Soft cells and the geometry of seashells
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 a...
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| Published in: | PNAS nexus Vol. 3; no. 9; p. pgae311 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
US
Oxford University Press
01.09.2024
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| Subjects: | |
| ISSN: | 2752-6542, 2752-6542 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 Competing Interest: The authors declare no competing interest. |
| ISSN: | 2752-6542 2752-6542 |
| DOI: | 10.1093/pnasnexus/pgae311 |