Compound of twelve pentagonal antiprisms

Compound of twelve pentagonal antiprisms
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gadsid
Elements
Components	12 pentagonal antiprisms
Faces	120 triangles, 24 pentagons as 12 stellated decagons
Edges	120+120
Vertices	120
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	$2(5+2{\sqrt {5}})\approx 18.94427$
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	12
Related polytopes
Army	Semi-uniform Grid
Regiment	Gadsid
Dual	Compound of twelve pentagonal antitegums
Conjugate	Compound of twelve pentagrammic retroprisms
Convex core	Dodecahedron
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great disnub dodecahedron, gadsid, or compound of twelve pentagonal antiprisms is a uniform polyhedron compound. It consists of 120 triangles and 24 pentagons (which fall in pairs in the same plane and combine into 12 stellated decagons), with one pentagon and three triangles joining at a vertex.

This compound has rotational freedom, represented by an angle θ. We start at θ = 0° with all the pentagonal antiprisms inscribed in an icosahedron, and rotate pairs of antiprisms in opposite directions. At θ = 36° the antiprisms coincide by pairs, resulting in a double cover of the great snub dodecahedron.

Vertex coordinates

The vertices of a great disnub dodecahedron of edge length 1 and rotation angle θ are given by all even permutations of:

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

External links

• Bowers, Jonathan. "Polyhedron Category C8: Antiprismatics" (#49).

• Klitzing, Richard. "gadsid".
• Wikipedia contributors. "Compound of twelve pentagonal antiprisms with rotational freedom".