Pentagram

The pentagram is a non-convex polygon with 5 sides and the simplest star regular polygon. A regular pentagram has equal sides and equal angles.

Pentagram
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Star
Coxeter diagram	x5/2o ()
Schläfli symbol	{5/2}
Elements
Edges	5
Vertices	5
Vertex figure	Dyad, length (√5–1)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573$
Inradius	$\sqrt{\frac{5-2\sqrt5}{20}} ≈ 0.16246$
Area	$\frac{\sqrt{25-10\sqrt5}}{4} ≈ 0.40615$
Angle	36°
Central density	2
Number of pieces	10
Level of complexity	2
Related polytopes
Army	Peg
Dual	Pentagram
Conjugate	Pentagon
Convex core	Pentagon
Abstract properties
Flag count	10
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H2, order 10
Convex	No
Nature	Tame

This is the only stellation of the pentagon. The only other polygons with a single non-compound stellation are the octagon, the decagon, and the dodecagon.

Pentagrams occur as faces in two of the four Kepler-Poinsot solids, namely the small stellated dodecahedron and great stellated dodecahedron.

Vertex coordinates

Coordinates for the vertices of a pentagram of unit edge length, centered at the origin, are:

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

Representations

A regular pentagram has the following Coxeter diagrams:

• x5/2o
• ß5o (as holosnub pentagon)

In vertex figures

The regular pentagram appears as a vertex figure in two uniform polyhedra, namely the great icosahedron (with an edge length of 1) and the great dodecahedron (with an edge length of (1+√5)/2). Irregular pentagrams further appear as the vertex figures of some snub polyhedra.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Klitzing, Richard. "Polygons"
• Nan Ma. "Pentagram {5/2}".

• Wikipedia Contributors. "Pentagram".