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]

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][4]

 {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}