UMASS LOWELL/UMASS DARTMOUTH - CyberEd
92.419 Computer Algebra with Mathematica
Kenneth M. Levasseur
Department of Mathematical Sciences
University of Massachusetts Lowell
Lowell, MA 01852

Wheels on Wheels on Wheels

Subject

Mathematics

Topic

If a wheel is turning and its center is attached to a wheel that is also turning and the second wheel is attached to a third wheel and that third wheel is turning, a point of the first wheel can trace out some very interesting paths. Depending on the rates at which the wheels are turning, the path can have a one of a variety symmetries or be totally asymmetrical.

Reference(s)

  1. Frank A. Farris, "Wheels on Wheels on Wheels - Surprising Symmetry," Mathematics Magazine, 69(1996) 185-189.

Project Idea(s)

  1. Reproduce the curves in [1] and then use the theory in the paper to produce curves with other symmetries. Some of [1] uses Fourier series, to pursue this idea. A facilty in Fourier series is not absolutely necessary.
  2. Examine wheels on wheels on other conic sections.

Prerequisite Mathematics

vector calculus, familiarity with complex numbers, elementary number theory. Fourier series a plus.

Required Programming Level

No advanced programming skill would seem to be required. Mathematica number theory functions and ParametricPlot.

Key Words

Geometry, Graphics, mechanics, symmetry

Reviewer

K. M. Levasseur (Kenneth_Levasseur@uml.edu)

Archive

Two students, Mike Pawlisheck and Ed McCoy, have done projects that have done the basic graphics and examined a few other facets of this topic. If you would like to do this project, request copies of their material. You would be expected to extend their work. One possibility is to pursue the Fourier series angle mentioned above.Return to Project Info. Page