Faces	96+96+96 squares, 64+64 hexagons, 48 octagons
Edges	192+192+192+192
Vertices	384
Vertex figure	Irregular tetrahedron, edge lengths √2 (3), √3 (2), and √2+√2 (1)
Measures (edge length 1)
Circumradius	$\sqrt{8+3\sqrt2} ≈ 3.49895$
Hypervolume	$2(131+92\sqrt2) ≈ 522.21530$
Dichoral angles	Toe–6–hip: 150°
	Op–4–hip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Toe–4–op: 135°
	Girco–8–op: 135°
	Girco–4–hip: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$
	Girco–6–toe: 120°
Central density	1
Number of pieces	80
Level of complexity	24
Related polytopes
Army	Gidpith
Regiment	Gidpith
Dual	Tetrahedral triacosioctacontatetrachoron
Conjugate	Great quasidisprismatotesseracti-hexadecachoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B4, order 384
Convex	Yes
Nature	Tame

The great disprismatotesseractihexadecachoron, or gidpith, also commonly called the omnitruncated tesseract or omnitruncated hexadecachoron, is a convex uniform polychoron that consists of 32 hexagonal prisms, 24 octagonal prisms, 16 truncated octahedra, and 8 great rhombicuboctahedra. 1 of each type of cell join at each vertex. It is the omnitruncate of the BC₄ family of uniform polychora.

The great disprismatotesseractihexadecachoron can be vertex-inscribed into a prismatorhombated icositetrachoron.

This polychoron can be alternated into a snub tesseract, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pyritosnub tesseract, which is also nonuniform. If the octagons are alternated alternately, then the resulting polychoron is a truncated icositetrachoron.

Gallery

• 

  B4 Coxeter plane projection

• 

  Stereographic projection

• 

  Net

• 

  Cross-sections

• 

  Cross-section animation

Vertex coordinates

Vertex coordinates for a great disprismatotesseractihexadecachoron of edge length 1 are given by all permutations of:

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

Representations

A great disprismatotesseractihexadecachoron has the following Coxeter diagrams:

• x4x3x3x (full symmetry)
• xxxwwxxx4xxuxxuxx3xuxxxxux&#xt (BC3 axial, great rhombicuboctahedron-first)
• wx3xx3xw *b3xx&#zx (D₄ symmetry)

Semi-uniform variant

The great disprismatotesseractihexadecachoron has a semi-uniform variant of the form a4b3c3d that maintains its full symmetry. This variant uses 8 great rhombicuboctahedra of form a4b3c, 16 great rhombitetratetrahedra of form b3c3d, 24 ditetragonal prisms of form d a4b, and 32 ditrigonal prisms of form a c3d as cells, with 4 edge lengths.

With edges of length a, b, c, and d (such that it forms a4b3c3d), its circumradius is given by $\sqrt{\frac{2a^2+3b^2+2c^2+d^2+4bc+2bd+2cd+(3ab+2ac+ad)\sqrt2}{2} }$.

It has coordinates given by all permutations of:

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

Related polychora

A great disprismatotesseractihexadecachoron can be augmented by joining truncated octahedron atop great rhombicuboctahedron segmentochora to its great rhombicuboctahedral cells. If all eight great rhombicuboctahedra are augmented, the result is the prismatorhombated icositetrachoron, as the octagonal prisms merge with square cupolas coming from the augments.

External links

• Bowers, Jonathan. "Category 9: Omnitruncates" (#331).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "gidpith".

• Quickfur. "The Omnitruncated Tesseract".

• Wikipedia Contributors. "Omnitruncated tesseract".