Tour of Symmetry Groups

This document is being developed to provide a guided tour of symmetry groups using the Kali program. Currently, only frieze groups are covered by the tour.

How to use this document

The pages in this document may be viewed with a WWW browser but may be more useful if printed out in PostScript from a WWW browser.

A PostScript version of each section will eventually be provided to avoid the hassle of having to save or print each page as PostScript from the WWW browser.

The first two sections, Types of Symmetry and Symmetry in Frieze Groups, cover information that students should be familiar with before they come into the Geometry Center for a tour. Teachers may use the materials provided here or other equivalent material in their classrooms before bringing the students into the Geometry Center, so that the visit will be more productive.

The third section, Using Kali to Explore Frieze Groups, contains a bunch of exercises that students may do while using the Kali program at the Geometry Center. The exercises are divided into groups based on their level of difficulty: beginner, intermediate and advanced. The last set of exercises, the Pattern Gallery, contains exercises in all ranges of difficulty.

The fourth section contains a very brief mention of symmetries in the plane. This section may or may not be developed into exercises at a later date.


  1. Types of Symmetry

  2. Symmetry in Frieze Groups

  3. Using Kali to Explore Frieze Groups

  4. Symmetry in Wallpaper Groups

Lori Thomson<>