Regular decagram  

Type  Regular star polygon 
Edges and vertices  10 
Schläfli symbol  {10/3} t{5/3} 
Coxeter–Dynkin diagrams  
Symmetry group  Dihedral (D_{10}) 
Internal angle (degrees)  72° 
Properties  star, cyclic, equilateral, isogonal, isotoxal 
Dual polygon  self 
Star polygons 


In geometry, a decagram is a 10point 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 combines a numeral prefix, deca, with the Greek suffix gram. The gram suffix derives from γραμμῆς (grammēs) meaning a line.^{[2]}
For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.
Decagrams have been used as one of the decorative motifs in girih tiles.^{[3]}
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 doublewound of a pentagon {5}, and last is a variation of a doublewound 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}} 
A regular decagram is a 10sided 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 tenvertex 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} 
{10/2} can be seen as the 2D equivalent of the 3D compound of dodecahedron and icosahedron and 4D compound of 120cell and 600cell; that is, the compound of two pentagonal polytopes in their respective dual positions.
{10/4} can be seen as the twodimensional equivalent of the threedimensional compound of small stellated dodecahedron and great dodecahedron or compound of great icosahedron and great stellated dodecahedron through similar reasons. It has six fourdimensional analogues, with two of these being compounds of two selfdual star polytopes, like the pentagram itself; the compound of two great 120cells and the compound of two grand stellated 120cells. 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 vertextransitive (any two vertices can be transformed into each other by a symmetry of the figure).^{[6]}^{[7]}^{[8]}
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} 