Group Theory of Wallpaper Patterns, an Epilogue



Back to
Tohsuke Urabe Laboratory 		Back to the beginning of the group theory

Why do only seventeen cases of symmetry occur in the previous table? Perhaps, you can understand the basic reason using the lemma below.

Lemma
Under the assumption of the conditions (1) and (2), G contains

  1. only rotations of 60, 90, 120 or 180 degrees.
  2. a translation parallel to the axis of the reflection for every reflection. Thus, every axis of a reflection is also an axis of a glide reflection.
  3. a translation perpendicular to the axis of the reflection or the glide reflection for every reflection and for every glide reflection.
To understand wallpaper groups, we recommend you to demonstrate the figures of each of the seventeen pattern types for each of the following: (1) all axes of reflections, (2) all axes of glide reflections, and (3) all centers of rotations for each value of angles 60, 90, 120, and 180 degrees. It may be hard to show all seventeen patterns, but it becomes interesting to find the unexpected axes of glide reflections and unexpected centers of rotations after confirming them for several cases.

A long time ago, people may have noticed there were seventeen ways to repeat patterns. We have several opinions on when and by whom the first mathematical verification was given. One of the oldest was given by Ergraf Stepanovic Fedorov in 1891. It may be true that since then, we have had repeated re-discoveries.

We have used the word "group" without any explanation. If a set with a multiplication defined satisfies three axioms: the associative law, the existence of a unit element, and the existence of an inverse element, then we call this set a group. The concept of groups plays an essential role in modern mathematics. It seems the theory of wallpaper groups explained here represents a part of the profundity of group theory.

To further your study, we recommend Chapter II of the following book as this literature contains these exact verifications:

Robert Bix, "Topics in Geometry," Academic Press (1994)
(Other chapters of this book are also very interesting.)

We can consider the three-dimensional version of the theory of wall paper groups. The three-dimensional version is very important in the study of the physics of the structure of crystals. Therefore, the three-dimensional version of a wallpaper group is called the crystal group. To change to three dimensions in the above condition (1) the words "two different directions" should be replaced by the words "three directions not contained in any plane." It is known that the crystal groups satisfying this replaced condition and the same condition (2) are classified into 230 types. This result was determined by Fedorov and Arthur Schoenflies in the period 1885 -- 1891. These two studied independently in the beginning. Having noticed that it is impossible to erase all mistakes by only one human being, they seem to have exchanged information.

In the case of four dimensions, groups are classified into 4895 types. This was shown in 1974 by five people: H. Brown, R. Bülow, J. Neubüser, H. Wondratscheck, and H. Zassenhaus.

In writing this page, I have been strongly influenced by the lecture for general citizens, "Geometry of patterns," given by Makoto Namba, a professor at Department of Mathematics, Osaka University, Japan (the lecture was held at Fukko Kinen Kan in Sendai on September 30, 1995, as a part of the autumn meeting of Japanese Mathematical Society). Another strong reference is in Japanese: "Geometry of Beauty" by Koji Fushimi, Mitsumasa Anno and Gisaku Nakamura (Chuko Shinsyo 554).

References:
http://www.clarku.edu/~djoyce/wallpaper/
http://neon.chem.ox.ac.uk/vrchemistry/sym/splash.htm
http://xahlee.org/Wallpaper_dir/c0_WallPaper.html
http://clowder.net/hop/17walppr/17walppr.html
http://www.oswego.edu/~baloglou/103/seventeen.html


Back to
Tohsuke Urabe Laboratory

Back to the colored page
of the Japanese patterns

Send your opinions and impressions to .