Star polygon

In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple and star polygons. Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons. The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram. One definition of a star polygon, used in turtle graphics, is a polygon having 2 or more turns (turning number and density), like in spirolaterals. Star polygon 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. For example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή (grammḗ) meaning a line. A "regular star polygon" is a self-intersecting, equilateral equiangular polygon. A regular star polygon is denoted by its Schläfli symbol {p/q}, where p (the number of vertices) and q (the density) are relatively prime (they share no factors) and q ≥ 2. The density of a polygon can also be called its turning number, the sum of the turn angles of all the vertices divided by 360°. The symmetry group of {n/k} is dihedral group Dn of order 2n, independent of k. Regular star polygons were first studied systematically by Thomas Bradwardine, and later Johannes Kepler. Regular star polygons can be created by connecting one vertex of a simple, regular, p-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. Alternatively 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.