Tilings with nonflat squares: a characterization

TL;DR

This paper characterizes the arrangements of nonflat squares in three dimensions, revealing that their geometric patterns are periodic and one-dimensional, and that nonflatness leads to complex bending and wrinkling behaviors.

Contribution

It provides a complete geometric characterization of nonflat square arrangements, highlighting the impact of nonflatness on their structural configurations.

Findings

Arrangements are periodic and one-dimensional.

Nonflatness induces bending, wrinkling, and rolling.

The geometry is fully characterized by mutual orientation patterns.

Abstract

Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual orientations of the squares and that these patterns are periodic and one-dimensional. In contrast to the flat case, the nonflatness of the tiles gives rise to nontrivial geometries, with configurations bending, wrinkling, or even rolling up in one direction.