Truncated great faceted hexacosichoron

Truncated great faceted hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Tigfix
Coxeter diagram	o5o5/2x3x ()
Elements
Cells	120 small stellated dodecahedra 120 truncated great icosahedra
Faces	1440 pentagrams 1200 hexagons
Edges	720+3600
Vertices	1440
Vertex figure	Pentagonal pyramid, edge lengths (√5–1)/2 (base) and √3 (side)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {23-9{\sqrt {5}}}{2}}}\approx 1.19904$
Hypervolume	$15\left(235-101{\sqrt {5}}\right)\approx 137.35701$
Dichoral angle	Tiggy–6–tiggy: 120° Sissid–5/2–tiggy: 108°
Central density	76
Number of external pieces	15960
Level of complexity	48
Related polytopes
Army	Semi-uniform Tex, edge lengths ${\frac {7-3{\sqrt {5}}}{2}}$ (icosahedra), $2({\sqrt {5}}-2)$ (surrounded by truncated tetrahedra)
Regiment	Tigfix
Conjugate	Truncated faceted hexacosichoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	57600
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Flag orbits	4
Convex	No
Nature	Tame

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.

Cross-sections

Vertex coordinates

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

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

Related polychora

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

External links

• Bowers, Jonathan. "Category 2: Truncates" (#28).

• Klitzing, Richard. "tigfix".