# Great hexacosihecatonicosachoron

Great hexacosihecatonicosachoron
Rank4
TypeUniform
SpaceSpherical
Bowers style acronymGixhi
Coxeter diagramo5/2x3x3o ()
Elements
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), (5–1)/2 (base 2) and 3 (sides)
Cells600 truncated tetrahedra, 120 truncated great icosahedra
Faces1200 triangles, 720 pentagrams, 2400 hexagons
Edges3600+3600
Vertices3600
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{59-25\sqrt5}{2}} ≈ 1.24465}$
Hypervolume${\displaystyle 5\frac{2726\sqrt5-5845}{4} ≈ 313.15163}$
Dichoral anglesTiggy–6–tut: ${\displaystyle \arccos\left(-\frac{\sqrt{7-3\sqrt5}}{4}\right) ≈ 97.76124^\circ}$
Tiggy–5/2–tiggy: 72°
Tut–3–tut: ${\displaystyle \arccos\left(\frac{3\sqrt5-1}{8}\right) ≈ 44.47751^\circ}$
Central density191
Euler characteristic0
Number of pieces122640
Level of complexity494
Related polytopes
ArmySemi-uniform Srix
RegimentGixhi
ConjugateHexacosihecatonicosachoron
Convex coreHexacosichoron
Topological properties
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The great hexacosihecatonicosachoron, or gixhi, is a nonconvex uniform polychoron that consists of 600 truncated tetrahedra and 120 truncated great icosahedra. 2 of each join at each vertex.

It is the medial stage of the truncation series between a great grand stellated hecatonicosachoron and its dual grand hexacosichoron, which makes it the bitruncation of either of these polychora.

## Vertex coordinates

Coordinates for the vertices of a great hexacosihecatonicosachoron of edge length 1 are given by all permutations of:

• ${\displaystyle \left(0,\,0,\,±(\sqrt5-1),\,±\frac{7-3\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{5-\sqrt5}{4},\,±\frac{5-\sqrt5}{4},\,±\frac{5\sqrt5-9}{4},\,±\frac{5\sqrt5-9}{4}\right),}$

together with all even permutations of:

• ${\displaystyle \left(0,\,±\frac12,\,±\frac{13-5\sqrt5}{4},\,±\frac{5\sqrt5-7}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac12,\,±3\frac{\sqrt5-1}{4},\,±\frac{7\sqrt5-13}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt5-1}{4},\,±5\frac{3-\sqrt5}{4},\,±\frac{2\sqrt5-3}{2}\right),}$
• ${\displaystyle \left(0,\,±3\frac{3-\sqrt5}{4},\,±\frac{5-2\sqrt5}{2},\,±\frac{11-3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt5-1}{2},\,±\frac{7\sqrt5-13}{4},\,±\frac{5-\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt5-1}{2},\,±5\frac{3-\sqrt5}{4},\,±\frac{7-3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±1,\,±\frac{5\sqrt5-11}{4},\,±\frac{5\sqrt5-9}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{3-\sqrt5}{2},\,±\frac{5\sqrt5-11}{4},\,±\frac{11-3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{4},\,±1,\,±\frac{\sqrt5-2}{2},\,±\frac{7\sqrt5-13}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{4},\,±\frac{7-3\sqrt5}{4},\,±\frac{5\sqrt5-9}{4},\,±\frac{11-3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{2},\,±\frac{2\sqrt5-3}{2},\,±\frac{5\sqrt5-9}{4},\,±3\frac{3-\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{2},\,±\frac{7-3\sqrt5}{4},\,±\frac{5\sqrt5-7}{4},\,±\frac{5-2\sqrt5}{2}\right),}$
• ${\displaystyle \left(±1,\,±\frac{\sqrt5-1}{2},\,±\frac{7-3\sqrt5}{2},\,±\frac{3-\sqrt5}{2}\right),}$
• ${\displaystyle \left(±1,\,±\frac{\sqrt5-2}{2},\,±\frac{13-5\sqrt5}{4},\,±3\frac{3-\sqrt5}{4}\right),}$
• ${\displaystyle \left(±1,\,±\frac{5-\sqrt5}{4},\,±\frac{5\sqrt5-11}{4},\,±\frac{5-2\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-2}{2},\,±(\sqrt5-1),\,±\frac{5\sqrt5-11}{4},\,±\frac{7-3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-2}{2},\,±\frac{3-\sqrt5}{2},\,±\frac{5\sqrt5-7}{4},\,±\frac{5\sqrt5-9}{4}\right),}$
• ${\displaystyle \left(±3\frac{\sqrt5-1}{4},\,±\frac{3-\sqrt5}{2},\,±\frac{2\sqrt5-3}{2},\,±\frac{5\sqrt5-11}{4}\right),}$
• ${\displaystyle \left(±\frac{5-\sqrt5}{4},\,±\frac{\sqrt5-2}{2},\,±5\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{5-\sqrt5}{4},\,±3\frac{\sqrt5-1}{4},\,±\frac{13-5\sqrt5}{4},\,±\frac{7-3\sqrt5}{4}\right).}$