Pentagonal antiprism

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Pentagonal antiprism
Bowers style acronymPap
Coxeter diagrams2s10o ()
Conway notationA5
Faces10 triangles, 2 pentagons
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, (1+5)/2
Measures (edge length 1)
Dihedral angles3–3:
Central density1
Number of external pieces12
Level of complexity4
Related polytopes
DualPentagonal antitegum
ConjugatePentagrammic retroprism
Abstract & topological properties
Flag count80
Euler characteristic2
Symmetry(I2(10)×A1)/2, order 20

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[edit | edit source]

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

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:

Representations[edit | edit source]

A pentagonal antiprism has the following Coxeter diagrams:

General variant[edit | edit source]

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 .

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 and side edges of length .

Related polyhedra[edit | edit source]

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.

External links[edit | edit source]