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} Regular Star Polygons-en.svg
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.

Contents

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). [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 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] [1]

Regular star polygon 5-2.svg
{5/2}
Regular star polygon 7-2.svg
{7/2}
Regular star polygon 7-3.svg
{7/3}
Regular star polygon 8-3.svg
{8/3}
Regular star polygon 9-2.svg
{9/2}
Regular star polygon 9-4.svg
{9/4}
Regular star polygon 10-3.svg
{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...
Regular star figure 2(3,1).svg
{6/2}=2{3}
Regular star figure 3(3,1).svg
{9/3}=3{3}
Regular star figure 4(3,1).svg
{12/4}=4{3}
Regular star figure 2(4,1).svg
{8/2}=2{4}
Regular star figure 3(4,1).svg
{12/3}=3{4}
Regular star figure 2(5,1).svg
{10/2}=2{5}
Regular star figure 2(5,2).svg
{10/4}=2{5/2}
Regular star figure 3(5,2).svg
{15/6}=3{5/2}

See also

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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.