Prismatorhombated hexacosichoric prism

Prismatorhombated hexacosichoric prism
File:Prismatorhombated hexacosichoric prism.png
Rank	5
Type	Uniform
Notation
Bowers style acronym	Prixip
Coxeter diagram	x x5x3o3x ()
Elements
Tera	1200 triangular-square duoprisms, 600 cuboctahedral prisms, 720 square-decagonal duoprisms, 120 truncated dodecahedral prisms, 2 prismatorhombated hexacosichora
Cells	2400+2400+2400 triangular prisms, 3600+3600 cubes, 1200 cuboctahedra, 1440+1440 decagonal prisms, 720 truncated dodecahedra
Faces	4800+4800 triangles, 3600+7200+7200+7200+7200 squares, 2880 decagons
Edges	7200+7200+14400+14400
Vertices	14400
Vertex figure	Skew rectangular pyramidal pyramid, edge lengths 1, √2, √10+2√5/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {193+84{\sqrt {5}}}}{2}}\approx 9.75743$
Hypervolume	$25{\frac {3367+1475{\sqrt {5}}}{4}}\approx 41657.5017$
Diteral angles	Cope–trip–tisdip: $\arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }$
	Squadedip–cube–tisdip: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	Cope–cube–squipdip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{10}}}\right)\approx 166.71747^{\circ }$
	Tiddip–dip–squadedip: 162°
	Sriddip–trip–tuttip: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Prix–tid–tiddip: 90°
	Prix–co–cope: 90°
	Prix–dip–squadeddip: 90°
	Prix–trip–tisdip: 90°
Height	1
Central density	1
Number of external pieces	2642
Level of complexity	80
Related polytopes
Army	Prixip
Regiment	Prixip
Dual	Rhombipyramidal heptachilliadiacosichoric tegum
Conjugate	Quasiprismatorhombated grand hexacosichoric prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H4×A1, order 28800
Convex	Yes
Nature	Tame

The prismatorhombated hexacosichoric prism or prixip is a prismatic uniform polyteron that consists of 2 prismatorhombated hexacosichora, 120 truncated dodecahedral prisms, 600 cuboctahedral prisms, 720 square-decagonal duoprisms, and 1200 triangular-square duoprisms. 1 prismatorhombated hexacosichoron, 1 truncated dodecahedral prism, 1 cuboctahedral prism, 2 square-decagonal duoprisms, and 1 triangular-square duoprism join at each vertex. As the name suggests, it is a prism based on the prismatorhombated hexacosichoron, which also makes it a convex segmentoteron.

Vertex coordinates

The vertices of a prismatorhombated hexacosichoric prism of edge length 1 are given by all permutations of the first four coordinates of:

• $\left(0,\,\pm 1,\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$

Plus all even permutations of the first four coordinates of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {17+9{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {13+7{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {19+7{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {17+7{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {19+7{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {17+9{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {17+7{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {13+7{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right),$
• $\left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {17+9{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {9+4{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {19+7{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {17+7{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {13+7{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {1}{2}}\right).$

External links

• Klitzing, Richard. "Prixip".