# Small tritrigonary prismatohecatonicosidishexacosichoron

Small tritrigonary prismatohecatonicosidishexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymStut phiddix
Coxeter diagramx3o3o5/2x3*b ()
Elements
Cells600 tetrahedra, 720 pentagrammic prisms, 600 cuboctahedra, 120 small ditrigonary icosidodecahedra
Faces2400+2400 triangles, 3600 squares, 1440 pentagrams
Edges3600+7200
Vertices2400
Vertex figureTriangular cupola, edge lengths 1 (top triangle and 3 edges of base ditrigon), (5–1)/2 (remaining base edges), and 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {8+3{\sqrt {5}}}}\approx 3.83513}$
Hypervolume${\displaystyle 5{\frac {960+281{\sqrt {5}}}{4}}\approx 1985.41888}$
Dichoral anglesCo–4–stip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{10}}}\right)\approx 166.71747^{\circ }}$
Tet–3–co: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Sidtid–5/2–stip: 162°
Sidtid–3–co: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Central density2
Number of external pieces3840
Related polytopes
ArmySemi-uniform sidpixhi, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (dodecahedra), 1 (tetrahedra)
RegimentStut phiddix
ConjugateGreat tritrigonary prismatohecatonicosidishexacosichoron
Convex coreSemi-uniform grix
Abstract & topological properties
Flag count144000
Euler characteristic–600
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The small tritrigonary prismatohecatonicosidishexacosichoron, or stut phiddix, is a nonconvex uniform polychoron that consists of 600 regular tetrahedra, 720 pentagrammic prisms, 600 cuboctahedra, and 120 small ditrigonary icosidodecahedra. 1 tetrahedron, 3 pentagrammic prisms, 3 cuboctahedra, and 1 small ditrigonary icosidodecahedron join at each vertex.

It can be constructed from the small ditetrahedronary hexacosihecatonicosachoron by expanding its small ditrigonary icosidodecahedral cells outward.

The small tritrigonary prismatohecatonicosidishexacosichoron contains the vertices of an inscribed great rhombicosidodecahedral prism.

## Vertex coordinates

The vertices of a small tritrigonary prismatohecatonicosidishexacosichoron of edge length 1 are all permutations of:

• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {\sqrt {5}}{2}}\right),}$

along with the even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right).}$

## Related polychora

The small tritrigonary prismatohecatonicosidishexacosichoron is the colonel of a regiment that contains 79 uniform members as well as 30 fissary uniforms.