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 (D_{10}) 
Internal angle (degrees)  72° 
Dual polygon  self 
Properties  star, cyclic, equilateral, isogonal, isotoxal 
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]}
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]}
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 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:
 {10/5} is a compound of five degenerate digons 5{2}
 {10/4} is a compound of two pentagrams 2{5/2}
 {10/2} is a compound of two pentagons 2{5}.^{[4]}^{[5]}
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} 
See also
References
 ^ Barnes, John (2012), Gems of Geometry, Springer, pp. 28–29, ISBN 9783642309649.
 ^ γραμμή, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus
 ^ Sarhangi, Reza (2012), "Polyhedral Modularity in a Special Class of Decagram Based Interlocking Star Polygons", Bridges 2012: Mathematics, Music, Art, Architecture, Culture (PDF), pp. 165–174.
 ^ Regular polytopes, p 9395, regular star polygons, regular star compounds
 ^ Coxeter, Introduction to Geometry, second edition, 2.8 Star polygons p.3638
 ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum.
 ^ *Coxeter, Harold Scott MacDonald; LonguetHiggins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 246 (916): 411. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 00804614. JSTOR 91532. MR 0062446.CS1 maint: ref=harv (link)
 ^ Coxeter, The Densities of the Regular polytopes I, p.43 If d is odd, the truncation of the polygon {p/q} is naturally {2n/d}. But if not, it consists of two coincident {n/(d/2)}'s; two, because each side arises from an original side and once from an original vertex. Thus the density of a polygon is unaltered by truncation.