Pentagram

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Pentagram
Rank2
TypeRegular
Notation
Bowers style acronymStar
Coxeter diagramx5/2o ()
Schläfli symbol{5/2}
Elements
Edges5
Vertices5
Vertex figureDyad, length (5–1)/2
Measures (edge length 1)
Circumradius
Inradius
Area
Angle36°
Central density2
Number of external pieces10
Level of complexity2
Related polytopes
ArmyPeg, edge length
DualPentagram
ConjugatePentagon
Convex corePentagon
Abstract & topological properties
Flag count10
Euler characteristic0
OrientableYes
Properties
SymmetryH2, order 10
Flag orbits1
ConvexNo
NatureTame

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.

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[edit | edit source]

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

  • ,
  • ,
  • .

Representations[edit | edit source]

A regular pentagram has the following Coxeter diagrams:

  • x5/2o ()
  • ß5o () (as holosnub pentagon)

In vertex figures[edit | edit source]

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[edit | edit source]