Wheels on Wheels on Wheels

Consider a wheel of radius 1 rotating about its axis with a frequency of 1 turn per second. Now imagine that there is a second, smaller wheel, the center of which is attached to a point on the circumference of the first. This second wheel is rotating on its own axis at a frequency of 7 turns per second. Finally, imagine a third, yet smaller wheel, whose center is attached to a point on the circumference of the second. This third wheel is also rotating on its own axis but in the other direction to the other two and at a frequency of 15 turns per second. A pencil put on the circumference of this third wheel would trace out a closed pattern of loops with two-fold symmetry.
In this Demonstration, the wheels have radii 1, 1/2, and 1/3 units, respectively. You can change each wheel's frequency of rotation using the sliders—positive values give anticlockwise rotation and negative values give clockwise rotation. The equation that models this system is shown above the plot and contains three terms—all complex exponentials. If you multiply any of these terms by then you change the phase of the wheel's rotation or, to put it another way, you change its starting position relative to the others.



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It was shown in a paper by Farris that the resulting curve exhibits -fold symmetry if the three frequencies are congruent (mod ).
Reference: F. A. Farris, "Wheels on Wheels on Wheels—Surprising Symmetry," Mathematics Magazine 69(3), 1996 pp. 185–189.
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