Before
starting our long working week, let's relax with this story of a
bicycle with square wheels. No, it's not a joke. And it even rides
smoothly. But there is a trick: the road must have a specific shape.
The Math Trek
section of Science News Online tells us more about this strange bicycle
-- actually a tricycle with one front wheel and two back wheels.
Stan Wagon,
a mathematician at Macalester College in St. Paul, Minn., 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.
Here is Stan Wagon riding his tricycle (Credit: Stan Wagon).
A square wheel can roll smoothly, keeping the axle moving
in a straight line and at a constant velocity, if it travels over
evenly spaced bumps of just the right shape. This special shape is
called an inverted catenary.
A catenary is the curve describing a rope or chain hanging
loosely between two supports. At first glance, it looks like a
parabola. In fact, it corresponds to the graph of a function called the
hyperbolic cosine. Turning the curve upside down gives you an inverted
catenary -- just like each bump of Wagon's road.
In fact, the idea is not new, and Wagon picked it after seeing an exhibit about square wheels
at the Exploratorium in San Francisco. But Wagon went further by
exploring the relationship between all kinds of wheels and road shapes.
Just as a square rides smoothly across a roadbed of linked
inverted catenaries, other regular polygons, including pentagons and
hexagons, also ride smoothly over curves made up of appropriately
selected pieces of inverted catenaries. As the number of a polygon's
sides increases, these catenary segments get shorter and flatter.
Ultimately, for an infinite number of sides (in effect, a circle), the
curve becomes a straight, horizontal line.
Here is the conclusion of the article.
So far, no one has found a road-and wheel combination in
which the road has the same shape as the wheel. That's an intriguing
challenge for mathematicians.
So why don't you try to solve this math puzzle?
Source: Ivars Peterson, Science News Online, Week of April 3, 2004
11:48:16 AM
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