Pentagrammic antiprism

Pentagrammic antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Stap
Coxeter diagram	s2s10/2o ()
Elements
Faces	10 triangles, 2 pentagrams
Edges	10+10
Vertices	10
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, (√5–1)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {15+{\sqrt {5}}}{40}}}\approx 0.65643$
Volume	${\frac {\sqrt {5{\sqrt {5}}}}{6}}\approx 0.55728$
Dihedral angles	5/2–3: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{3}}}\right)\approx 114.80110^{\circ }$
	3–3: $\arccos \left({\frac {2-{\sqrt {5}}}{3}}\right)\approx 94.51323^{\circ }$
Height	${\sqrt {\frac {{\sqrt {5}}-1}{2}}}\approx 0.78615$
Central density	2
Number of external pieces	32
Level of complexity	11
Related polytopes
Army	Semi-uniform Pip, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (base), ${\sqrt {\frac {{\sqrt {5}}-1}{2}}}$ (sides)
Regiment	Stap
Dual	Pentagrammic antitegum
Conjugate	Pentagonal retroprism
Convex core	Pentagonal bifrustum
Abstract & topological properties
Flag count	80
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Convex	No
Nature	Tame

The pentagrammic antiprism, or stap, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagrams. Each vertex joins one pentagram and three triangles. As the name suggests, it is an antiprism based on a pentagram. It is one of two pentagrammic antiprisms, the other one being the pentagrammic retroprism. In this case, the pentagrams are aligned with one another.

Vertex coordinates

A pentagrammic antiprism of edge length 1 has vertex coordinates given by:

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1}{8}}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1}{8}}}\right),$
• $\left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1}{8}}}\right).$

Related polyhedra

Two non-prismatic uniform polyhedron compounds are composed of pentagrammic antiprisms:

• Small snub dodecahedron (6)
• Small disnub dodecahedron (12)

There are an infinite amount of prismatic uniform compounds that are the antiprisms of compounds of pentagrams.

In vertex figures

Pentagrammic antiprisms appear as vertex figures of four uniform polychora: the small prismatohecatonicosachoron, pentagrammal antiprismatoverted hexacosihecatonicosachoron, small pentagrammal antiprismatoverted dishecatonicosachoron, and great pentagrammal antiprismatoverted dishecatonicosachoron.

External links

• Klitzing, Richard. "stap".
• Wikipedia contributors. "Pentagrammic antiprism".
• McCooey, David. "Pentagrammic Antiprism"