# Great disnub dishexacosichoron

Great disnub dishexacosichoron
Rank4
TypeUniform
Notation
Elements
Cells4800 octahedra as 2400 golden hexagrammic antiprisms, 1200 great icosahedra as 600 small retrosnub disoctahedra
Faces7200+14400 triangles, 9600 triangles as 4800 golden hexagrams
Edges7200+7200+14400
Vertices3600
Vertex figureBlend of two pentagrammic prisms, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {5-2{\sqrt {5}}}}\approx 0.72654}$
Hypervolume0
Dichoral anglesGike–3–oct: ${\displaystyle \arccos \left(-{\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 97.76124^{\circ }}$
Oct–3–oct: ${\displaystyle \arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }}$
Number of external pieces1058400
Level of complexity3018
Related polytopes
ArmySemi-uniform Srix
ConjugateSmall disnub dishexacosichoron
Convex coreTriangular-antitegmatic dischiliatetracosichoron
Abstract & topological properties
Euler characteristic–1800
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The great disnub dishexacosichoron, or gadsadox, is a nonconvex uniform polychoron that consists of 4800 regular octahedra (falling in pairs into the same hyperplane, thus forming 2400 golden hexagrammic antiprisms) and 1200 regular great icosahedra (also falling in pairs in the same hyperplane, forming 600 small retrosnub disoctahedra). 8 octahedra and 4 great icosahedra join at each vertex.

This polychoron can be obtained as the blend of 10 rectified grand hexacosichora. In the process some of the octahedra blend out fully, while the other cells compound as noted above. In addition the vertex figure would in turn be a blend of two pentagrammic prismatic vertex figures of the rectified grand hexacosichoron.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}},\,\pm {\frac {2{\sqrt {10}}-3{\sqrt {2}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm 3{\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {7{\sqrt {2}}-3{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm 3{\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {5{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {5{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {7{\sqrt {2}}-3{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}},\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {4{\sqrt {2}}-{\sqrt {10}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}}\right),}$

together with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {5{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm 3{\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {2{\sqrt {10}}-3{\sqrt {2}}}{4}},\,\pm {\frac {5{\sqrt {2}}-{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{8}},\,\pm {\frac {4{\sqrt {2}}-{\sqrt {10}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}},\,\pm 3{\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}},\,\pm {\frac {7{\sqrt {2}}-3{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm {\frac {7{\sqrt {2}}-3{\sqrt {10}}}{8}},\,\pm {\frac {2{\sqrt {10}}-{\sqrt {2}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm 3{\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {5{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm {\frac {2{\sqrt {10}}-3{\sqrt {2}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}},\,\pm {\frac {4{\sqrt {2}}-{\sqrt {10}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}},\,\pm {\frac {7{\sqrt {2}}-3{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}},\,\pm 3{\frac {{\sqrt {10}}-{\sqrt {2}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm {\frac {5{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{8}},\,\pm {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}}\right).}$