Voronoi Diagrams Generated by the Archimedes Spiral

TL;DR

This paper explores Voronoi diagrams generated from seed points on Archimedes Spirals, analyzing their entropy, cell shapes, and aesthetic qualities, with implications for decorative arts and pattern design.

Contribution

It introduces a novel method of generating Voronoi mosaics from spiral-based seed points and analyzes their structural and aesthetic properties.

Findings

Hexagonal cells are predominant in equidistant and non-equidistant spiral patterns.

Voronoi entropy varies with the number of seed points and pattern type.

Ordered patterns can have Voronoi entropy exceeding 1.71, higher than random distributions.

Abstract

Voronoi mosaics inspired by the seed points placed on the Archimedes Spirals are reported. Voronoi entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi mosaics built from cells of equal size which are of primary importance for 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 the seed points located on the Archimedes Spirals is reported. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed.