 # Decagram (geometry)

Regular decagram A regular decagram
TypeRegular star polygon
Edges and vertices10
Schläfli symbol{10/3}
t{5/3}
Coxeter diagram        Symmetry groupDihedral (D10)
Internal angle (degrees)72°
Dual polygonself
Propertiesstar, 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}.

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

## 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. ## Isotoxal variations

An isotoxal polygon has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom of one angle. The first is a variation of a double covering of a pentagon {5}, and last is a variation of a double covering of a pentagram {5/2}. The middle is a variation of a regular decagram, {10/3}. {(5/2)α} {(5/3)α} {(5/4)α}

## 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:

{10/2} can be seen as the 2D equivalent of the 3D compound of dodecahedron and icosahedron and 4D compound of 120-cell and 600-cell; that is, the compound of two pentagonal polytopes in their respective dual positions.

{10/4} can be seen as the two-dimensional equivalent of the three-dimensional compound of small stellated dodecahedron and great dodecahedron or compound of great icosahedron and great stellated dodecahedron through similar reasons. It has six four-dimensional analogues, with two of these being compounds of two self-dual star polytopes, like the pentagram itself; the compound of two great 120-cells and the compound of two grand stellated 120-cells. A full list can be seen at Polytope compound#Compounds with duals.

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).