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. All polygons are polygrams, but they can also include disconnected sets of edges, called a compound polygon. For example, 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 star polygons (for gcd(p,q) = 1, q > 1) or in a set of regular polygon compounds (if gcd(p,q) > 1).

Etymology

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 γραμμή (grammē) meaning a line.

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.


{5/2}
{7/2}
{7/3}
{8/3}
{9/2}
{9/4}
{10/3}...

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...

{6/2}=2{3}
{9/3}=3{3}
{12/4}=4{3}
{8/2}=2{4}
{12/3}=3{4}
{10/2}=2{5}
{10/4}=2{5/2}
{15/6}=3{5/2}

See also

References

  1. 1 2 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.