Curves in the Sand: Algorithmic Drawing

The field of ethnomathematics is the study of mathematics in the works of art of various cultures [2, 3, 9, 13]. The concepts in this paper are inspired by the visual art of sand drawings that has developed independently in different forms in diverse cultures. Generally speaking, the artist draws a set of dots on some flat surface (usually in the sand or in powder on the floor) and then draws one continuous curve that surrounds the dots and crosses itself repeatedly. Although not universally the case, we focus on drawings in which there is exactly one dot per bounded face (and no dots in the outside face). In particular, sand drawings made by the Tshokwe people in the West Central Bantu area of Africa are called sona. Sona drawings have been considered in the field of topology under an equivalent guise as generic planar closed curves (immersions of the unit circle into the plane). Several topological invariants about such curves are proved by Arnol′d [1], who also enumerated all sona drawings on small numbers of dots. Calvarho [5] considers curves that are “maximally looped”. Ozawa [11] considers the number of bitangents, tangents shared by different points on the curve. From a graph-theoretic perspective, sona drawings can be viewed as 4-regular planar maps with the additional property that some Eulerian cycle “goes straight” at every vertex. This class of underlying graphs is called Gaussian graphs; the name is attributed to an observation made by Carl Gauss in 1830 [8] that was proved by Julius v. Sz. Nagy almost a hundred years later [12]. More recently, Michael Gargano and John Kennedy [7] introduced a more formal notion of Gaussian graphs, which was later generalized by John Kennedy and Brigitte and Herman Servatius [10]. The connection between Gaussian graphs, generic closed curves, and sona