Polygram (geometry)

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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[edit]

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 star polygon 5-2.svg
Regular star polygon 7-2.svg
Regular star polygon 7-3.svg
Regular star polygon 8-3.svg
Regular star polygon 9-2.svg
Regular star polygon 9-4.svg
Regular star polygon 10-3.svg

Regular compound polygons[edit]

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...
Regular star figure 2(3,1).svg
Regular star figure 3(3,1).svg
Regular star figure 4(3,1).svg
Regular star figure 2(4,1).svg
Regular star figure 3(4,1).svg
Regular star figure 2(5,1).svg
Regular star figure 2(5,2).svg
Regular star figure 3(5,2).svg

See also[edit]


  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.
  • Cromwell, P.; Polyhedra, CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. (1999), ISBN 0-521-66405-5. p.175
  • Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
  • Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)
  • Robert Lachlan, An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893, p. 83 polygrams.
  • Branko Grünbaum, Metamorphoses of polygons, published in The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994)