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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.
