Softening locally polyhedral tilings

Abstract

We call a cell $C\subset {\mathbb{R}}^d$ soft if every point of its boundary lies on a smooth curve contained in $\partial C$. A tiling of the space is called completely soft if all of its cells are soft. In their 2024 article, Domokos, Goriely, G. Horváth and Regős conjectured that every polyhedral tiling of ${\mathbb{R}}^3$ satisfying mild regularity assumptions can be locally deformed into a completely soft tiling. By constructing an algorithm that first bends the edges emanating from each vertex and then extends this transformation to a sufficiently smooth deformation, they proved the conjecture for polyhedral tilings satisfying a certain combinatorial condition. In the present paper, we precisely describe a new edge-bending algorithm that establishes a more general version of this conjecture: every locally polyhedral tiling of ${\mathbb{R}}^3$ can be completely softened. We also give a short proof of an earlier result of Domokos, G. Horváth, and Regős stating that every suitably nondegenerate polygonic tiling of the plane has, on average, at least two points per cell at which the softness criterion is violated.

23 pages; revised version