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[IMAGE 071-073_Veteran_Neophyte1.GIF]

Symmetry is more interesting than you might think. At first glance there doesn't seem to be much to it, but if you look a little closer you'll find that symmetry runs swift and cold and deep through many human pursuits. Symmetry concepts are found at the heart of topics ranging from the passionately artistic to the coolly scientific, and from the trivial to the fundamental.

I learned a lot about symmetry while trying to learn how to create tile shapes. I've always been intrigued and tantalized by M. C. Escher's periodic drawings, the ones that use lizards or birds or fish or little people as jigsaw puzzle pieces, interlocking and repeating forever in a systematic way to completely tile a surface (mathematicians call thistessellation of a plane). My own halting attempts to draw tessellations have met with only tepid success. Especially hard is creating tiles that are recognizably something other than meaningless abstract shapes.

To accomplish this feat of tiling a plane, you have to apply a set of constraints to everything you draw. Every line serves multiple purposes. In one of Escher's prints, for example, the same line that forms the left arm of one lizard also forms the tail of an adjacent lizard. That line is also repeated ad infinitum across the plane;every lizard's left arm and tail is defined by that same line shape. Now think about drawing a line like that. Not only are you drawing two shapes with one line (which is difficult enough), but you're also drawing innumerable identical lines simultaneously. They sort of spin out from the point of your pencil in a dazzling dancing tracery of lines. Trying to hold all that complexity and interrelatedness in your head is very, very difficult.

Being a basically lazy person with too much time on my hands, I decided to write a program that would handle it all for me. I envisioned a direct manipulation kind of thing: as I changed a line, all the other corresponding lines in the pattern would change simultaneously. I figured it would be easy to draw little people and leaves and fishes that perfectly interlocked, if only I didn't have to keep all those interdependencies and constraints in mind and could just draw. Also, I thought maybe that by interactively "doodling" and being able to watch the whole pattern change on the fly, I could get some sort of gut feeling for the constraints.

All this was way back in 1990. To learn more, I bought a book calledHandbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions by Peter S. Stevens. The book is a sort of systematic catalog of hundreds of regular patterns, including many of Escher's, and also has a great introduction to the mathematics of symmetry (which turns out to figure heavily in this tiling business). Unfortunately, after an intense but superficial examination and an evening or two playing with pencil and paper and little dime store pocket mirrors (bought in a frenzy of excitement the day after I bought the book), I decided that the program would beway too hard to write to make it worth it, and shelved the whole thing.

Well, last month I finally picked up the idea again. QuickDraw GX was getting close to being released, and it had features that made it relatively easy to implement what I wanted: very flexible transformation and patterning capabilities, and excellent hit testing, which makes implementing direct manipulation of lines a snap. So I dusted off Stevens's book and my little mirrors and got to work, trying to figure out the constraints on the tiles and implement the program.

Here's a basic fact about tiling a plane that I still find thoroughly remarkable three years after I first learned about it: there are only 17 possible arrangements of tiles. "But wait!" I hear you cry in your many-throated voice, "How can that be? Surely there are a very large number -- nay, an infinite number -- of possible tile shapes?"

Well, yes, that's true. But the way they fit together, the underlying structure, will always be one of only 17 possibilities. This applies toany two-dimensional pattern made up of regularly repeating motifs, not just seamless tilings. The motif that's repeated, of course, can be anything: a leaf, a loop, or a lizard; a frog, a flower, or a fig -- it makes no difference. There are still only 17 ways to build a regularly repeating 2-D pattern. This was proved conclusively in 1935 by a mathematician named von Franz Steiger. (Yes, that's his name; I checked twice.)

To see why, you need to learn a little about the fundamental symmetry operations and how they combine with one another to breed other symmetry operations. I'll gloss over most of the details (see Stevens's book, or any introductory text on crystallography, for more info), but the gist of it is that when you sit down and begin to repeat some motif by repeatedly applying fundamental symmetry operations -- like reflection and rotation -- you find an interesting thing: combining symmetry operations with one another often causes other types of symmetry to sort of spring into existence. And the operations always seem to gather themselves into the same few groups.

Figure 1 shows a very simple example. We start with a simple motif (a comma shape) and repeat it by applying a transformation to it, in this case by reflecting it across a vertical line. Then we reflect the whole thing again, this time across a line perpendicular to the first one. The resulting pattern of four commas possesses mirror symmetry in two directions, meaning that a reflection of theentire pattern across either one of the lines leaves the pattern unchanged. But if you study it, you'll find another symmetry embedded in the pattern that we didn't explicitly specify. In particular, it showsrotational symmetry: rotating the pattern 180º about its center leaves it unchanged, too.

