Runcicantic 5-cube

Runcicantic 5-cube
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Girhin
Coxeter diagram	x3x3x3o *b3o ()
Elements
Tera	16 decachora, 16 great rhombated pentachora, 10 tesseractihexadecachora
Cells	80 triangular prisms, 40+80+80 truncated tetrahedra, 80 truncated octahedra
Faces	160+160 triangles, 240 squares, 160+320 hexagons
Edges	240+480+480
Vertices	480
Vertex figure	Skewed tetragonal disphenoidal pyramid, edge lengths 1 (bases of disphenoid), √2 (lacings to base 1 edge) and √3 (sides of disphenoid and lacings to base 2)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {122}}{4}}\approx 2.76134$
Hypervolume	${\frac {29993{\sqrt {2}}}{120}}\approx 353.47089$
Diteral angles	Deca–tut–grip: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Tah–tut–deca: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Tah–toe–grip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Grip–trip–grip: $\arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }$
	Tah–tut–tah: 90°
Central density	1
Number of external pieces	42
Level of complexity	30
Related polytopes
Army	Girhin
Regiment	Girhin
Dual	Disphenoidal pyramidal tetracosioctacontateron
Conjugate	None
Abstract & topological properties
Flag count	57600
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	D5, order 1920
Convex	Yes
Nature	Tame

The runcicantic 5-cube, also called the great rhombidemipenteract or girhin, is a convex uniform 5-polytope. It consists of 16 decachora, 16 great rhombated pentachora, and 10 tesseractihexadecachora. One decachoron, 2 tesseractihexadecachora, and 2 great rhombated pentachora join at each vertex. It can be formed from an alternated faceting of the runcicantellated 5-cube, or as a bitruncation of the 5-demicube.

Vertex coordinates

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

• $\left({\frac {5{\sqrt {2}}}{4}},\,{\frac {5{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$.

Representations

A runcicantic 5-cube has the following Coxeter diagrams:

• x3x3x3o *b3o () (full symmetry)
• s4o3x3x3o () (as alternated faceting)

Gallery

• 

  B₅ orthographic projection

• 

  D₄

• 

  D₃

• 

  A₃

External links

• Bowers, Jonathan. "Category 10: Greater Truncates" (#732).

• Klitzing, Richard. "girhin".
• Wikipedia contributors. "Runcicantic 5-cube".