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鈥淪nowflakes ring鈥攁re you listening? Down the lane, bells are
glistening;

A beautiful sight, we鈥檙e鈥︹��

Hold it. Snowflakes ring? Shouldn鈥檛 it be 鈥渟leighbells鈥�? Well, not
necessarily. Believe it or not, a bunch of mathematicians has been listening
carefully to snowflakes. Not the sound they make as they land on the
ground鈥攚hich I imagine would be a gentle thud or a slushy squelch,
depending on where you are鈥攂ut the vastly more elegant sound that comes
from their beautiful shape. Sounds a tad off the wall? Not a bit of it. Ringing
snowflakes have a host of very down-to-earth applications. They could help us to
understand, for instance, why a rocky coastline dissipates rough seas better
than a smooth one, or why the foliage of trees is so resistant to the wind. They
could even tell us why the thud of our beating heart doesn鈥檛 wreck our system of
elastic-walled veins, arteries and capillaries.

What鈥檚 all that got to do with listening to snowflakes? Well, it started
around 120 years ago, when a small group of maverick mathematicians annoyed the
rest of the profession by inventing a series of bizarre shapes with the sole
purpose of showing up the limitations of classical maths: a curve that fills an
entire region of space (like Santa in his sleigh), a curve that crosses itself
at every point (ditto), and a curve of infinite length that encloses a finite
area.

This last example of geometric weirdness, invented by Helge von Koch in 1906,
is the so-called snowflake curve. Real snowflakes, of course, have many diverse
shapes: this one has a rather special geometry. Begin with an equilateral
triangle and add triangular promontories to the middle of each side to create a
six-pointed star. Then add smaller promontories to the middle of the star鈥檚
twelve sides, and keep going forever. Because of its sixfold symmetry, the end
result looks like an intricate snowflake, and it makes an excellent decoration
for the top of the Christmas tree.

Monster maths

Mainstream mathematicians promptly denounced these oddities as 鈥減athological鈥�
and a 鈥済allery of monsters鈥�. Gradually, though, interest in the 鈥渮oo鈥� grew and,
in the 1960s, the theoretical monsters were unexpectedly let out and headed in
the direction of applied science. Benoit Mandelbrot, then a researcher at IBM鈥檚
Thomas J. Watson Research Center, realised that these monstrous curves were
clues to a far-reaching theory of irregularities in nature. He renamed them, and
their more sophisticated brethren, 鈥渇ractals鈥�. Until then, science had been
happy to stick to traditional geometric forms such as rectangles and spheres,
but Mandelbrot insisted that this approach was far too restricting. The natural
world is littered with complex and irregular structures鈥攃oastlines,
mountains, clouds, trees, glaciers, river systems, ocean waves, craters,
cauliflowers鈥攁bout which traditional geometry remains mute. And so the
science of fractals was born.

Anyway, so much for the shape of this mathematical snowflake. What about its
sound? Let鈥檚 think about bells, jingling all the way. A bell jingles because the
metal that it is made from vibrates when hit by a clapper. If you made a
slow-motion movie of a jingling bell and looked closely, you would see that it
wobbles in and out like a jellyfish. The bell鈥檚 vibration frequency determines
the pitch of the note that is produced: high frequencies create high pitches,
low frequencies low pitches.

In fact, a bell can produce several distinct notes. The same effect is more
easily demonstrated on a guitar. Pluck an open string: it produces a single
note, the 鈥渇undamental鈥�. Now rest your finger gently against the exact middle of
the string, pluck again, and quickly lift your finger off. You will hear a
high-pitched 鈥減ing鈥�, exactly one octave higher up the musical scale than the
fundamental. If you place your finger one-third of the way along the string, you
can create an even higher note, and so on. Your finger is selecting various
vibrational 鈥渕odes鈥� of the string. When the string sounds its fundamental, it
forms a single standing wave. The ends are fixed, but the rest of the string
moves up and down in a regular, repetitive fashion. For a note an octave higher,
two such waves fit into the length of the string, and when one goes up, the
other goes down. A guitar string can鈥攊n theory鈥攙ibrate with any
whole number of waves. So as well as the fundamental frequency, it has
vibrational frequencies that are twice, three or four times, and so on. Its
acoustic spectrum鈥攖he list of possible frequencies鈥攃onsists of all
whole number multiples of the fundamental.

