# Decagram (geometry)

Regular decagram
A regular decagram
Type Regular star polygon
Edges and vertices 10
Schläfli symbol {10/3}
t{5/3}
Coxeter diagram
Symmetry group Dihedral (D10)
Internal angle (degrees) 72°
Dual polygon self
Properties star, cyclic, equilateral, isogonal, isotoxal

In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.[1]

The name decagram combine a numeral prefix, deca-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs) meaning a line.[2]

## Regular decagram

For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.

## Applications

Decagrams have been used as one of the decorative motifs in girih tiles.[3]

## Related figures

A regular decagram is a 10-sided polygram, represented by symbol {10/n}, containing the same vertices as regular decagon. Only one of these polygrams, {10/3} (connecting every third point), forms a regular star polygon, but there are also three ten-vertex polygrams which can be interpreted as regular compounds:

Form Convex Compound Star polygon Compounds
Image
Symbol {10/1} = {10} {10/2} = 2{5} {10/3} {10/4} = 2{5/2} {10/5} = 5{2}

Deeper truncations of the regular pentagon and pentagram can produce intermediate star polygon forms with ten equally spaced vertices and two edge lengths that remain vertex-transitive (any two vertices can be transformed into each other by a symmetry of the figure).[6][7][8]

Isogonal truncations of pentagon and pentagram
Quasiregular Isogonal Quasiregular
Double covering

t{5} = {10}

t{5/4} = {10/4} = 2{5/2}

t{5/3} = {10/3}

t{5/2} = {10/2} = 2{5}