# Small disnub dishexacosichoron

Small disnub dishexacosichoron
Rank4
TypeUniform
Notation
Elements
Cells4800 octahedra as 2400 golden hexagrammic antiprisms, 1200 icosahedra as 600 snub disoctahedra
Faces7200+14400 triangles, 9600 triangles as 4800 golden hexagrams
Edges7200+7200+14400
Vertices3600
Vertex figureBlend of two pentagonal prisms, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {5+2{\sqrt {5}}}}\approx 3.07768}$
Hypervolume0
Dichoral anglesOct–3–oct: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Ike–3–oct: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Number of external pieces148200
Level of complexity397
Related polytopes
ArmySemi-uniform Srix
ConjugateGreat disnub dishexacosichoron
Convex coreIcosakis truncated hexacosichoron
Abstract & topological properties
Euler characteristic–1800
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The small disnub dishexacosichoron, or sadsadox, 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 icosahedra (also falling in pairs in the same hyperplane, forming 600 snub disoctahedra). 8 octahedra and 4 icosahedra join at each vertex.

This polychoron can be obtained as the blend of 10 rectified hexacosichora, positioned in a similar way to the compound of 10 hexacosichora known as the snub decahecatonicosachoron. 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 pentagonal prismatic vertex figures of the rectified hexacosichoron.

## Vertex coordinates

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

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

together with all even permutations of:

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

## Related polychora

The regiment of the small disnub dishexacosichoron, known as the "sidtaps", contains 9 uniform members, 11 scaliform members, 3 fissary scaliforms, and a number of uniform compounds.