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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.04190 |
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| _version_ | 1866908306255118336 |
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| author | Domokos, Gábor Goriely, Alain Horváth, Ákos G. Regős, Krisztina |
| author_facet | Domokos, Gábor Goriely, Alain Horváth, Ákos G. Regős, Krisztina |
| contents | 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, non-flat 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 \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Soft cells and the geometry of seashells Domokos, Gábor Goriely, Alain Horváth, Ákos G. Regős, Krisztina Applied Physics Metric Geometry 05B45 52C20 52C22 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, non-flat 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 \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{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. |
| title | Soft cells and the geometry of seashells |
| topic | Applied Physics Metric Geometry 05B45 52C20 52C22 |
| url | https://arxiv.org/abs/2402.04190 |