Striking shapes

Every shape has an acoustic spectrum. Physically, you can observe the
spectrum by making the shape from metal and hitting it鈥攐r by doing
something equivalent, such as making the shape from a soap film and watching it
wobble, or carving the shape as a cavity in a lump of metal and filling it with
microwaves.

This is how, back in 1910, the physicist Hendrik Lorentz from Leiden in the
Netherlands worked out that the set of frequencies in an object鈥檚 spectrum is
related to its size鈥攖he smaller the object, the higher the frequencies.
Lorentz鈥檚 conjecture worked just for a two-dimensional cavity鈥攁n area in
the plane. But less than two years later, Hermann Weyl from G枚ttingen
showed that Lorentz鈥檚 conjecture works for higher dimensions, too. So for a
three-dimensional object, the frequencies in the spectrum are related to the
object鈥檚 volume, and if the object gets smaller, the frequencies increase.

The next step was to apply this principle to fractal objects, such as the
snowflake. In 1979 the physicist Michael Berry from Bristol University was
thinking about light scattering from irregular surfaces, and he came up with a
conjectured improvement to Weyl鈥檚 formula. This improvement involved an extra
鈥渃orrection鈥� term, proportional to the 鈥渇ractal dimension鈥� of the object鈥檚
boundary.

Every object has a number associated with it that measures how 鈥渞ough鈥� it is.
For nice tidy objects, such as circles or spheres, this number is equal to the
classical dimension鈥攁 one-dimensional smooth curve has roughness 1, a
two-dimensional area has roughness 2, and so on. But for fractals the number
is in between鈥攆or instance, the roughness of the snowflake is about 1.26.
So it makes sense to refer to the roughness as a 鈥渇ractal dimension鈥�, even when
it鈥檚 not a whole number. According to Berry, the correction term in Weyl鈥檚
formula contains both the frequency and the fractal dimension of the
boundary.

Strange sounds

The upshot is that if you 鈥渓isten鈥� to the spectrum of an object, you can
often 鈥渉ear鈥� its shape. (鈥淏eating out the shape of a drum,鈥� 13 June 1992, p 26).
Why would you want to? Well, there are many cases where it is impossible to
observe an object directly, but far easier to observe its vibrations. Children
rattle wrapped presents to try to work out what鈥檚 inside. A more serious example
is seismology. You can infer the inner structure of the Earth from vibrations
generated by earthquakes. You can also do the same thing for the Sun using
helioseismology. Oil companies use sound waves created by surface explosions to
look for oil deposits deep underground.

It鈥檚 more than just hearing sounds which identify the shape though. Strange
shapes have strange spectra鈥攚ith interesting potential applications. Take
the 鈥渟quareflake鈥�, the snowflake鈥檚 nerdy cousin, made from squares rather than
triangles. In the early 1990s, Bernard Sapoval, Thierry Gobron, and A. Margolina
from the Ecole Polytechnique in Palaiseau, France, etched a steel plate with a
laser to produce a squareflake-shaped groove. They deposited a soap bubble on
the plate, so that its boundary coincided with the groove, and pointed a
loudspeaker at the bubble. When the frequency of the soundwaves directed at it
belongs to the bubble鈥檚 spectrum, it vibrates sympathetically, like a champagne
glass does in response to an opera diva. By watching its behaviour, Sapoval鈥檚
team discovered effects that don鈥檛 occur for more traditional shapes. For
example, it looks as though wave motion in regions with fractal boundaries can
be localised鈥� small regions of the object vibrate noticeably, but the rest
hardly moves. For waves, space and time are often somewhat interchangeable and,
sure enough, the experiments showed that some of the vibrations died out
extremely quickly, with the squareflake effectively soaking up the vibrational
energy.

