Truncated great faceted hexacosichoron

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Truncated great faceted hexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymTigfix
Coxeter diagramo5o5/2x3x ()
Elements
Cells
Faces
Edges720+3600
Vertices1440
Vertex figurePentagonal pyramid, edge lengths (5–1)/2 (base) and 3 (side)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {23-9{\sqrt {5}}}{2}}}\approx 1.19904}$
Hypervolume${\displaystyle 15\left(235-101{\sqrt {5}}\right)\approx 137.35701}$
Dichoral angleTiggy–6–tiggy: 120°
Sissid–5/2–tiggy: 108°
Central density76
Number of external pieces15960
Level of complexity48
Related polytopes
ArmySemi-uniform Tex, edge lengths ${\displaystyle {\frac {7-3{\sqrt {5}}}{2}}}$ (icosahedra), ${\displaystyle 2({\sqrt {5}}-2)}$ (surrounded by truncated tetrahedra)
RegimentTigfix
ConjugateTruncated faceted hexacosichoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic–480
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits4
ConvexNo
NatureTame

The truncated great faceted hexacosichoron, or tigfix, is a nonconvex uniform polychoron that consists of 120 small stellated dodecahedra and 120 truncated great icosahedra. One small stellated dodecahedron and five truncated great icosahedra join at each vertex. As the name suggests, it can be obtained by truncating the great faceted hexacosichoron.

Vertex coordinates

The vertices of a truncated great faceted hexacosichoron of edge length 1 are given by all even permutations of:

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

Related polychora

The truncated great faceted hexacosichoron is the colonel of a two-member regiment that also includes the truncated grand hexacosichoron.