FORTIFICATIONS don’t have to be pretty, but they often are. A
few kilometres south of the Belgian border, the French
town of Maubeuge
is surrounded by a seven-pointed star. The moat,
overlooked by geometric walkways that lead up and out
into thin air, seems more like an eccentric’s playground
than a landscape tailored by Vauban to keep out
invaders. War is ugly, but its constraints evidently can
beautify.
Constraints do that everywhere. Which is why those who study the
subject of constraints (mathematicians) sometimes feel a
kinship with those who are in the business of making
beautiful things (artists). At the eighth New York
Digital Salon, which opened last week in the city’s
School of Visual Arts and
online they compete for your appreciation. “Scenic
views abound”, a picture of tourists on the moon, is
shown alongside “Extruded Hilbert Curve over a charged
Hexagonal Truchet Pattern”, a mathematical image from a
world that is in some ways more real than an imaginary
lunar tourist trap. At a recent conference held just
inside Maubeuge’s walls, the two worlds met for three
days of abstract curves, weird shapes on screens and, of
course, equations on blackboards.
For, given any mathematical statement, a shape that the eye can
savour is never far away. Take addition: 3+4=7 is
trivial, but x+y=7 is an equation. Even better, it also
defines a line: values for x and y are now constrained
to certain combinations such as 3+4, 5+2 and an infinite
number of other pairs. In a graph of x against y they
form a line thinner and straighter than any pencil can
draw.
Such graphs may help a student to fathom what the formula is
about. In higher mathematics, graphs are often
indispensable in allowing researchers to deal with
baffling theoretical structures—as they always have
been. Konrad
Polthier, of the Technical University of Berlin,
pointed out the lengths to which his predecessors in the
study of “minimal surfaces” were prepared to go to
obtain pictures
of what they were doing. In the
late 19th
century, geometers from Göttingen would
commission plaster models of certain minimal surfaces,
costing exorbitant sums that Dr Polthier would be lucky
to extract from his university’s treasurer. Fortunately
for him, there now exist “three-dimensional printers” that,
at a cost of a few hundred dollars, will convert one of
the complicated surfaces he draws on his computer into
something he can grasp. The latest models even do full
colour, allowing mathematicians to get an even better
feel for their subjects and opening new avenues for
artists who come along for the ride.
(....)
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