# Great disprismatohexacosihecatonicosachoron

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Great disprismatohexacosi-hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymGidpixhi
Coxeter diagramx5x3x3x ()
Elements
Cells1200 hexagonal prisms, 720 decagonal prisms, 600 truncated octahedra, 120 great rhombicosidodecahedra
Faces3600+3600+3600 squares, 2400+2400 hexagons, 1440 decagons
Edges7200+7200+7200+7200
Vertices14400
Vertex figureIrregular tetrahedron, edge lengths 2 (3), 3 (2), and (5+5)/2 (1)
Measures (edge length 1)
Circumradius$\displaystyle \sqrt{83+36\sqrt5} \approx 12.78665$
Hypervolume$\displaystyle 75(845+366\sqrt5) \approx 124755.066$
Dichoral anglesToe–6–hip: $\displaystyle \arccos\left(-\frac{\sqrt6+\sqrt{30}}{8}\right) \approx 172.23876^\circ$
Dip–4–trip: $\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) \approx 169.18768^\circ$
Toe–4–dip: $\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) \approx 166.71747^\circ$
Grid–10–dip: 162°
Grid–4–hip: $\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) \approx 159.09484^\circ$
Grid–6–toe: $\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) \approx 157.76124^\circ$
Central density1
Number of external pieces2640
Level of complexity24
Related polytopes
ArmyGidpixhi
RegimentGidpixhi
DualTetrahedral myriatetrachiliatetracosichoron
ConjugateGreat quasidisprismatohexacosihecatonicosachoron
Abstract & topological properties
Flag count345600
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits24
ConvexYes
NatureTame

The great disprismatohexacosihecatonicosachoron, or gidpixhi, also commonly called the omnitruncated 120-cell, is a convex uniform polychoron that consists of 1200 hexagonal prisms, 720 decagonal prisms, 600 truncated octahedra, and 120 great rhombicosidodecahedra. 1 of each type of cell join at each vertex. It is the omnitruncate of the H4 family of uniform polychora, and could also be considered to be the omnitruncated 600-cell. It is therefore the most complex of the non-prismatic convex uniform polychora.

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

## Vertex coordinates

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

• $\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{12+5\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{11+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{11+6\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac32,\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm1,\,\pm2(2+\sqrt5),\,\pm(5+2\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{5+\sqrt5}{2},\,\pm2(2+\sqrt5),\,\pm2(2+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac{4+\sqrt5}{2},\,\pm\frac{4+\sqrt5}{2},\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+2\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{7+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm(2+\sqrt5),\,\pm(2+\sqrt5),\,\pm(3+2\sqrt5),\,\pm2(2+\sqrt5)\right)$ ,

plus all even permutations of:

• $\displaystyle \left(\pm\frac12,\,\pm5\frac{3+\sqrt5}{4},\,\pm\frac{15+7\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{17+7\sqrt5}{4},\,\pm\frac{17+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm1,\,\pm\frac{7+5\sqrt5}{4},\,\pm\frac{23+11\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm3\frac{7+3\sqrt5}{4},\,\pm(3+2\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{25+9\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{23+9\sqrt5}{4},\,\pm\frac{11+7\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{11+6\sqrt5}{2},\,\pm\frac{4+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\, \pm\frac{7+\sqrt5}{4},\,\pm\frac{17+9\sqrt5}{4},\,\pm2(2+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{25+9\sqrt5}{4},\,\pm(3+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{23+11\sqrt5}{4},\,\pm\frac{5+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm(1+\sqrt5),\,\pm\frac{23+9\sqrt5}{4},\,\pm\frac{13+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm3\frac{3+\sqrt5}{4},\,\pm\frac{17+9\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm(2+\sqrt5),\,\pm\frac{19+9\sqrt5}{4},\,\pm\frac{17+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{11+6\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{19+9\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{25+9\sqrt5}{4},\,\pm\frac{11+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{23+9\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{12+5\sqrt5}{2},\,\pm\frac{11+3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{4+\sqrt5}{2},\,\pm\frac{17+9\sqrt5}{4},\,\pm\frac{17+7\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{3+2\sqrt5}{2},\,\pm3\frac{7+3\sqrt5}{4},\,\pm5\frac{3+\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{13+5\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm2(2+\sqrt5),\,\pm\frac{17+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac32,\,\pm(2+\sqrt5),\,\pm\frac{23+11\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{11+6\sqrt5}{2},\,\pm\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{4+\sqrt5}{2},\,\pm(1+\sqrt5),\,\pm\frac{23+11\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{11+3\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2},\,\pm(5+2\sqrt5),\,\pm\frac{17+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac32,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{12+5\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac32,\,\pm\frac{3+\sqrt5}{2},\,\pm\frac{19+9\sqrt5}{4},\,\pm\frac{15+7\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac32,\,\pm\frac{5+3\sqrt5}{4},\,\pm3\frac{7+3\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm(3+\sqrt5),\,\pm2(2+\sqrt5),\,\pm3\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{17+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{11+6\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{7+\sqrt5}{4},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{23+11\sqrt5}{4}\right)$ ,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm (1+{\sqrt {5}}),\,\pm {\frac {12+5{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{2}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {17+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {11+4{\sqrt {5}}}{2}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm (3+2{\sqrt {5}}),\,\pm {\frac {17+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {11+7{\sqrt {5}}}{4}},\,\pm 2(2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {12+5{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm (5+2{\sqrt {5}}),\,\pm {\frac {15+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {11+4{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {7+4{\sqrt {5}}}{2}},\,\pm {\frac {17+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {25+9{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {11+4{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm (5+2{\sqrt {5}}),\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm (3+2{\sqrt {5}}),\,\pm {\frac {15+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {11+7{\sqrt {5}}}{4}},\,\pm {\frac {7+4{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {25+9{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {7+4{\sqrt {5}}}{2}},\,\pm 2(2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {11+4{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {23+9{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm 3{\frac {7+3{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm (1+{\sqrt {5}}),\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {7+4{\sqrt {5}}}{2}},\,\pm {\frac {15+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm (1+{\sqrt {5}}),\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm 2(2+{\sqrt {5}}),\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {19+9{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {23+9{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm (3+2{\sqrt {5}}),\,\pm {\frac {9+4{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm 3{\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {11+7{\sqrt {5}}}{4}},\,\pm (5+2{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {19+9{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm (2+{\sqrt {5}}),\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {17+9{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm (2+{\sqrt {5}}),\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {11+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {17+9{\sqrt {5}}}{4}}\right)}$.

## Semi-uniform variant

The great disprismatohexacosihecatonicosachoron has a semi-uniform variant of the form a5b3c3d that maintains its full symmetry. This variant uses 120 great rhombicosidodecahedra of form a5b3c, 600 great rhombitetratetrahedra of form b3c3d, 720 dipentagonal prisms of form d a5b, and 1200 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 a5b3c3d), its circumradius is given by ${\displaystyle {\sqrt {\frac {14a^{2}+21b^{2}+10c^{2}+3d^{2}+33ab+22ac+11ad+28bc+14bd+10cd+(6a^{2}+9b^{2}+4c^{2}+d^{2}+15ab+10ac+5ad+12bc+6bd+4cd){\sqrt {5}}}{2}}}}$.