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. (For example, what tiles can tile a plane). Pattern theory concerns the classification of geometric patterns in the plane (For example, 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):

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 Buy at amazon by Israel Grossman and Wilhelm Magnus. (1964) Another title you should read is sections 9 to 11 of Geometry and the Imagination Buy at 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 Buy at 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

Geometry Center. (University of Minnesota)
Tiling and Symmetry
High school level discussion of tiling and symmetry. Very detailed and full of illustrations. Estimated: probably more than 40 pages. ★
Dror Bar-Natan'
A photographic exhibition of symmetry. Beautifully done. ★
Alok Bhushan, Kendrick Kay, Eleanor Williams
Totally Tessellated
A short discussion on tilings by polygons. Other parts of the site touchs on related history and art. This site is huge with over 100 dynamic pages and quality illustrations, photographs, and links. Superbly well done. ★
Carol Bier and Melissa June Dershewitz
Symmetry and Pattern, The Art of Oriental Carpets
A non-technical discussion and gallery on plane symmetry and oriental carpets. Complete with photographs of oriental rugs. Estimated: more than 50 pages. ★
Hop David
17 Wallpaper Groups
Illustration of the 17 wallpapers. Particular noteworthy is the animations showing every symmetry of any of the 17 pattern. ★
Chaim Goodman-Strauss
Symmetry and the Shape of Space
High school level intro to symmetry and shape of space. In depth discussion. Estimated: more than 50 pages. ★
URL:, 2007-02-28
Heidi Burgiel and others
Symmetry and Patterns
Instruction based introduction to symmetry and orbifolds. This is probably the best site to learn about orbifolds. About 30 pages
URL:, 2007-02-28.
John Conway, Peter Doyle, Jane Gilman, and Bill Thurston
Geometry and the Imagination
Lecture notes of a summer workshop. There are several pages relevant to wallpaper groups. Their instructions are childishly simple but the ideas are very advanced. The notes are chaotically organized and scanty. If you do read them, you should read the following selections in order, and carefully.
Tohsuke Urabe
Math Museum
Example of the 17 wallpaper groups in Japanese culture, plus about 5 pages of explanation on the math.
URL:, 2007-02-28
Allan Bergmann Jensen
Symmetries, patterns and tesselations with Geometers Sketchpad
Downloadable Geometer's Sketchpad files for the 17 wallpaper patterns. About 7 pages.
URL:, 2007-02-28
Suzanne Alejandre
Tessellation Tutorials
A grade school level activity oriented tutorial on tesselation. Estimated: 30+ pages.
URL:, 2007-02-28
Steve Edwards
Tiling Plane and Fancy
This site focuses on tilings. It is a brief informal intro to tilings, enough to whet your appitite. The sites contains tilings form diffirent cultures, and a slide show of several tilings demostrating important properties. Dozens of small pages.
URL:, 2007-02-28

Web Sites, Technical

Wallpaper groups
A excellent encyclopedic article. ★
David Joyce
Wallpaper Groups
Undergraduate level mathematical exposition on wallpaper groups. Includes illustration, table, short history, and bibliography. The history section is particularly worth reading. About 25 pages.
Silvio Levy, CRC Press staff
Geometry Formulas and Facts
A excerpt of CRC Press' reference work, including sections on plane symmetry and wallpaper groups. The latter explains the orbifold notation.
David Eppstein
Geometry Junkyard
This site collects all web resources on symmetry and tilings, as well as other geometry related topics.

Printed References, Non-technical

Visions of Symmetry Buy at amazon
Doris Schattschneider
A coffee-table Book. This 28cmx22cm book contains a comprehensive collection of artist M. C. Escher's art work on plane symmetry. This book is printed on glossy paper and full color. Both hardcover and paper back are available.
Symmetries of Islamic Geometrical Patterns Buy at amazon
Syed Jan Abas, Amer Shaker Salman
World Scientific
This book is written for the general reader and is non-mathematical. It contains a collection of about 250 Islamic patterns, perhaps the most complete collection in print. The author Syed Jan Abas has a home pageat:
Symmetry (a unifying concept) Buy at amazon
Istvan Hargittai, Magdolna Hargittai
Shelter Pub.
$19. A picture book. A collection of photographs of symmetric objects: buildings, decorations, visual arts, sculptures, designs, …etc. A table-top book suitable for all ages. Part of the book is on-line at the publisher's web site at
Parquet deformations: patterns of tiles that shift gradually in one dimension
Douglas R. Hofstadter
Journal: Scientific American 1983/07, p.14-20.
A 6-page commentary on a special form of decorative pattern pioneered by architect William S. Huff. The boundary of the pattern are long rectangular shaped (a long strip), and the pattern gradually transforms from one end to the other. The most famous example is artist M. C. Escher's Metamorhposis III. Huff's patterns are purely geometrical. The article includes 13 examples of such patterns with comments by the author on some of them. The author also touchs on the possibility of formalization of creativity. I have scanned these articles.

