Compound of twelve pentagrammic antiprisms

Compound of twelve pentagrammic antiprisms
(OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Sadsid
Elements
Components	12 pentagrammic antiprisms
Faces	120 triangles, 24 pentagrams as 12 stellated decagrams
Edges	120+120
Vertices	60
Vertex figure	Isosceles trapezoid, edge length 1, 1, 1, (√5–1)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {15+{\sqrt {5}}}{40}}}\approx 0.65643$
Volume	$2{\sqrt {5{\sqrt {5}}}}\approx 6.68740$
Dihedral angles	5/2–3: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{3}}}\right)\approx 114.80110^{\circ }$
	3–3: $\arccos \left({\frac {2-{\sqrt {5}}}{3}}\right)\approx 94.51323^{\circ }$
Central density	24
Number of external pieces	1380
Level of complexity	82
Related polytopes
Army	Semi-uniform Grid
Regiment	Sadsid
Dual	Compound of twelve pentagrammic antitegums
Conjugate	Compound of twelve pentagrammic antiprisms
Convex core	Disdyakis triacontahedron
Abstract & topological properties
Flag count	960
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small disnub dodecahedron, sadsid, or compound of twelve pentagrammic antiprisms is a uniform polyhedron compound. It consists of 120 triangles and 24 pentagrams (which fall in pairs into the same planes combining into 12 stellated decagrams), with one pentagram and three triangles joining at a vertex.

It can be formed by combining the two chiral forms of the small snub dodecahedron.

Its quotient prismatic equivalent is the pentagrammic antiprismatic dodecadakoorthowedge, which is fourteen-dimensional.

Vertex coordinates

The vertices of a small disnub dodecahedron of edge length 1 are given by all even permutations of:

• $\left(\pm {\sqrt {\frac {{\sqrt {5}}+{\sqrt {5({\sqrt {5}}-2)}}}{20}}},\,\pm {\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}-2{\sqrt {10({\sqrt {5}}-1)}}}{40}}}\right),$
• $\left(\pm {\sqrt {\frac {3{\sqrt {5}}-5}{40}}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {\sqrt {5{\sqrt {5}}}}{10}}\right),$
• $\left(\pm {\sqrt {\frac {2{\sqrt {5}}+{\sqrt {10({\sqrt {5}}-1)}}}{20}}},\,\pm {\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}+2{\sqrt {10({\sqrt {5}}-1)}}}{40}}}\right),$
• $\left(\pm {\sqrt {\frac {{\sqrt {5}}-{\sqrt {5({\sqrt {5}}-2)}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}+2{\sqrt {15({\sqrt {5}}-2)}}}{20}}}\right),$
• $\left(\pm {\sqrt {\frac {{\sqrt {5}}+2{\sqrt {5({\sqrt {5}}-2)}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}-2{\sqrt {5({\sqrt {5}}-2)}}}{20}}}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category C8: Antiprismatics" (#53).

• Klitzing, Richard. "sadsid".
• Wikipedia contributors. "Compound of twelve pentagrammic antiprisms".