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.
