Spirolateral

Spirolaterals are a class of polygons, mostly self-intersecting, that generally have a single internal angle and a sequence of edge lengths formed from repetitions of the sequence 1, 2, ..., n.

Simple spirolateral
A simple spirolateral is constructed as follows: draw a 1-unit line segment, turn &phi; degrees clockwise, draw a 2-unit line segment, turn &phi; degrees clockwise, ... draw an n-unit line segment, turn &phi; degrees clockwise, repeat from the beginning. The process is immediately halted when one returns to the original point, forming a closed polygon that is usually self-intersecting. Defining &theta; = 180° - &phi; as the internal angle, every simple spirolateral is uniquely identified by n and &theta;, and notated as n&theta;. The regular polygons can be obtained by setting n=1.

General spirolateral
A general spirolateral allows turns of a single angle in either direction, provided that the directions of the turns repeat every n turns. That is, the turn directions are determined by a word of length n where each symbol is "counterclockwise" or "clockwise."

General spirolaterals are notated n&theta;a1, ..., ak where 1 &le; ai &le; n. For each ai, the aith turn is counterclockwise rather than clockwise, and all other turns are clockwise.

Classification
If a spirolateral closes after exactly n steps, it is an unexpected closed spirolateral. Golygons are a special case of unexpected closed general spirolaterals where &theta; = 90°.