Rank4
TypeUniform
Notation
Bowers style acronymGidpith
Coxeter diagramx4x3x3x ()
Elements
Cells32 hexagonal prisms, 24 octagonal prisms, 16 truncated octahedra, 8 great rhombicuboctahedra
Faces96+96+96 squares, 64+64 hexagons, 48 octagons
Edges192+192+192+192
Vertices384
Vertex figureIrregular tetrahedron, edge lengths 2 (3), 3 (2), and 2+2 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {8+3{\sqrt {2}}}}\approx 3.49895}$
Hypervolume${\displaystyle 2(131+92{\sqrt {2}})\approx 522.21530}$
Dichoral anglesToe–6–hip: 150°
Op–4–hip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Toe–4–op: 135°
Girco–8–op: 135°
Girco–4–hip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Girco–6–toe: 120°
Central density1
Number of external pieces80
Level of complexity24
Related polytopes
ArmyGidpith
RegimentGidpith
DualTetrahedral triacosioctacontatetrachoron
Abstract & topological properties
Flag count9216
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
ConvexYes
NatureTame

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 B4 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.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+3{\sqrt {2}}}{2}}\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A great disprismatotesseractihexadecachoron has the following Coxeter diagrams:

• x4x3x3x () (full symmetry)
• xxxwwxxx4xxuxxuxx3xuxxxxux&#xt (B3 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 ${\displaystyle {\sqrt {\frac {2a^{2}+3b^{2}+2c^{2}+d^{2}+4bc+2bd+2cd+(3ab+2ac+ad){\sqrt {2}}}{2}}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {a+(b+c+d){\sqrt {2}}}{2}},\,\pm {\frac {a+(b+c){\sqrt {2}}}{2}},\,\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a}{2}}\right)}$.

## 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.