Voronoi Diagrams Generated by the Archimedes Spiral: Fibonacci Numbers, Chirality and Aesthetic Appeal

Abstract

Voronoi mosaics inspired by seed points placed on the Archimedes Spirals are reported. Voronoi (Shannon) entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi tessellations generated by the seeds located on the Archimedes spiral and separated by linearly growing radial distance demonstrate a switch in their chirality. Voronoi mosaics built from cells of equal size, which are of primary importance for the decorative arts, are reported. The pronounced prevalence of hexagons is inherent for the patterns with an equidistant and non-equidistant distribution of points when the distance between the seed points is of the same order of magnitude as the distance between the turns of the spiral. Penta- and heptagonal “defected” cells appeared in the Voronoi diagrams due to the finite nature of the pattern. The ordered Voronoi tessellations demonstrating the Voronoi entropy larger than 1.71, reported for the random 2D distribution of points, were revealed. The dependence of the Voronoi entropy on the total number of seed points located on the Archimedes Spirals is reported. Voronoi tessellations generated by the phyllotaxis-inspired patterns are addressed. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed.