Pentagonal antiprism

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:
 * $$\left(±\frac{1+\sqrt5}{2},\,+\frac12,\,0\right),$$
 * $$\left(0,\,±\frac{1+\sqrt5}{2},\,±\frac12\right),$$
 * $$±\left(\frac12,\,0,\,\frac{1+\sqrt5}{2}\right).$$

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:


 * $$±\left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,\sqrt{\frac{5+\sqrt5}{40}}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5+\sqrt5}{40}}\right),$$
 * $$±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5+\sqrt5}{40}}\right).$$

Representations
A pentagonal antiprism has the following Coxeter diagrams:


 * s2s10o (alternated decagonal prism)
 * s2s5s (alternated dipentagonal prism)
 * xo5ox&#x (bases considered separately)

General variant
The pentagonal antiprism has a general isogonal variant of the form xo5ox&#y that maintains its full symmetry. This veriant 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 $$\sqrt{l^2-b^2\frac{5-\sqrt5}{10}}$$.

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 $$\sqrt{\frac{5+\sqrt5}{2}}$$ and side edges of length $$\sqrt2$$.

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:


 * Great snub dodecahedron (6)
 * Great disnub dodecahedron (12)

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