Three-periodic nets and tilings: minimal nets

Abstract

The 15 3-periodic minimal nets of Beukemann & Klee [Z. Kristallogr. (1992), 201, 37-51] have been examined. Seven have collisions in barycentric coordinates and are self-entangled. The other eight have natural tilings. Five of these tilings are self-dual and the nets are the labyrinth nets of the P, G, D, H and CLP minimal surfaces of genus 3. Twelve ways have been found for subdividing a cube into smaller tiles without introducing new vertices. Duals of such tilings with one vertex in the primitive cell have nets that are one of the minimal nets. Minimal nets without collisions are uniform.