Spirolateral

Simple spirolaterals

390° (4 cycles)	3108° (5 cycles)	990° ccw spiral	990° (4 cycles)
100120° spiral		100120° (4 cycles)

In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,...,n which repeat until the figure closes. The number of repeats needed is called its cycles.^[1] A simple spirolateral has only positive angles. A simple spiral approximates of a portion of an archimedean spiral. A general spirolateral allows positive and negative angles.

A spirolateral which completes in one turn is a simple polygon, while requiring more than 1 turn is a star polygon and must be self-crossing.^[2] A simple spirolateral can be an equangular simple polygon <p> with p vertices, or an equiangular star polygon <p/q> with p vertices and q turns.

Spirolaterals were invented and named by Frank C. Odds as a teenager in 1962, as square spirolaterals with 90° angles, drawn on graph paper. In 1970, Odds discovered triangular and hexagonal spirolateral, with 60° and 120° angles, can be drawn on isometric^[3] (triangular) graph paper.^[4] Odds wrote to Martin Gardner who encouraged him to publish the results in Mathematics Teacher^[5] in 1973.^[3]

The process can be represented in turtle graphics, alternating turn angle and move forward instructions, but limiting the turn to a fixed rational angle.^[2]

The smallest golygon is a spirolateral, 7_90°⁴, made with 7 right angles, and length 4 follow concave turns. Golygons are different in that they must close with a single sequence 1,2,3,..n, while a spirolateral will repeat that sequence until it closes.