MAMIKON'S THEOREM (only for Ti bike
A recently publicized mathematical
theorem provides an interesting insight into the way that bikes move through turns. You
have probably noticed that in a turn the tracks of the front and rear wheels separate,
then come back together at the end of the turn. This is particularly noticeable when
riding on soft earth or with wet tires on dry pavement.
You may also have noticed that the distance between the
two tracks is greater for a tandem than for a bike with a shorter wheelbase (the distance
between front and rear axles). If you think about it, as long as the front wheel is
cocked, the frame of the bike is moving partly sideways and the area that the frame sweeps
over during the turn is exactly the area between the tracks of the two wheels.
Mamikon's Theorem tells us that the bike's
"sweep" (the area between the tracks of the front and rear wheel) depends only
on the wheelbase and the angle through which the bike turns. In particular, if we let T be
the angle of the turn (in radians) and W be the wheelbase, then the sweep S is given by
S = T * W * W / 2
Remarkably, the area of the sweep doesn't depend on
whether the bike turns sharply or gradually or at a variable rate. The sweep that results
from a given change in direction is the same whatever the bike's trajectory.This result is
one of a number of remarkable findings that follow from a geometric theorem discovered by
Armenian mathematician Mamikon K. Mnatsakanian, who now teaches at the University of
While it is traditional to give mathematical theorems the
last name of their discoverers, we can be thankful that a less formal approach was adopted
in this case.
Of course, you might reasonably ask what use a bike rider
can make of this new information. One possible application would be when racing through a
turn in a criterium. If another rider yells at you to "Watch you line!", you
might reasonably yell back "According to Mamikon's Theorem my sweep is independent of
how tightly I turn!" By the time the other rider finishes puzzling over what you just
said, you may have successfully broken away on the far side of the turn.
There are some other remarkable results that follow from
Mamikon's Theorem, including a simple proof of the Pythagorean Theorem and a neat way to
calculate the area under a cycloid, which is the trajectory of any given point on the
outside of a bike wheel as it rolls along.
The general form of Mamikon's Theorem is "The area
of a tangent sweep to a space curve is equal to its tangent cluster." To see an
explanation and illustrations of what that means, go to http://www.its.caltech.edu/~mamikon/calculus.html
and click on the Introduction. There are also some animated illustrations there.
- Les Earnest (original