Soft tilings

By means of constructing a new edge-bending algorithm, we prove that every locally polyhedral tiling of ${\mathbb{R}}^3$ can be completely softened. A weaker form of this statement, for polyhedral space tilings, was conjectured by Domokos, Goriely, G. Horváth and Regős in 2024. We also provide a short proof for a result of Domokos, G. Horváth, and Regős, stating that in a balanced polygonic tiling of the plane, the average number of spikes is at least 2 per cell.