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It is well established that crystalline 3-periodic nets can be constructed and systematically catalogued using Delaney-Dress tiling theory. One approach does this by enumerating two-dimensional hyperbolic (
) tilings and then projects those patterns into three-dimensional Euclidean space (
) via triply periodic minimal surfaces. We extend this to an investigation of three-dimensional patterns that emerge from a the systematic enumeration of ribbon-like free tilings of .
) tilings and then projects those patterns into three-dimensional Euclidean space (
) via triply periodic minimal surfaces. We extend this to an investigation of three-dimensional patterns that emerge from a the systematic enumeration of ribbon-like free tilings of .research papers
In this article, rod packings are generalized to include arrays of more general one-dimensional curvilinear cylinders. Examples are built by projecting free tilings of two-dimensional hyperbolic space into three-dimensional Euclidean space via genus-3 triply periodic minimal surfaces, forming three-dimensional weavings of filaments, which are tightened to a canonical form.
research papers
It is shown that the 15 minimal nets can serve as the labyrinth graph of one or other of the five known minimal surfaces of genus 3 and that the enumeration of such surfaces is likely to be complete.
research papers
Examples are given of nets describing the topology of real crystal structures in which groups of vertices collide in barycentric coordinates and in high-symmetry embeddings.
