Compound of twelve pentagrammic retroprisms

Compound of twelve pentagrammic retroprisms
(OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gidasid
Elements
Components	12 pentagrammic retroprisms
Faces	120 triangles, 24 pentagrams as 12 stellated decagrams
Edges	120+120
Vertices	120
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	$2(5-2{\sqrt {5}})\approx 1.05573$
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	36
Related polytopes
Army	Semi-uniform Grid
Regiment	Gidasid
Dual	Compound of twelve pentagrammic concave antitegums
Conjugate	Compound of twelve pentagonal antiprisms
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great inverted disnub dodecahedron, gidasid, or compound of twelve pentagrammic retroprisms is a uniform polyhedron compound. It consists of 120 triangles and 24 pentagrams (whcih fall in pairs in the same plane and combine into 12 stellated decagrams), with one pentagram 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 pentagrammic retroprisms inscribed in a great icosahedron, and rotate pairs of retroprisms in opposite directions. At θ = 36° the retroprisms coincide by pairs, resulting in a double cover of the great inverted snub dodecahedron.

Vertex coordinates

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

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

External links

• Bowers, Jonathan. "Polyhedron Category C8: Antiprismatics" (#51).

• Klitzing, Richard. "gidasid".

• Wikipedia Contributors. "Compound of twelve pentagrammic crossed antiprisms with rotational freedom".