A contribution for a mathematical classification of square tiles

TL;DR

This paper investigates the geometric properties of square tiles used in plane tilings with vertices forming a square lattice, classifying tiles with a rotation center of order 4 and identifying exceptions.

Contribution

It provides a classification of non-trivial square tiles with a 4-fold rotation center, identifying four main types and eight exceptions, inspired by a historical pattern tile.

Findings

Tiles belong to four main types with specific geometric properties.

Eight exceptional tiles do not follow the general classification.

The work is inspired by a 1966 pattern tile by Eduardo Nery.

Abstract

In this article we study some geometric properties of a non-trivial square tile (a non-trivial square tile is a non-constant function on a square). Consider infinitely many copies of this single square tile and cover the plane with them, without gaps and without overlaps (a tiling of the plane), with the vertices making a square point lattice. The question we ask ourselves in this article is the following: if there is a rotation center of order 4 what kind of geometric properties has the drawing in the tile? We prove that the tiles of this kind that one can see belong to one of four types. However, there are eight exceptions to these general rules. This work was inspired by a pattern tile, with remarkable properties, designed in 1966 by the portuguese artist Eduardo Nery. It can be seen in many beautiful panels in several locations in Portugal.