Runcic 5-cube

Runcic 5-cube
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Sirhin
Coxeter diagram	x3o3x3o *b3o ()
Elements
Tera	16 rectified pentachora, 16 small rhombated pentachora, 10 rectified tesseracts
Cells	40+80 tetrahedra, 80 triangular prisms, 80 octahedra, 80 cuboctahedra
Faces	160+160+320 triangles, 240 squares
Edges	240+480
Vertices	160
Vertex figure	Triangular fastegium, edge lengths 1 (base triangular prism and top triangle) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {42}}{4}}\approx 1.62019$
Hypervolume	${\frac {1811{\sqrt {2}}}{120}}\approx 21.34284$
Diteral angles	Rap–oct–srip: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Rit–tet–rap: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Rit–co–srip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Srip–trip–srip: $\arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }$
	Rit–tet–rit: 90°
Central density	1
Number of external pieces	42
Level of complexity	18
Related polytopes
Army	Sirhin
Regiment	Sirhin
Dual	Tegmatonotched hecatonhexecontateron
Conjugate	None
Abstract & topological properties
Flag count	34560
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	D5, order 1920
Convex	Yes
Nature	Tame

The runcic 5-cube, also called the small rhombidemipenteract or sirhin, is a convex uniform 5-polytope. It consists of 16 rectified pentachora, 16 small rhombated pentachora, and 10 rectified tesseracts. One rectified pentachoron, 2 rectified tesseracts, and 3 small rhombated pentachora join at each vertex. It can be formed from an alternated faceting of the runcinated 5-cube, or as a birectification of the 5-demicube.

Vertex coordinates

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

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

Representations

A runcic 5-cube has the following Coxeter diagrams:

• x3o3x3o *b3o () (full symmetry)
• s4o3o3x3o ()
• xxoo3oxxo3ooxx *b3xoox&#xt (D₄ axial, rectified tesseract-first)

Gallery

• 

  B₅ orthographic projection

• 

  D₄

• 

  D₃

• 

  A₃

Related polytopes

The regiment of the runcic 5-cube includes a total of 37 members plus one fissary member.

External links

• Klitzing, Richard. "sirhin".
• Wikipedia contributors. "Runcic 5-cube".