Quasicelligreatorhombated penteract

Quasicelligreatorhombated penteract
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Quacgarn
Coxeter diagram	x4/3x3x3o3x ()
Elements
Tera	80 cuboctahedral prisms, 80 triangular-octagrammic duoprisms, 40 quasitruncated cuboctahedral prisms, 32 prismatorhombated pentachora, 10 great quasirhombated tesseracts
Cells	320+320+320 triangular prisms, 480 cubes, 320 hexagonal prisms, 160 truncated tetrahedra, 240 octagrammic prisms, 160 cuboctahedra, 80 quasitruncated qcuboctahedra
Faces	640+640 triangles, 960+960+960+960 squares, 640 hexagons, 240 octagrams
Edges	960+960+1920+1920
Vertices	1920
Vertex figure	Skew rectangular pyramidal pyramid, base edge lengths 1 and √2, remaining edge lengths √2 (6), √3 (2), and √2–√2 (1)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {41-16{\sqrt {2}}}}{2}}\approx 2.14316$
Hypervolume	${\frac {29863{\sqrt {2}}-41080}{30}}\approx 38.42199$
Diteral angles	Cope–trip–tistodip: 150°
	Quitcope–stop–tistodip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Quitcope–cube–cope: 135°
	Gaqrit–quitco–quitcope: 135°
	Gaqrit–trip–cope: 120°
	Gaqrit–tut–prip: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$
	Prip–hip–quitcope: $\arccos \left({\frac {\sqrt {10}}{5}}\right)\approx 50.76848^{\circ }$
	Prip–trip–tistodip: $\arccos \left({\frac {\sqrt {15}}{5}}\right)\approx 39.23152^{\circ }$
	Prip–co–cope: $\arccos \left({\frac {2{\sqrt {5}}}{5}}\right)\approx 26.56505^{\circ }$
Related polytopes
Army	Semi-uniform Cogrin, edge lengths ${\sqrt {2}}-1$ (octagons), $2-{\sqrt {2}}$ (remaining edges of great rhombicuboctahedra), 1 (triangles of duoprisms)
Regiment	Quacgarn
Conjugate	Celligreatorhombated penteract
Abstract & topological properties
Flag count	307200
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	No
Nature	Tame

The quasicelligreatorhombated penteract, or quacgarn, is a nonconvex uniform polyteron. It consists of 80 cuboctahedral prisms, 80 triangular-octagrammic duoprisms, 40 quasitruncated cuboctahedral prisms, 10 great quasirhombated tesseracts, and 32 prismatorhombated pentachora. 1 prismatorhombated pentachoron, 1 cuboctahedral prism, 1 great quasirhombated tesseract, 1 triangular-octagrammic duoprism, and 2 quasitruncated cuboctahedral prisms join at each vertex. It is the quasistericantitruncation of the penteract.

Vertex coordinates

The vertices of a quasicelligreatorhombated penteract of edge length 1 are given by all permutations of:

• $\left({\frac {3{\sqrt {2}}-1}{2}},\,{\frac {2{\sqrt {2}}-1}{2}},\,{\frac {2{\sqrt {2}}-1}{2}},\,{\frac {{\sqrt {2}}-1}{2}},\,\pm {\frac {1}{2}}\right).$

Related polytopes

The quasicelligreatorhombated penteract is the colonel of a three-member regiment that also includes the great retrocelligreatorhombated penteract and great lepidocelligreatorhombated penteract.

External links

• Bowers, Jonathan. "Category 13: Prismatorhombates and Kin" (#856).

• Klitzing, Richard. "quacgarn".