Wheels on Wheels on Wheels
A circle is rotating at a rate determined by the setting of
the upper scrollbar. A second circle of half the radius is attached
to the first; the center of the second circle is on the circumference
of the first. The second circle rotates at a rate
determined by the second scrollbar. And there is a third circle
rotating at a rate determined by the third scrollbar. The third circle's center is attached
to the second circle.
For example, if the scrollbar settings are 2, -5 and -19, then
as the first circle rotates 2 times, the second will rotate
5 times and the third will rotate 19 times.
To understand why the values 2, -5 and -19 produce 7-fold symmetry (count the loops!),
we can follow the argument given by Frank A. Farris, "Wheels on
Wheels on Wheels - Surprising Symmetry", Mathematics Magazine,
Volume 69, No. 3, June, 1996, pp. 185-189.
2, -5 and -19 are all congruent to 2 (mod 7). Using complex
numbers, we can write the equation of the curve as
Consider what happens to any of the wheels when time is advanced
by 1/7th of a period. The contribution of any wheel is
so that the wheel is back where it started, but rotated 2/7-ths
of the way around. Each wheel has the same behavior, so we get
7-fold symmetry.
In general, we get m-fold symmetry if the three numbers are congruent
(mod m).
Be patient - it takes a few seconds to initialize the applet.
Source code
This page is maintained by: Robert F. Rossa who can be reached at
rossa@quapaw.astate.edu
Last revised on: 04/17/97