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Soft cells and the geometry of seashells
Gábor Domokos1,2, Alain Goriely3, Ákos G Horváth2,4
1Department of Morphology and Geometric Modeling, Budapest University of Technology and Economics, Budapest, 1111, Hungary.
PNAS Nexus
|September 11, 2024
Summary
This study introduces "soft cells" and "soft tilings" to bridge geometric shapes and natural forms. These new shapes smoothly deform from traditional tilings, appearing in nature from cells to shells.
Area of Science:
- Geometry
- Mathematical Physics
- Natural Sciences
Background:
- Classical geometry relies on sharp-cornered shapes like polygons and polyhedra for space tiling.
- Natural formations often exhibit curved edges and non-flat faces, deviating from ideal geometric tilings.
Purpose of the Study:
- To connect classical sharp tilings with softer, naturally occurring shapes.
- To introduce and define a new class of geometric shapes called "soft cells" and their spatial arrangements, "soft tilings".
Main Methods:
- Developing a mathematical framework to smoothly deform polyhedral tilings into soft tilings.
- Constructing soft versions of Dirichlet-Voronoi cells for point lattices in 2D and 3D.
Main Results:
- Proving that an infinite class of polyhedral tilings can be continuously transformed into soft tilings.
- Demonstrating the existence and construction of soft tilings derived from point lattices.
Conclusions:
- Soft cells and soft tilings provide a geometric model for understanding natural shapes with curved boundaries.
- These idealized soft shapes, derived from geometric principles, are prevalent in various natural structures, including biological cells and shells.
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