Enumerating tilings of triply-periodic minimal surfaces with rotational symmetries

International audienceWe present a technique for the enumeration of all isotopically distinct ways of tiling, with disks, a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group of the tiling under consideration as a set of algebraic operations on certain generators of the symmetry group. We derive explicit descriptions of certain subgroups of mapping class groups and of tilings as embedded graphs on orbifolds. We further use this explicit description to present an algorithm that we illustrate by producing an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the prominent Primitive, Diamond and Gyroid triply-periodic minimal surfaces, outlining how the approach yields an unambiguous enumeration. We also present the corresponding 3-periodic graphs on these surfaces

Three-dimensional entanglement: knots, knits and nets

Three-dimensional entanglement, including knots, periodic arrays of woven filaments (weavings) and periodic arrays of interpenetrating networks (nets), forms an integral part of the analysis of structure within the natural sciences. This thesis constructs a catalogue of 3-periodic entanglements via a scaffold of Triply-Periodic Minimal Surfaces (TPMS). The two-dimensional Hyperbolic plane can be wrapped over a TPMS in much the same way as the two-dimensional Euclidean plane can be wrapped over a cylinder. Thus vertices and edges of free tilings of the Hyperbolic plane, which are tilings by tiles of infinite size, can be wrapped over a TPMS to represent vertices and edges of an array in three-dimensional Euclidean space. In doing this, we harness the simplicity of a two-dimensional surface as compared with 3D space to build our catalogue. We numerically tighten these entangled flexible knits and nets to an ideal conformation that minimises the ratio of edge (or filament) length to diameter. To enable the tightening of periodic entanglements which may contain vertices, we extend the Shrink-On-No-Overlaps algorithm, a simple and fast algorithm for tightening finite knots and links. The ideal geometry of 3-periodic weavings found through the tightening process exposes an interesting physical property: Dilatancy. The cooperative straightening of the filaments with a fixed diameter induces an expansion of the material accompanied with an increase in the free volume of the material. Further, we predict a dilatant rod packing as the structure of the keratin matrix in the corneocytes of mammalian skin, where the dilatant property of the matrix allows the skin to maintain structural integrity while experiencing a large expansion during the uptake of water

Towards a combinatorial algorithm for the enumeration of isotopy classes of symmetric cellular embeddings of graphs on hyperbolic surfaces

Based on the recent mathematical theory of isotopic tilings, we present the, to the best of our knowledge, first algorithm for the enumeration of isotopy classes of cellular embeddings of graphs invariant under a given symmetry group on hyperbolic surfaces. To achieve this, we substitute the isotopy classes with combinatorial objects and propose different techniques, guided by structural results on the mapping class group of an orbifold and notions from computational group theory that ensure that the algorithm is computationally tractable. Furthermore, we extend data structures of combinatorial tiling theory to isotopy classes that lead to an actual implementation of the algorithm for symmetry groups generated by rotations. \\From the enumerated combinatorial objects, we produce a range of simple graphs on hyperbolic surfaces represented as symmetric tilings in the hyperbolic plane, illustrating the enumeration with examples and experimentally demonstrating the feasibility of the approach. These tilings are finally projected onto a family of triply-periodic surfaces that are relevant for the natural sciences