Truncated hexacosichoric prism

Truncated hexacosichoric prism
File:Truncated hexacosichoric prism.png
Rank	5
Type	Uniform
Notation
Bowers style acronym	Texip
Coxeter diagram	x o5o3x3x ()
Elements
Tera	600 truncated tetrahedral prisms, 120 icosahedral prisms, 2 truncated hexacosichora
Cells	2400 triangular prisms, 1200 hexagonal prisms, 1200 truncated tetrahedra, 240 icosahedra
Faces	4800 triangles, 720+3600 squares, 2400 hexagons
Edges	1440+1440+7200
Vertices	2880
Vertex figure	Pentagonal pyramidal pyramid, edge lengths 1, √3 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {25+9{\sqrt {5}}}{2}}}\approx 4.74998$
Hypervolume	${\frac {4025+2000{\sqrt {5}}}{4}}\approx 2124.28399$
Diteral angles	Tuttip–hip–tuttip: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Ipe–trip–tuttip: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Tex–ike–ipe: 90°
	Tex–tut–tuttip: 90°
Height	1
Central density	1
Number of external pieces	722
Level of complexity	20
Related polytopes
Army	Texip
Regiment	Texip
Dual	Dodecakis hecatonicosachoric bipyramid
Conjugate	Truncated great hexacosichoric prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H4×A1, order 28800
Convex	Yes
Nature	Tame

The truncated hecatonicosachoric prism or texip is a prismatic uniform polyteron that consists of 2 truncated hexacosichora, 120 icosahedral prisms, and 600 truncated tetrahedral prisms. 1 truncated hexacosichoron, 1 icosahedral prism, and 5 truncated tetrahedral prisms join at each vertex. As the name suggests, it is a prism based on the truncated hexacosichoron, which also makes it a convex segmentoteron.

Vertex coordinates

The vertices of a truncated hexacosichoron of edge length 1 are given by all even permutations of the first four coordinates of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {3+3{\sqrt {5}}}{4}},\,\pm {\frac {9+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {9+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {9+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right).$

External links

• Klitzing, Richard. "Texip".