"Soft cells and the geometry of seashells" (2024)
https://arxiv.org/abs/2402.04190 :
> 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.
Schema:NewsArticles about said schema.org/ScholarlyArticle:
-
https://www.popularmechanics.com/science/math/a46973545/soft-cells-secret-geometry-of-life/
https://www.aol.com/lifestyle/mathematicians-discovered-secret-geometry-life-174500998.html
:
> The team believes that they’ve solved the problem of dimensions with
this new “infinite class of polyhedral tilings” that can smoothly deform
into soft tiles and construct soft versions of cells generally associated
with point lattices in both two and three dimensions. [...]
>
> In two dimensions, these soft shell shapes are pretty easy to
describe—according to the paper, they are “cells with curved boundaries
which have only two corners.” In the three-dimensional space, things get a
little more complicated, but the goal is the same: let things be bendy and
minimize the amount of “corners” present. In 3D, a soft cell shape can have
no corners at all.
>
> “We found that architects have found these kinds of shapes intuitively
when they wanted to avoid corners,” Domokos said.
A math thing to be Python'd.
What [Python,] geometry software could do or does 2D, 3D, and N-D infinite
polyhedral tilings like this?
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