# Decagram

Decagram Rank2
TypeRegular
SpaceSpherical
Bowers style acronymDag
Info
Coxeter diagramx10/3o
Schläfli symbol{10/3}
SymmetryI2(10), order 20
ArmyDec
Elements
Edges10
Vertices10
Measures (edge length 1)
Circumradius$\frac{\sqrt5-1}{2} ≈ 0.61803$ Inradius$\frac{\sqrt{5-2\sqrt5}}{2} ≈ 0.36327$ Area$5\frac{\sqrt{5-2\sqrt5}}{2} ≈ 1.81636$ Angle72°
Central density3
Euler characteristic0
Number of pieces20
Level of complexity2
Related polytopes
DualDecagram
ConjugateDecagon
Convex coreDecagon
Properties
ConvexNo
OrientableYes
NatureTame

The decagram, or dag, is a star polygon with 10 sides. A regular decagram has equal sides and equal angles.

It is the second stellation of the decagon, and the only one that is not a compound. The only other polygons with a single non-compound stellation are the pentagon, the octagon, and the dodecagon.

It is the uniform quasitruncation of the pentagram, and as such appears frequently in uniform polyhedra and polychora. It is the largest uniform star polygon to appear in a non-prismatic uniform polytope in 3 or 4 dimensions.

## Vertex coordinates

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

• $\left(±\frac12,\,±\frac{\sqrt{5-2\sqrt5}}2\right),$ • $\left(±\frac{3-\sqrt5}4,\,±\sqrt{\frac{5-\sqrt5}8}\right),$ • $\left(±\frac{\sqrt5-1}2,\,0\right).$ ## Representations

A decagram has the following Coxeter diagrams:

• x10/3o (full symmetry)
• x5/3x (H2 symmetry)