Polygram (geometry)

In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but 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 "γραμμῆς" ("grammos") 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 "q"th point out of "p" points regularly spaced in a circular placement.
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.