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

The team believes that they’ve solved the problem of dimensions with

"Soft cells and the geometry of seashells" (2024) https://arxiv.org/abs/2402.04190 : 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-ge... https://www.aol.com/lifestyle/mathematicians-discovered-secret-geometry-life... : 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?