Polygram (geometry)

Regular polygrams {n/d}, with red lines showing constant d, and blue lines showing compound sequences k{n/d}

In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides, so a regular pentagram, {5/2}, has 5 sides, and the regular hexagram, {6/2} or 2{3} has 6 sides divided into two triangles.

A regular polygram {p/q} can either be in a set of regular polygons (for gcd(p,q)=1, q>1) or in a set of regular polygon compounds (if gcd(p,q)>1).[1]


The polygram names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The -gram suffix derives from γραμμῆς (grammos) meaning a line.[2]

Generalized regular polygons

A regular polygram, as a general regular polygon, is denoted by its Schläfli symbol {p/q}, where p and q are relatively prime (they share no factors) and q  2. For integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement.[3][4]








Regular compound polygons

In other cases where n and m have a common factor, a polygram is interpreted as a lower polygon, {n/k,m/k}, with k = gcd(n,m), and rotated copies are combined as a compound polygon. These figures are called regular compound polygons.

Some regular polygon compounds
Triangles... Squares... Pentagons... Pentagrams...









See also


  1. Weisstein, Eric W. "Polygram". MathWorld.
  2. γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  3. Coxeter, Harold Scott Macdonald (1973). Regular polytopes. Courier Dover Publications. p. 93. ISBN 978-0-486-61480-9.
  4. Weisstein, Eric W. "Polygram". MathWorld.
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