Decagon

The decagon is a polygon with 10 sides. A regular decagon has equal sides and equal angles. It has the most sides of any polygon that occurs as a face of a non-prismatic uniform polytope, although dodecagons appear in some tilings and hexadecagons appear in scaliform polychora.

Decagon
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Dec
Coxeter diagram	x10o ()
Schläfli symbol	{10}
Elements
Edges	10
Vertices	10
Vertex figure	Dyad, length √(5+√5)/2
Measures (edge length 1)
Circumradius	$\frac{1+\sqrt5}{2} ≈ 1.61803$
Inradius	$\frac{\sqrt{5+2\sqrt5}}{2} ≈ 1.53884$
Area	$5\frac{\sqrt{5+2\sqrt5}}{2} ≈ 7.69421$
Angle	144°
Central density	1
Number of external pieces	10
Level of complexity	1
Related polytopes
Army	Dec
Dual	Decagon
Conjugate	Decagram
Abstract & topological properties
Flag count	20
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10), order 20
Convex	Yes
Nature	Tame

The combining prefix in BSAs is da-, as in dadip.

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

It can be constructed as the uniform truncation of the regular pentagon. This is how it appears in many uniform polyhedra and polychora.

Naming

The name decagon is derived from the Ancient Greek δέκα (10) and γωνία (angle), referring to the number of vertices.

Other names include:

• Dec, Bowers style acronym, short for "decagon".

Vertex coordinates

Coordinates for a regular decagon of unit edge length, centered at the origin are all sign changes of:

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

Representations

A regular decagon can be represented by the following Coxeter diagrams:

• x10o (regular),
• x5x (H2 symmetry, generally a dipentagon),
• to5ot&#zx (t=√(5+√5)/2, generally a pentambus),
• xFV Tto&#zx (rectangular symmetry, t as above, T=ft),
• xFVFx&#xt (axial edge-first),
• otTTto&#xt (axial vertex-first).

Variations

Two main variants of the decagon have pentagon symmetry: the dipentagon, with two alternating side lengths and equal angles, and the dual pentambus, with two alternating angles and equal edges. Other less regular variations with chiral pentagonal, rectangular, central inversion, mirror, or no symmetry also exist.

Stellations

• 1st stellation: Stellated decagon (compound of two pentagons)
• 2nd stellation: Decagram
• 3rd stellation: Stellated decagram (compound of two pentagrams)

External links

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

• Klitzing, Richard. "Polygons"
• Wikipedia Contributors. "Decagon".
• Hi.gher.Space Wiki Contributors. "Decagon".

• Hartley, Michael. "{10}*20".