# Compound of twelve pentagonal antiprisms

Compound of twelve pentagonal antiprisms
Rank3
TypeUniform
Notation
Elements
Components12 pentagonal antiprisms
Faces120 triangles, 24 pentagons as 12 stellated decagons
Edges120+120
Vertices120
Vertex figureIsosceles trapezoid, edge length 1, 1, 1, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Volume${\displaystyle 2(5+2{\sqrt {5}})\approx 18.94427}$
Dihedral angles3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
5–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Central density12
Related polytopes
ArmySemi-uniform Grid
DualCompound of twelve pentagonal antitegums
ConjugateCompound of twelve pentagrammic retroprisms
Convex coreDodecahedron
Abstract & topological properties
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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:

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