Soft tilings

TL;DR

This paper introduces a new edge-bending algorithm to prove that all locally polyhedral tilings of three-dimensional space can be fully softened, confirming a conjecture and providing new insights into tiling structures.

Contribution

It presents a novel edge-bending algorithm to prove the softening of locally polyhedral tilings and offers a short proof for a related planar tiling result.

Findings

Every locally polyhedral tiling of R^3 can be completely softened.

A weaker form of the conjecture for polyhedral space tilings is proven.

In balanced planar tilings, the average number of spikes per cell is at least 2.

Abstract

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. Horv\'ath and Reg\H{o}s in 2024. We also provide a short proof for a result of Domokos, G. Horv\'ath, and Reg\H{o}s, stating that in a balanced polygonic tiling of the plane, the average number of spikes is at least 2 per cell.