Truncated great hecatonicosachoron

Truncated great hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Tighi
Coxeter diagram	x5x5/2o5o ()
Elements
Cells	120 small stellated dodecahedra 120 truncated great dodecahedra
Faces	1440 pentagrams 720 decagons
Edges	720+3600
Vertices	1440
Vertex figure	Pentagonal pyramid, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (base) and ${\sqrt {\frac {5+{\sqrt {5}}}{2}}}$ (side)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {13+5{\sqrt {5}}}{2}}}\approx 3.47709$
Hypervolume	${\frac {2925+1260{\sqrt {5}}}{2}}\approx 2871.22283$
Dichoral angle	Sissid–5/2–tigid: 144° Tigid–10–tigid: 144°
Central density	6
Number of external pieces	3720
Level of complexity	15
Related polytopes
Army	Semi-uniform Tex, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (icosahedra), 1 (surrounded by truncated tetrahedra)
Regiment	Tighi
Conjugate	Quasitruncated grand stellated hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	57600
Euler characteristic	–960
Orientable	Yes
Properties
Symmetry	H4, order 14400
Flag orbits	4
Convex	No
Nature	Tame

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

Cross-sections

Vertex coordinates

The vertices of a truncated great hecatonicosachoron of edge length 1 are given by all permutations of:

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

plus all even permutations of:

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

Related polychora

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

External links

• Bowers, Jonathan. "Category 2: Truncates" (#25).

• Klitzing, Richard. "tighi".