Curves in the Sand: Algorithmic Drawing

1 IntroductionEthnomathematics is the study of mathematics in the worksof art of various cultures [3, 4, 10, 14]. The concepts in thispaper are inspired by the visual art of sand drawings that hasdeveloped independently in different forms in diverse cul-tures. Generally speaking, the artist draws a set of dots onsome flat surface (usually in the sand or in powder on thefloor) and then draws one continuous curve that surroundsthe dots and crosses itself repeatedly. Although not univer-sally the case, we focus on drawings in which there is exactlyone dot per bounded face (and no dots in the outside face).In particular, sand drawings made by the Tshokwe people inthe West Central Bantu area of Africa are called sona.Sona drawings have been considered in the field of topol-ogy under an equivalent guise as generic planar closedcurves (immersions of the unit circle into the plane). Sev-eral topological invariants about such curves are proved byArnol