Worth
every penny, August 30, 2000
Reviewer: A. Jesse Davis from Redmond, WA USA
I attended a lecture by Jean-Marc at Microsoft's MOSAIC (Microsymposium
on Analysis and Synthesis of Intuitively Conceived Geometrical Art).
Castera's work is unique and brilliant. The book is worth it for
the hundreds, maybe thousands of giant color plates of Morroccan
mosaics alone, and Castera's exposition of the artistic technique
and culture of North Africa is excellent.
The core of his work, though, is the mathematical analysis of
mosque mosaics, and his rigorous but accessible explanation of
tiling rules is fantastic. He goes all the way from how to construct
a Solomon's Seal on graph paper to multi-dimensional group theory
and back again. I can't say enough good things about Castera's
work and this book.
Consider it an investment in a lifetime fascination.
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SCIENCE
20APRIL2001 VOL 292 pp445-446
BOOKS: MATHEMATIC AND ART
Algorithms of Boundless Beauty
Gregory Buck (1)
Most readers of Science are primarily
interested in the truth, as supported by the scientific method.
Me too. But all truths are not equal. I recall well when I first
grokked Newton's arguments giving the spécial properties
of the inverse square law. I was so moved by the elegance of the
constructions, I found myself wip-ing away tears. Now why should
this be ? Why should aesthetic appeal have anything to do with
the truth? The question is particularly interesting to mathematicians.
We make many of our working decisions on aesthetic principles
accepting ugly proofs like an obedient child takes brussel sprouts,
but reaching for pretty results like they are a plate of cookies.
Answer the question and you'll get a chair in philosophy at Harvard.
The artists and craftsmen of Morocco have never
found any conceptual difficulty here. They saw the beauty of geometrical
and topological algorithms a thousand years before Escher and
Mandlebrot. The French artist and mathematician Jean-Marc Castéra
has produced a book worthy of their efforts, a nearly impossible
feat. Arabesques comprises nearly five hundred full-color
pages of perfectly composed photographs by Françoise Peuriot
and Philippe Ploquin that present decorative and architectural
works of eye-numbing beauty. This extensive sampling of designs
from Moroccan mosques, palaces, and, cities is accompanied by
Castéra's knowledgeable and insightful analysis.
Every bit of the featured work is mathematical.
I have only space and time to mention rudiments. Thèse
craftsmen understand planar tilings the way Kepler understood
trajectories. They are happy to swim in the complex mathematics,
as adept with symmetry groups as a salesman is with a cell phone.
The designers evidently understand the fondamental facets of mathematical
knot theory (which we now apply to DNA). They comprehend the symmetry
properties some knots have (and others lack), and they clearly
have command of the concept of alternating knots- if you trace
any strand you will find that it travels under one crossing strand
then over the next, and so on. This understanding is shared with
another magnificent tradition of decorative topology, Celtic knotting
(see especially the illuminated manuscripts of the Lindesfarne
Gospels and the Book of Kells). Castéra does a fine job
explaining some of the myriad approaches taken with the geometry.
He also provides a pretty solution to one of the basic problems
of such work: how does one even get started? The designs are so
complex, the symmetries so demanding, that a novice could easily
be flummoxed from the start.
There are many wonderful details in this book.
One of my favorites is a photograph of a collection of some of
the complex tiles the artists use. (Home renovators: don't bother
look for these at Tile World.) Castéra shows that several
of the designs consist of a central motif of one symmetry group,
a border of another symmetry group, and a transition zone which
must be appealing but cannot carry either symmetry exactly. Such
transitions are a recurring problem in aesthetics. Consider an
analogous situation in music, where opposing tonal demands ask
us to divide the octave in différent ways; the hard part
is to make the whole sound good anyway. As the illustrations in
Castéra's book demonstrate, the Moroccan artists make their
whole designs look great.
One cannot help but be impressed by the magnitude
of thé artists' efforts. These design elements can take
years to complete. Possibly the mathematical ambition is focused
by the traditional Islamic prohibition on representations of humans
or animals. Think of the song lyrics which might have been written
if there were a ban on references to sex and drugs. (Well, maybe
that analogy doesn't obtain: l'm not sure that Britney Spears
would be singing about, say, non-euclidean geometry under any
circumstances.) There is a sense of elation when one first realizes
that these patterns are algorithmic. Castéra includes the
proof in the book. His computer efforts produce designs that are
theoretical équivalents of those created by the craftsmen,
and even give them a pretty good run as images on the page. In
one light, this is a near miracle. Perhaps we ail can fashion
this sort of beauty; perhaps such handmade beauty need not cost
a lifetime of bent backs and dusty lungs. But for me this elation
is matched by a tactile melancholy. Why do we have hands if not
to use them? Castéra observes that the Moroccan craftsmen
seem happy. It is easy to imagine that they would be. How would
you feel if your job was creating works of transcendent beauty
to grace the public places of your community?
It is trite but true to say that we are the
most scientifically and technologically advanced society in the
history of Earth. Yet when we look out the window, what do we
see? I suppose it depends on your particular window, but my guess
is that your view doesn't offer the aesthetic equal of these Moroccan
works. If you see human artifice, you likely see linear blocks.
I imagine this seems barbarie to someone from Morocco. After all,
the mathematics of our art and architecture we teach to ten-year-olds;
the mathematics of their decorations, we study in graduate school.
It should corne as no surprise that the great mathematical designer,
M. C. Escher (the officiai artist of graduate student offices
and mathematics departments), made his own trip to Morocco. Perhaps
the reason Escher's work seems so surprising lies in our bias-he
isn't of the western tradition. But by Moroccan lights his is
the logical next step. If you want to understand Escher, call
your travel agent or buy this book.
Let's look out the window again. We see our
artifice, regular as crystals. We see nature in its glory, the
flora and fauna, and the fauna we're most fond of-us. The Moroccan
craftsmen cannot represent the human form. But if they tried,
we would not have this art; on the large scale, we do not appear
algorithmic. However, one of the great technological achievements
of the last century is the exponential improvement in our acuity.
We can now see the very small. And in the universe of the very
small, in the molecules and atoms that are us, we see this same
algorithmic magnificence we create for our own pleasure. Thus
it seems that the Moroccan artists and craftsmen render us after
all. Their work and this book make me wonder: perhaps what exists
in our imagination is as wonderful and beautiful as anything nature
can create. You pick Arabesques up fascinated; you put
it down humbled. It is one of the most beautiful books I have
ever seen.
(1) The author is in the Department of Mathematics,
Saint Anselm College, Box 1641, 100 Saint Anselm
Drive, Manchester, NH 03102-1310, USA. E-mail:
gbuck@anselm.edu
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