Pentagram

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

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:


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

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+$\sqrt{5}$)/2). Irregular pentagrams further appear as the vertex figures of some snub polyhedra.