A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two-Dimensional Spaces and Some Simple Polyhedral, Planar and Three-Periodic Examples

Abstract

We analyse two-component “weavings” made of a pair of dual (p,q) and (q,p) nets that undulate on both sides of the sphere, the plane and the hyperbolic plane. Families of weavings are described, sharing a common parent net. The examples describe zero-, two- and three-periodic weavings in three-space. We derive all edge-2-transitive weavings with (p,q)=(3,3), (4,3), (4,4), (4,6) using Delaney-Dress tiling theory, described in detail. The two-dimensional hyperbolic weavings are mapped into (euclidean) three-space to form a pair of catenated crystalline nets. The examples suggest generalisations to other weavings on surfaces, including weavings of filaments. A simple hyperbolic weavings of filaments is derived, analogous to the common warp-andweft filament weaving in the plane. The resulting threeperiodic pattern is related to the molecular-scale weaving in the synthetic COF-505 material synthesized by Liu et al. [Science, vol. 351: 365–369, 2016].