Category: Students

Soft cells, Kelvin foam and the minimal surfaces of Schwarz
Gábor Domokos; Alain Goriely; Ákos G. Horváth; & Krisztina Regős

Abstract: We study a class of geometric shapes termed soft cells tiling three-dimensional (3D) space without sharp corners. A special class of soft tilings, called standard soft tilings, can be obtained by an algorithm transforming any convex polyhedral tiling into at least one combinatorially equivalent soft tiling. Natural examples of such shapes include, among others, cell tissues, corals and chambers in nautilus shells. However, this construction leads to sharp, highly curved edges. Here, we generalize this construction to produce not just a single standard soft tiling but all soft tilings corresponding to a given polyhedral configuration. Unlike standard soft cells, these non-standard soft cells do not exhibit protruding edges. Notably, some non-standard soft cells are the fundamental building blocks within triply periodic minimal surfaces (TPMSs) such as Schwarz surfaces and gyroid structures, which are critical in modelling the nanoscale architecture of various polymers and carbon-based materials. These shapes also appear at the nanoscale as fundamental models of biological structures. Finally, we identify a family of intermediate space-filling cells that bridge two distinct soft-cell morphologies, providing a previously unrecognized connection between Schwarz surfaces and encompassing the Kelvin cell, a structure of enduring importance in materials science.

Calcitic coverings of the lower jaw of Jurassic ammonites exhibit Bouligand-like structures
Attila Demény, László Bujtor, Gábor Domokos, András A. Sipos, Csaba Farkas, Eszter Ferencz, Boglárka Topa, Máté Leskó & Norbert Zajzon

Abstract: Well-preserved layered structures have been reported for Jurassic ammonites’ aptychi, but the role, function, and origin of the lamination has been unclear. Here, we applied electron backscatter diffraction imaging to detect crystal fabric patterns in the laminated layer of Laevaptychi (calcitic coverings of the lower jaw of Aspidoceratoidea ammonites) and their diagenetic alteration, which was used to interpret variations in stable oxygen isotope compositions. We combined electron backscatter diffraction imaging with discrete and combinatorial geometry to model the calcite crystal deposition patterns. A new probabilistic approach was developed to gain a fundamental 3D description of the tiling based on information from 2D sections. The analyses revealed that the lamination results from variable deposition of elongated calcite crystals arranged in rotating bundles, which may be interpreted as a Bouligand-like structure. Our finding is one of the earliest occurrences of Bouligand-like structure in non-chitinous, carbonate-based organs of molluscs.

Soft cells and the geometry of seashells
Gábor Domokos, Alain Goriely, Ákos G. Horváth, Krisztina Regős.

Abstract: A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, nonflat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the soft cells, minimizing the number of sharp corners and filling space as soft tilings. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet–Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells.