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MatheMUSEments
Square Wheel
By Ivars Peterson
Muse, February
1999, p. 29.
Riding around on a flat tire is no
fun. It feels
really bumpy.
But a square wheel may be the ultimate flat tire. There's no way
it can roll over a flat, smooth road without jolting the rider
again and again.
Stan Wagon, a mathematician at
Macalester College in
St. Paul,
Minnesota, has a bicycle with square wheels. It's a weird
contraption, but he can ride it perfectly smoothly. His secret is
the shape of the road over which the wheels roll.
A square wheel can roll smoothly if
it travels over
evenly
spaced bumps of just the right shape. That special shape is
called an inverted catenary. A catenary is the curve formed by a
chain or rope hanging loosely between two supports.
Turn the curve upside down, and you
get an inverted
catenary--just like one of the bumps in Wagon's road. Make the
road out of a whole bunch of those bumps all in a row, and you
can take your square-wheeled bike for a quick spin.
It turns out that for every
possible wheel shape,
there's a
road that produces a smooth ride. And for every road, there's an
appropriate wheel.
Wagon has used mathematics and
computers to
investigate many
wheel-and-road combinations without actually building them.
Wheels shaped like pentagons (five sides), hexagons (six sides),
and octagons (eight sides) also roll smoothly over bumps made up
of pieces of inverted catenaries.
In fact, as the wheel you use gets
more and more
sides, the
catenary segments required for the road get shorter and flatter.
Ultimately, the wheel has so many sides that it looks like a
circle and its road is practically flat. It rides just like a
normal wheel.
That's not all. You can find roads
for other wheel
shapes,
including an ellipse (which looks like a flattened circle), a
cardioid (like a heart with a rounded tip), and a rosette (like a
flower with four petals). You can also start with a road profile
and find the wheel shape that rolls smoothly across it.
There's one little problem. A
weird-wheeled bike has
to travel
in a straight line along its special road. You can't go left or
right!
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