The Open Gate
 Symmetry
is everywhere around us, from the twin spiral of the Milky Way up in the
midnight sky to the shapes of the lowliest single celled plants down
here on earth. There is symmetry in the blossom of every flower in
spring, and most of all there is symmetry in the arts of man. We walk on
symmetrically decorated rugs, live in symmetrical constructed houses,
and even symmetrically adorn our food. Yet, what do we really know about
symmetry? It seems obvious that there is no limit to the number of
different symmetric rugs that one could weave, but if we look, not at
the art of a rug but rather at its underlying pattern of repetitions, is
there some limit to the number of ways that symmetry can be created? It
turns out that there is.
For hundreds of years, learned men tried to discover all of the possible
kinds of mathematical symmetry. Leonardo da Vinci was the first to
catalog the dozen or so interesting kinds of "point symmetry." These are
patterns which are symmetrical around a fixed point in space, like the
picture on the facing page. Sunflowers and starfish also have point
symmetry, and Leonardo himself used such symmetry in the design of some
cathedral chapels that look like the petals of a lily.
There are just two more classes of two-dimensional symmetry ----
patterns with "line symmetry", and patterns with "plane symmetry." The
border around the quotation at the bottom of this page has a type of
line symmetry, as do the geometric borders of everything from ancient
vases to cowboy belts. It is hard to believe that there are just seven
possible ways to arrange an artistic element symmetrically in a line,
but this is true. When you strip away all of the art and ornament, and
get down to just what makes a border symmetric, there are only seven
ways that it can be done! Likewise, in 1935, von Franz Steiger proved
that there are just 17 kinds of plane symmetry ---- patterns that
symmetrically fill a flat plane. The pictures on pages 34 to 44 exhibit
various types of plane symmetry, as do geometric tilings and rugs, most
kinds of wallpaper, and much of the work of M. C. Escher. Thus we see
that all of the kinds of point, line and plane symmetry have been
identified and cataloged by those who came before us to the subject.
What else is left?
It may seem that the mathematicians have taken something away from us by
proving there are a finite number of ways of creating symmetry. But they
have not. If you have any doubt that infinite scope still remains in the
subject of symmetry, then go find at a copy of a book I recently
discovered. Written by Peter S. Stevens, Handbook of Regular Patterns is
a catalog of symmetry patterns that shows thousands of examples of
symmetric art. The art points to infinite possibilities, and will
stimulate you to create your own symmetric art. The book also includes
much interesting information on the underlying mathematics of symmetry.
This is a book to buy if you want to get serious about the subject.
M. C. Escher wrote his own book called Regelmatige vlakverdeling
(Regular Division of the Plane), which is quoted in a much more
accessible volume on his work called Visions of Symmetry, by Doris
Schattschneider. His metaphorical description of the domain of symmetry,
as translated in the second book, is quoted below. It shows the great
value that Escher placed on the work of the mathematicians.
A long time ago, I chanced upon this domain in
one of my wanderings; I saw a high wall and as I had a premonition of an
enigma, something that might be hidden behind the wall, I climbed over
with some difficulty. However, on the other side I landed in a
wilderness and had to cut my way through with great effort until ---- by
a circuitous route ---- I came to the open gate, the open gate of
mathematics. From there, well-trodden paths lead in every direction, and
since then I have often spent time there. Sometimes I think I have
covered the whole area, I think I have trodden all the paths and admired
all the views, and then I suddenly discover a new path and experience
fresh delights.
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Books:
Handbook of Regular Patterns: An Introduction to Symmetry in Two
Dimensions
Peter S. Stevens, Cambridge: MIT Press,
1980
M. C. Escher: Visions in Symmetry, Doris Schattschneider,
New York
W. H. Freeman and Co. 1990.
Many good books about Escher are available. This one
has the most to say about the
symmetrical aspects of his work.
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