Decagram

Decagram
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Dag
Coxeter diagram	x10/3o ()
Schläfli symbol	{10/3}
Elements
Edges	10
Vertices	10
Vertex figure	Dyad, length √(5–√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt5-1}{2} ≈ 0.61803$
Inradius	$\frac{\sqrt{5-2\sqrt5}}{2} ≈ 0.36327$
Area	$5\frac{\sqrt{5-2\sqrt5}}{2} ≈ 1.81636$
Angle	72°
Central density	3
Number of external pieces	20
Level of complexity	2
Related polytopes
Army	Dec
Dual	Decagram
Conjugate	Decagon
Convex core	Decagon
Abstract & topological properties
Flag count	20
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10), order 20
Convex	No
Nature	Tame

The decagram 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)

External links

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

• Klitzing, Richard. "Polygons"
• Wikipedia Contributors. "Decagram (geometry)".