## About QuasiTiler

QuasiTiler is an application that generates quasiperiodic tiling of the plane
by projecting sections of higher-dimensional integer lattices onto the plane.
Particular sections of a 5-dimensional lattice are **Penrose tilings**.
The approach used by this program was first used by the dutch mathematician
**N.G. de Bruijn** in 1981. Also this program provides an interesting way to interact with higher-dimensional spaces.
Originally, QuasiTiler was an interactive application written
for NeXTSTEP, to conveniently generate Penrose tilings for Marjorie Senechal,
a visitor at the Geometry Center during November 1993.

## Introduction

Penrose tilings of the plane come in various guises. In one of them, all tiles are copies of two rhombi, one "thick" and one "thin". The edge's lengths of both rhombi are equal; their smaller angles are 2Pi/5 and Pi/5, respectively. Penrose showed that if certain "matching rules" are obeyed, then every tiling with these rhombi is nonperiodic. Although the rules admit uncountably many different tilings, any finite region of any one of them appears infinitely many times, not only in that tiling but also in each of the others! Penrose tilings also have the substitution property: the tiles in a tiling can be grouped together into larger tiles, and the ensemble of larger tiles is again a Penrose tiling.
In 1981 de Bruijn showed that every Penrose tiling by rhombi, and also its matching rules and substitution property, can be obtained by projecting a certain 2-dimensional surface S in 5-dimensional Euclidean space onto a 2-dimensional plane E.

The surface S is composed of congruent squares whose vertices are particular points of the 5-dimensional integer lattice. The Penrose tiles are the projections of these squares on E, and their vertices are the projections of the incident lattice points.

The 2-dimensional plane E is chosen such that:

- it is orthogonal to the body diagonal of the unit 5-dimensional cube,
- and each of the squares forming the surface project as one of the "thick" or "thin" rhombi.

One of the most fascinating aspects of de Bruijn's construction is that the properties of the Penrose tiling are encoded in the orthogonal projection of the unit 5-cube into the 3-dimensional space orthogonal to E! The projection of the unit 5-cube into 3-dimensional space is a rhombic icosahedron, which we will call K. The surface S is then determined by all the lattice points that projects inside K. We have that all the projections of the lattice points on the surface S are all contained in at most four planes parallel to E, which determine four pentagonal cross-sections of K.
What happens if we translate the projection K in the space orthogonal to E? We get different surfaces, each with its corresponding projection into E as a tiling. DeBruijn showed that if K is translated by a vector orthogonal to the diagonal W we get another Penrose tiling. But if it is translated by a vector parallel to W, we get a tiling that is not a Penrose tiling! It is easy to see this, because the tiling will contain configurations not permitted by the Penrose rules.

QuasiTiler implements a generalization of the above idea. You can generate different tilings by choosing the ambient dimension, the generating plane and the translation of the projection of the unit cell in the orthogonal space. Some will be periodic, but must of them are quasiperiodic; this depends if the "slope" of the plane is rational or irrational. Chances are that you will get a quasiperiodic tiling must of the time

## About this document

This document was written by Prof. Marjorie Senechal, Dept. of Mathematics, Smith College, Northampton, MA, and edited by Eugenio Durand. She is the original reason QuasiTiler was written; she required a convinient way of generating Penrose tilings during her visit to
the Geometry Center
in November 1993. She provided the expertise on Penrose tilings required to write the program.