[IMAGE 071-073_Veteran_Neophyte2.GIF]

Figure 1 Building a Simple Symmetry Group

Figure 2 shows an alternative way to create the same pattern. This time we begin with the rotation (the point of rotation, orrotocenter , is shown by an oval). If we then run a mirror line through the rotocenter, we produce exactly the same structure, the samesymmetry group , as we did by combining two perpendicular reflections above. These three symmetry operations (two perpendicular reflections and a 180º rotation) come as a set. Combining any two automatically produces a pattern that also contains the third. This is where the constraints on the structure of regular 2-D patterns appear. No matter how you combine and recombine the fundamental operations to cover a plane, you find yourself generating the same 17 arrangements, the same 17 groups of operations. [IMAGE 071-073_Veteran_Neophyte3.GIF]

Figure 2 Another Way to Build the Group

By the way, this example group isn't one of the 17 plane groups. It's one of the 10point groups, groups whose constituent transformations operate around a single point. In case you're curious, there are also 7 line groups (ways to repeat motifs endlessly along a line) and 230space groups (ways to repeat a solid shape to fill three-dimensional space). I don't know if anyone has figured out the groups of higher-dimensional spaces. Knowing mathematicians, I don't doubt it.

So what about that computer program I was going to write? As this column goes to press, it's undergoing its second major overhaul, having suffered mightily from my "write it first,then design it" philosophy. So far I have 5 of the 17 groups implemented, and it's pretty cool. There's no telling how far I'll actually get before my deadline arrives, but I'll put the results, however clunky and raw they may be, on this issue's CD so that you can check it out.

I've learned a couple of things already: Even with the constraints automatically handled by the computer, it's still really hard to create representational shapes that will tile a plane, though creating abstract tile shapes is suddenly a piece of cake. Also, I still haven't gotten the kind of gut-level understanding of thestructure of the patterns that I was hoping for (though just watching them change as I doodle is very entertaining).

I've also learned along the way that symmetry concepts go far deeper than the simple plane groups I'm messing with. The rules of symmetry and of form are, in a sense, manifestations of the structure of space itself. It's an odd thought that spacehas a structure, isn't it? Normally we think of space as a sort of continuous nothingness, as anabsence of structure or as a formless container for structure. But space itselfdoes have a structure, and every single material thing must conform to that structure in order to exist.

Physicists, of course, have been trying very hard for a long time to describe precisely the nature of space. Einstein thought that there was really nothing in the worldexcept curved, empty space. Bend it this way, and you get gravity, tie it in a tight enough knot and you get a particle of matter, rattle it the right way and you get electromagnetic waves.

And there are other symmetries, symmetries even more fundamental. Einstein's theory of special relativity broke some of the central symmetries in physics, and thus called attention to therole of symmetry in science. Shortly afterward a mathematician named Emmy Noether established a remarkable fact: each symmetry principle in physics implies a physical conservation law. For instance, the familiar conservation of energy law is implied by symmetry in time -- energy is conserved because time is symmetric. (Of course, I'm greatly oversimplifying here. The symmetry of time is one that Einstein tarred and feathered and ran out of town on a rail. He showed that under extreme conditions time isnot symmetric, and energyisn't conserved. Reassuringly, he replaced these broken and bloodied false symmetries with fresh new ones, but they're well beyond the scope of this column and my poor addled brain.) The point is that symmetries seem to be part of the very fabric of the universe; they seem to be the warp and weft of existence itself.

Yes, it's heady stuff indeed, this symmetry business. I'm staying plenty busy just trying to understand the symmetries possible in a plane, thank you very much, so I'll leave worries about the symmetry of space-time or of K-meson decay to the pros. Once again, I find that by looking just beneath the surface of a seemingly innocuous topic, I find depth and complexity beyond measure. Ain't life grand?


  • Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions by Peter S. Stevens (MIT Press, 1981).
  • Patterns in Nature by Peter S. Stevens (Little, Brown & Company, 1974).
  • Where the Wild Things Are by Maurice Sendak (Harper & Row, 1963).

DAVE JOHNSON once thought that maybe computers contained the secret of life, but has since decided that no, it can't be found there, either. He's now beginning to look elsewhere. Compost piles (preferably hot, steaming, and active) are currently being eagerly investigated.*

Thanks to Jeff Barbose, Michael Greenspon, Bill Guschwan, Mark Harlan, Bo3b Johnson, Lisa Jongewaard, and Ned van Alstyne (aka Ned Kelly) for reviewing this column. *

Dave welcomes feedback on his musings. He can be reached at JOHNSON.DK on AppleLink, on the Internet, or 75300,715 on CompuServe.*


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