Compound of six pentagonal antiprisms

Compound of six pentagonal antiprisms
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gassid
Elements
Components	6 pentagonal antiprisms
Faces	60 triangles, 12 pentagons
Edges	60+60
Vertices	60
Vertex figure	Isosceles trapezoid, edge length 1, 1, 1, (1+√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106$
Volume	$5+2{\sqrt {5}}\approx 9.47214$
Dihedral angles	3–3: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	5–3: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }$
Central density	6
Number of external pieces	600
Level of complexity	33
Related polytopes
Army	Semi-uniform Tid, edge lengths ${\frac {\sqrt {5}}{5}}$ (triangles) and ${\frac {3{\sqrt {5}}-5}{10}}$ (between dipentagons)
Regiment	Gassid
Dual	Compound of six pentagonal antitegums
Conjugate	Compound of six pentagrammic retroprisms
Convex hull	Semi-uniform Tid
Convex core	Dodecahedron
Abstract & topological properties
Flag count	480
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great snub dodecahedron, gassid, or compound of six pentagonal antiprisms is a uniform polyhedron compound. It consists of 60 triangles and 12 pentagons, with one pentagon and three triangles joining at a vertex.

This compound can be formed by inscribing six pentagonal antiprisms within an icosahedron (each by removing one pair of opposite vertices) and then rotating each antiprism by 36º around its axis.

Its quotient prismatic equivalent is the pentagonal antiprismatic hexateroorthowedge, which is eight-dimensional.

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

Vertex coordinates

Coordinates for the vertices of a great snub dodecahedron of edge length 1 are given by all even permutations of:

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

External links

• Bowers, Jonathan. "Polyhedron Category C8: Antiprismatics" (#48).

• Klitzing, Richard. "gassid".
• Wikipedia contributors. "Compound of six pentagonal antiprisms".