Description
We call a cell C soft if every point of its boundary lies on a smooth curve contained in the boundary of 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 the 3-space satisfying mild regularity assumptions can be locally deformed into a completely soft tiling, and proved the conjecture for polyhedral tilings satisfying a certain combinatorial condition. By means of designing a new edge-bending algorithm, we prove and generalize the conjecture to locally polyhedral tilings. We also give a short proof for the earlier result 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.
This is a joint work with Dorottya Dancso.