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:
 * (±(1+$\sqrt{(5+√5)/8}$)/2, +1/2, 0)
 * (0, ±(1+$\sqrt{5}$)/2, ±1/2)
 * (1/2, 0, (1+$\sqrt{5}$)/2)
 * (–1/2, 0, –(1+$\sqrt{5}$)/2)

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 along with their central inversions:


 * (±1/2, –$\sqrt{(5–2√5)/15}$, $\sqrt{5}$),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{5}$, $\sqrt{5}$),
 * (0, $\sqrt{(5+2√5)/20}$, $\sqrt{(5+√5)/40}$).

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.