Printed References, Technical

Geometry and the Imagination Buy at amazon
David Hilbert, S. Cohn-vossen
Chelsea Pub. Co.,
A classic introduction to geometry in prose style. Topics include: conic sections, regular system of points, projective geometry, differential geometry, kinematics, and topology. You may want to read section 9 to 11 (about 25 pages), which basically cover the same thing as this site. Perhaps easier to follow. ★
Symmetries Buy at amazon
D. L. Johnson
Essentially group theory in geometry. Table of Contents: Preface.- 1. Metric Spaces and their Groups.- 2. Isometries of the Plane.- 3. Some Basic Group Theory.- 4. Products of Reflections.- 5. Generators and Reflections.- 6. Discrete Subgroups of the Euclidean Group.- 7. Plane Crystallographic Groups: OP Case.- 8. Plane Crystallographic Groups: OR Case.- 9. Tessellations of the Plane.- 10. Tessellations of the Sphere.- 11. Triangle Groups.- 12. Regular Polytopes.- Solutions to Exercises.- Guide to the Literature.- Index of Notation.- Index.
The Plane Symmetry Groups: Their recognition and notation
Journal: American Math. Monthly. vol 85, pp. 439-450.
Doris Schattschneider
American Math Society
A 10 page introduction to wallpaper groups. Includes figures of group's symmetries, generators, examples of 17 wallpapers, and a table comparing 7 notations used up to 1978. This article is not technical. The main value of this article is the table of past notations. I have scanned these articles.
The orbifold notation for surface groups
J. H. Conway
Cambridge Univ. Press
Notes: Series: Groups, combinatorics and geometry
The first publication of the orbifold notation by its inventor. About 10 pages. I have scanned these articles. See wikipedia: Conway's orbifold notation
Generators and Relations for Discrete Groups Buy at amazon
H.S.M. Coxeter, W.O.J. Moser
Springer-Verlag, N.Y.
A standard reference. It contains discussions and Cayley diagrams for the 17 wallpaper groups.
Regular Polytopes Buy at amazon
H.S.M. Coxeter
Notes: 3rd ed.
A standard reference. The book is mostly focused on polytopes. (polytope is the general name for regular solids of any dimension.) Several sections give a brief but significant discussion on symmetry groups.
Tiling and Patterns Buy at amazon
B. Grunbaum, G. C. Shephard
Tiling, Pattern
This is a comprehensive monograph on all mathematical aspects of tilings and patterns of the plane, treated in a systematic fashion. (However, it does not discuss groups.) The math here is rigorous, but is lively written and accessible to undergraduate. Half of the book comprise well-done illustrations. For example, in this book you'll find a database of tilings and patterns by various classifications schemes, all illustrated. This is a standard reference on the topic, and no other book approaches its scope or depth. This will probably remain so a few years after 2000. Prerequisite: To fully appreciate the book, one must be familiar with group theory and have math maturity of that level. Anyone can also benefit from the book by its vast illustrations. Some sections are suitable for nonprofessionals.
Symmetry in Moorish and Other Ornaments
Branko Grunbaum, Zdenka Grunbaum, G. C. Shephard
Journal: Comp. and Maths. with appls., Vol. 12B, Nos. 3/4. pp. 641-653, 1986
A 10 page article, about 4 pages are photos and illustrations. This article is non-mathematical. It look at traditional Moorish patterns through mathematics. For example, it discusses what symmetry patterns are found, how to analize them mathematically, and problems in analyzing them with symmetry. This is a good article. Prerequisite: group theory.
Interlace Patterns in Islamic and Moorish Art
Branko Grunbaum, G. C. Shephard
Journal: Leonardo, vol. 25, No. 3/4, pp. 331-339, 1992
A 9 page article, about 4 pages are illustrations. This article is mathematical. It gives a non-trivial theory in dealing with counting the number of strands (threads) and crossings in traditional Islamic/Moorish patterns. (decorative weavings) This is a must read for anyone interested in mathematical analysis of Islamic/Moorish patterns. Excellent. Prerequisite: group theory.

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

Chinese Lattice Designs Buy at amazon
Daniel S. Dye
Probably a ordinary collection of Chinese patterns. This book is listed in Doris Schattschneider's 1978 AMS article bibliography.
Symmetry Buy at amazon
Hermann Weyl
Princeton U. Press
A not very technical book on symmetry. Some group concept is discussed. Contain many illustration of symmetric plants or photographs of buildings…etc. A fine book. (1997/07)
Groups, A Path to Geometry Buy at amazon
R. P. Burn
Cambridge Univ. Press
$30. Paperback. It appears to be a excellent book. It's a upper division undergraduate level text on group theory and geometry. The book culminates in deriving all wallpaper groups. (1997/07)
Numbers and Symmetry (An intro to algebra) Buy at amazon
Bernard L. Johnston, Fred Richman
CRC Press Inc
lower division undergraduate
$30. Paperback. Informal college level text. It appears to be good book. p.149-171 discusses wallpaper groups. (Xah, 1997/07)
Sphere packings, lattices, and groups Buy at amazon
John Horton Conway
Springer-Verlag, New York
1998, 3rd ed.
Probably a standard reference on the subject. 3rd edition published on 1998.

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