Pentagonal antiprism

Pentagonal antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Pap
Coxeter diagram	s2s10o ()
Elements
Faces	10 triangles, 2 pentagons
Edges	10+10
Vertices	10
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, (1+√5)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106$
Volume	$\frac{5+2\sqrt5}{6} ≈ 1.57869$
Dihedral angles	3–3: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$
	5–3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°$
Height	$\sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065$
Central density	1
Number of pieces	12
Level of complexity	4
Related polytopes
Army	Pap
Regiment	Pap
Dual	Pentagonal antitegum
Conjugate	Pentagrammic retroprism
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	I2(10)×A1/2, order 20
Convex	Yes
Nature	Tame

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:

• $\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,0\right),$
• $\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac12\right),$
• $±\left(\frac12,\,0,\,\frac{1+\sqrt5}{4}\right).$

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:

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

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.

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#3 under ike).

• Klitzing, Richard. "pap".

• Quickfur. "The Pentagonal Antiprism".

• Wikipedia Contributors. "Pentagonal antiprism".
• McCooey, David. "Pentagonal Antiprism"