# Pentagonal antiprism

Pentagonal antiprism
Rank3
TypeUniform
Notation
Bowers style acronymPap
Coxeter diagrams2s10o ()
Conway notationA5
Elements
Faces10 triangles, 2 pentagons
Edges10+10
Vertices10
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Volume${\displaystyle {\frac {5+2{\sqrt {5}}}{6}}\approx 1.57869}$
Dihedral angles3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
5–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces12
Level of complexity4
Related polytopes
ArmyPap
RegimentPap
DualPentagonal antitegum
ConjugatePentagrammic retroprism
Abstract & topological properties
Flag count80
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(10)×A1)/2, order 20
ConvexYes
NatureTame

The pentagonal antiprism, or pap, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagons. Each vertex joins one pentagon and three triangles. As the name suggests, it is an antiprism based on a pentagon.

It can also be obtained as a diminishing of the regular icosahedron when two pentagonal pyramids are removed from opposite ends.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \pm \left({\frac {1}{2}},\,0,\,{\frac {1+{\sqrt {5}}}{4}}\right).}$

These coordinates are obtained by removing two opposite vertices from a regular icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points:

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

## Representations

A pentagonal antiprism has the following Coxeter diagrams:

## General variant

The pentagonal antiprism has a general isogonal variant of the form xo5ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by ${\displaystyle {\sqrt {l^{2}-b^{2}{\frac {5-{\sqrt {5}}}{10}}}}}$.

The bases of the pentagonal antiprism are rotated from each other by an angle of 36°. If this angle is changed the result is more properly called a pentagonal gyroprism.

A notable case occurs as the alternation of the uniform decagonal prism. This specific case has base edges of length ${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{2}}}}$ and side edges of length ${\displaystyle {\sqrt {2}}}$.

## Related polyhedra

A pentagonal pyramid can be attached to a base of the pentagonal antiprism to form the gyroelongated pentagonal pyramid. If a second pyramid is attached to the other base, the result is the gyroelongated pentagonal bipyramid, better known as the regular icosahedron.

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

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