Mathematical Wheels

according to Frank Farris

In the paper
Wheels on Wheels on Wheels -- Surprising Symmetry
in Vol. 69, No. 3 of Mathematics Magazine, June 1996,
Frank Farris studies some interesting properties of functions of the form

f(t)=a1*exp(2*pi*i*(t*n1+s1))+a2*exp(2*pi*i*(t*n2+s2))+a3*exp(2*pi*i*(t*n3+s3))+....ak*exp(2*pi*i*(t*nk+sk))

To study these functions Erich Neuwirth built an Excel model which allows to try out different values for the parameters ni and ai.

In these equation each of the terms a*exp(2*pi*i*(t*n+s)) is a "wheel" which turns while its center is fixed on a point on the circumference of the "previous" wheel (while this previous wheel is turning). aj is the radius of the wheel number j, nj is the number of rotations the wheel is performing, and sj is a "phase shift", giving the initial angle of wheel j when the system starts running.

Graphical representation of wheels

Frank's main result states that the curve f(t) has k-fold rotational symmetry if all the pairwise differences n2-n1, n3-n1 .... n4-n3 have k as their greatest common divisor. The symmetry is not affected by the values s1, s2...

You need Microsoft Excel 5.0 to be able to use this worksheet. If you have this version of the software, you may download the worksheet now

If you like what you see, you also may want to go to
Erich Neuwirth's Spreadsheets in Education Home Page.


Author: Erich Neuwirth
Last Change: 19960910
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