Steric 5-cube

Steric 5-cube
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Siphin
Coxeter diagram	x3o3o3x *b3o ()
Elements
Tera	16 pentachora, 40 tetrahedral prisms, 10 hexadecachora, 16 small prismatodecachora
Cells	80+80+80 tetrahedra, 80+160 triangular prisms
Faces	160+320 triangles, 240 squares
Edges	160+240
Vertices	80
Vertex figure	Stretched tetrahedron atop octahedron, edge lengths 1 (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {26}}{4}}\approx 1.27475$
Hypervolume	${\frac {467{\sqrt {2}}}{120}}\approx 5.50365$
Diteral angles	Hex–tet–tepe: 135°
	Spid–trip–tepe: $\arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }$
	Pen–tet–spid: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Hex–tet–spid: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Spid–trip–spid: $\arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }$
Central density	1
Number of external pieces	82
Level of complexity	15
Related polytopes
Army	Siphin
Regiment	Siphin
Dual	Tetrahedral-octahedral tegmoidal octacontateron
Conjugate	None
Abstract & topological properties
Flag count	28800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	D5, order 1920
Convex	Yes
Nature	Tame

The steric 5-cube, also called the small prismatodemipenteract or siphin, is a convex uniform 5-polytope. It consists of 16 regular pentachora, 40 tetrahedral prisms, 10 hexadecachora, and 16 small prismatodecachora. 1 pentachoron, 4 tetrahedral prisms, 4 small prismatodecachora, and 1 hexadecachoron join at each vertex. It can be formed from an alternated faceting of the stericated 5-cube, or by expanding the hexadecachoric facets of the 5-demicube outwards.

Vertex coordinates

The vertices of a steric 5-cube of edge length 1 are given by all permutations and even sign changes of:

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$.

Representations

A steric 5-cube has the following Coxeter diagrams:

• x3o3o3x *b3o () (full symmetry)
• s4o3o3o3x () (as alternated faceting)
• xxoo3oooo3ooxx *b3oxxo&#xt (D₄ axial, hexadecachoron-first)
• xx(uo)x3oo(oo)o3ox(ox)o3oo(ox)x&#xt (A₄ axial, pentachoron-first)
• o(xx)(uo)(xx)(uo)x x(uo)(xx)(uo)(xx)o o(ox)(xo)(ou)(ox)x3x(ox)(ou)(xo)(ox)o&#xt (A₂×A₁×A₁ axial, triangular prism-first)

Gallery

• 

  B₅ orthographic projection

• 

  D₄

• 

  D₃

• 

  A₃

Related polytopes

The regiment of the steric 5-cube contains a total of 56 members as well as 20 fissaries. It is the largest regiment of uniform polytera whose colonel has 5-demicubic symmetry. This regiment also has subsymmetric scaliform members.

External links

• Klitzing, Richard. "siphin".
• Wikipedia contributors. "Steric 5-cube".