Compound of six pentagrammic retroprisms

Compound of six pentagrammic retroprisms
(OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gissed
Elements
Components	6 pentagrammic retroprisms
Faces	60 triangles, 12 pentagrams
Edges	60+60
Vertices	60
Vertex figure	Crossed isosceles trapezoid, edge length 1, 1, 1, (√5–1)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779$
Volume	$5-2{\sqrt {5}}\approx 0.52786$
Dihedral angles	3–3: $\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }$
	5/2–3: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }$
Central density	18
Number of external pieces	1272
Level of complexity	83
Related polytopes
Army	Semi-uniform Tid, edge lengths ${\frac {5-{\sqrt {5}}}{10}}$ (triangles), ${\frac {5-2{\sqrt {5}}}{5}}$ (between dipentagons)
Regiment	Gissed
Dual	Compound of six pentagrammic concave antitegums
Conjugate	Compound of six pentagonal antiprisms
Convex core	Pentakis dodecahedron
Abstract & topological properties
Flag count	480
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great inverted snub dodecahedron, gissed, or compound of six pentagrammic retroprisms is a uniform polyhedron compound. It consists of 60 triangles and 12 pentagrams, with one pentagram and three triangles joining at a vertex.

This compound can be formed by inscribing six pentagrammic retroprisms within a great icosahedron (each by removing one pair of opposite vertices) and then rotating each retroprism by 36° around its axis.

Its quotient prismatic equivalent is the pentagrammic retroprismatic hexateroorthowedge, which is eight-dimensional.

A double cover of this compound occurs as a special case of the great inverted disnub dodecahedron.

Vertex coordinates

The vertices of a great inverted snub dodecahedron of edge length 1 are given by all even permutations of:

• $\left(\pm {\frac {5-2{\sqrt {5}}}{10}},\,0,\,\pm {\frac {5+3{\sqrt {5}}}{20}}\right),$
• $\left(\pm {\frac {\sqrt {5}}{5}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {\sqrt {5}}{10}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{20}},\,\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{10}}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category C8: Antiprismatics" (#50).

• Klitzing, Richard. "gissed".

• Wikipedia contributors. "Compound of six pentagrammic crossed antiprisms".