Tort-Decó: a “turtle geometry”-based package for drawing periodic designs

Abstract

We have developed a package called Tort-Decó. The aim of this project was to construct a package for drawing periodic tilings by computer and an AI aid for wallpaper group and minimal generator regions recognition. The three topics covered are cyclic and dihedral groups, freezes, and wallpaper groups. There are several packages allowing users to apply movements to a subset of the plane. Some of them are specifically designed to draw periodic designs (RepTyles, Tesselmania!, Kali …). However, we know of no exhaustive approach based upon the use of “Turtle Graphics”. This approach evolved from [1] (where the motifs were fixed) and makes it possible to describe and store the periodic designs in an extremely short form. Moreover, the strong hierarchy of the consecutive steps in our approach makes the change of group clear. Finally, the classification expert system for wallpaper groups included is a novelty. It is well known that wallpaper group and minimal generator regions recognition is an algorithmic problem. Nevertheless, we have found that our students and nonmathematicians (like geologists) have serious difficulties in using these algorithms. Therefore, we decided to construct an AI aid to guide users in their search (in a similar way to expert systems designed to repair car and locomotive breakdowns).