# References and Related Web Sites

This page is a annotated bibliography on 3 subjects: symmetry, plane tiling theory, plane pattern theory. Symmetry is the mathematical study of symmetry; this usually means group theory with geometry as focus. Tiling theory concerns tilings in the plane. (⁖ what tiles can tile a plane). Pattern theory concerns the classification of geometric patterns in the plane (⁖ pattern classifications, weavings, quasi-periodic patterns). The three subjects are related, but distinct. Among these are also aspects of computational geometry problems related to algorithms that plot patterns or weavings, or solve tiling problems. This is not a comprehensive list, but is fairly complete as a starting point for anyone interested in these subjects.

The majority of references cited here are accessible by undergraduate math majors. A large number of websites are at the level of high school students or nonprofessionals. Others are for mathematicians.

Both printed and web resources are included. It is worth noting that in general, the quality of printed publications are far superior than web resources. In particular, web resources lack depth and accuracy.

This bibliography is roughly grouped into the following 4 sections (click to jump to the section):

- Web Sites, Non-technical
- Web Sites, Technical
- Printed References, Non-technical
- Printed References, Technical

This bibliography contains over 30 titles. It may be overwhelming for beginning students. The following paragraphs will give a introduction to the literature for beginners.

An elementary understanding of group theory is either essential or extremely desirable to the study of symmetry or tilings or patterns. Thus, the first thing for serious students is to learn some group theory. If you are not that serious or do not wish to take a systematic learning approach, then you can visit some of the non-technical websites to get a feeling of the subjects first. When you starts to get the feeling that all the talk about symmetry, isometry…etc. are confusing and you are not getting a coherent picture, then it may be time for you to take the following advice.

Group theory is extremely abstract, and is usually taught in the 3rd year college to math majors. However, group theory itself doesn't have any prerequisites in the usual sense. It can be taught to grade level students. What you need is a curious mind and a strong desire to learn. It is often difficult to learn such abstract theory without a teacher. Because its advanced nature, there are very few books that introduce group theory to non-professionals. One of them is Groups and their Graphs amazon by Israel Grossman and Wilhelm Magnus. (1964) Another title you should read is sections 9 to 11 of Geometry and the Imagination amazon by David Hilbert and S. Cohn-vossen. (1932) Both books take a informal approach at high school level, and both are classics. It is by these two books that I (Xah Lee) learned group theory on my own. Once you've grasped the concepts of group theory, then you have many choices of where to go. You can get a deeper and formal understanding of group theory with emphasis to geometry, or you can start learning tiling theory or patterns theory.

There are too many books on group theory, since it is a very important and large topic in mathematics. You might start with a undergraduate exposition by yours truely at The Discontinuous Groups of Rotation and Translation in the Plane. This exposition ties some aspects of group theory and geometry in the plane. If your main interest is in tilings or patterns, then you should buy a copy of Tiling and Patterns amazon by B. Grunbaum and G. C. Shephard. This is the most significant book on this subject by far. It is a standard reference as of 2002, and probably will remain so for many years.

For software that draws tilings and patterns, please see Great Math Programs: Tilings, Patterns, Symmetry.

## Web Sites, Non-technical

- Main Page http://www.geom.umn.edu/docs/education/institute91/.
- More Paper-cutting Patterns.
- Symmetry and Orbifolds.
- Names for features of symmetrical patterns.
- The Orbifold Shop.
- Euler Characteristic of Orbifold.
- Positive and Negative Euler Characteristic.
- Explains John Conway's nomenclature of the 17 wallpaper groups, along with photocopy of Conway's manuscript. Orbifold Fieldtrip.

## Web Sites, Technical

- Main Page.
http://www.geom.umn.edu/docs/reference/CRC-formulas/book.html.
- Plane Symmetries or Isometries section.
- Wallpaper Groups section.

## Printed References, Non-technical

## Printed References, Technical

The following books I have not used, but are probably very valuable.

- Introduction to Symmetry (middle school level, one-page intro essay)
- Tiling and Patterns Image Gallery
- Selected software for Tilings, Patterns, Symmetry
- Plane Tiling Mathematica Package (software that generates planar symmetric images)