Small disprismatohexacosihecatonicosachoric prism

Small disprismatohexacosihecatonicosachoric prism
File:Small disprismatohexacosihecatonicosachoric prism.png
Rank	5
Type	Uniform
Notation
Bowers style acronym	Sidpixhip
Coxeter diagram	x x5o3o3x ()
Elements
Tera	600 tetrahedral prisms, 1200 triangular-square duoprisms, 720 square-pentagonal duoprisms, 120 dodecahedral prisms, 2 small disprismatohexacosihecatonicosachora
Cells	1200 tetrahedra, 2400+2400 triangular prisms, 3600 cubes, 1440+1440 pentagonal prisms, 240 dodecahedra
Faces	4800 triangles, 3600+3600+7200 squares, 2400 pentagons
Edges	2400+7200+7200
Vertices	4800
Vertex figure	Triangular antipodial pyramid, edge lengths 1, √2, (1+√5)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {57+24{\sqrt {5}}}}{2}}\approx 5.25989$
Hypervolume	${\frac {7025+3150{\sqrt {5}}}{4}}\approx 3517.15353$
Diteral angles	Tepe–trip–tisdip: $\arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }$
	Squipdip–cube–tisdip: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	Dope–pip–squipdip: 162°
	Sidpixhi–doe–dope: 90°
	Sidpixhi–tet–tepe: 90°
	Sidpixhi–pip–squipdip: 90°
	Sidpixhi–trip–tisdip: 90°
Height	1
Central density	1
Number of external pieces	2642
Level of complexity	40
Related polytopes
Army	Sidpixhip
Regiment	Sidpixhip
Dual	Triangular-antitegmatic dischilliatetracosichoric tegum
Conjugate	Quasidisprismatohexacosihecatonicosachoric prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H4×A1, order 28800
Convex	Yes
Nature	Tame

The small dis-prismato-hexacosi-hecatonicosa-choric prism or sidpixhip is a prismatic uniform polyteron that consists of 2 small disprismatohexacosihecatonicosachora, 120 dodecahedral prisms, 600 tetrahedral prisms, 720 square-pentagonal duoprisms, and 1200 triangular-square duoprisms. 1 small disprismatohexacosihecatonicosachoron, 1 dodecahedral prism, 1 tetrahedral prism, 3 square-pentagonal duoprisms, and 3 triangular-square duoprisms join at each vertex. As the name suggests, it is a prism of the small disprismatohexacosihecatonicosachoron, which also makes it a convex segmentoteron.

Vertex coordinates

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

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

along with the even permutations of the first four coordinates of:

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

External links

• Klitzing, Richard. "Sidpixhip".