Fine fractals

Some caution is needed in interpreting these experimental results, however. A
true mathematician鈥檚 fractal has structure on infinitely fine scales, whereas
for a real metal plate or bubble the detail is eventually blurred out. This
isn鈥檛 really a new problem: no experimental apparatus can reproduce a
mathematical shape exactly, not even a square or a circle. How sure can we be,
though, that the phenomena found in approximate fractals also arise in
infinitely delicate mathematical fractals?

One approach is to develop the mathematical theory in more depth. In 1995
Michel Lapidus from the University of California at Riverside and Michael Pang,
visiting from the University of Missouri at Columbia, performed a rigorous
mathematical analysis of the lowest frequency vibration, or fundamental mode of
the snowflake curve. Imagine a drumskin whose boundary is the snowflake curve.
The researchers discovered that for this mode, the vibrating 鈥渄rumskin鈥漜an
become infinitely steep near certain points of the boundary鈥攎ainly those
where the angles are greater than 90 degrees. In a physical analogy, the
vibrating fractal drum experiences 鈥渋nfinite stress鈥� at such points. Stuck long
enough in this mode, a real drum would tear.

Another way forward is to exploit the power of computers, which brings us
back to the 1996 work on snowflake harmonics and computer graphics. What does a
vibrating fractal really look like? Using an Onyx computer from Silicon
Graphics, and some cunning numerical methods, Lapidus鈥檚 team has drawn the first
fifty 鈥渟nowflake harmonics鈥濃�攖he vibrational modes of a snowflake
drumskin.

In the fundamental mode, as predicted by general theory, the whole drumskin
moves up or down at the same time鈥攖here are no 鈥渘odal curves鈥� where the
drumskin is stationary, except at the boundary. The extremely steep gradient
near obtuse-angle boundary points is clearly visible. Conversely, the gradient
near acute angles is zero鈥攖he membrane is flat, the stress is zero. The
computer-generated spectra confirm many of the features seen in the squareflake
experiments, and throw up a whole new set of puzzles. For instance, some of the
vibration patterns, such as the fundamental, are highly symmetrical, while
others have essentially no symmetry. Lapidus and his colleagues are now
struggling to understand why the modes behave the way they do.

It is not simply an academic question. Though the snowflake is really just a
mathematical toy, it could make plenty of real problems that involve more
complicated fractals easier to solve. Think about waves hitting a rocky coast. A
coastline is much more like a fractal than a smooth curve, so the action of
ocean waves on a rocky coast should be much closer to the snowflake model than
to classical analyses of waves hitting a flat boundary. A flat boundary reflects
most of the waves鈥� energy back out to sea, but a fractal boundary seems able to
absorb wave energy, or isolate it into small patches. Engineers have been using
rugged boundaries for years to repel the sea and soak up its energy, without
quite knowing why they work so much better than smooth ones. Work such as that
by Lapidus could explain why, and might even end up predicting what shapes would
be even more efficient.

Similarly, the leaves and branches of trees are fractal, and this may be why
they make better barriers to the wind than a simple flat fence鈥攁 fact that
modern agriculture is starting to rediscover after decades of rooting out hedges
to produce vast, but windswept, fields. And the human circulatory system, with
its repeated branching pattern of ever-smaller veins and arteries, is fractal.
As well as conducting blood effectively to all parts of our bodies, this fractal
structure may also help the blood vessels to absorb and dissipate the stress
caused by the heavy thumps of a beating heart, in the same way as a snowflake鈥檚
boundary dissipates the energy of waves when they run into it.

So, as you knock back your seasonal beverages and send your pulse rate up the
chimney, bear in mind if it wasn鈥檛 for the unusual, energy-absorbing properties
of fractals, maybe none of us would be here. Remember that, the next time you
hear bells jingle in the snow.