PolygonFlux for the Math-Minded

Polygon Billiards poses many interesting questions. Contributions Welcome!

Warning -- some problems are harder than others, some are insanely difficult!


The exercises we have so far are directly or indirectly about 'orbits'. An orbit is when the point makes a complete circuit bouncing around and starts retracing the same exact path.

  1. Orbits can be of period 2, such as in a square, where the point just bounces back and forth between two sides. Set the number of sides to 4 and reduce bouncing down to just a few bounces. Rotate the emitter until you achieve the period 2 orbit. (Try 0°, 90°, 180°, or 270°!). See what happens when the minutes or seconds are changed slightly.

    Now find a period 2 orbit in a hexagon, and one in an octagon.

  2. Select the triangle and form an orbit of period 3. Hint: It's triangle-shaped. :^) Do the same with a hexagon.
  3. An orbit of period 4 can form inside a triangle when the point bounces off one side at a right angle and bounces back toward whatever side is "behind" it:

    Orbit of period 4 in triangle

    Explore this kind of orbit with PolygonFlux by dragging the Origination Point around. Prove to yourself that the angles in the figure add up as they should.

  4. When you have any kind of orbit, change the angle slightly. Increase the bouncing, while adjusting the angle to keep the the now-spiraling orbit lines "together". Use different Themes to help visualize the fluxagon structure.
  5. Find an orbit of period 6 in a triangle. Experiment by dragging the origination point around.
  6. Find two [fundamentally different] orbits of period 20 on the equilateral triangle. (Period 20 is the lowest period for which two different orbits exist.) Thanks to Dr Andrew Baxter at Penn State University!
  7. Look at orbits in the triangle with period less than 20. Are they are in fact unique as suggested in the previous exercise? Make a catalog of them.
  8. Experiment with the square, and you'll see that there are quite a few orbits of various periods. Can you characterize the periods of orbits possible in a square? Are there a finite number of orbits?
  9. Can you characterize the orbits possible in a pentagon? (See Pentagons paper in Links below)
  10. Consider the 9-sided fluxagon below. The initial angle was 40°. What angles do you suppose lead to orbits? Can you write a formula that gives angles that produce orbits? Does your formula cover all such angles or a subset of such angles?

    9-sided fluxagon, initial angle 40°

  11. Do all angles go into an orbit if you bounce long enough? Do some angles just keep passing through all the points in the area inside the polygon, and never orbit? Explain you answers.
  12. As bouncing increases, the intersection of lines increases. What can you say about the patterns formed by the intersections? Can you think of a way to describe what these patterns look like, or predict what they will look like? (The Neutrino, Electron, and Chromodynamics themes are probably the best themes to use for this exercise — their lines get finer and finer as bouncing increases into the thousands.)
  13. It's fun to set Bounces to some low number and then chase after the termination point with the emitter. If you can align the first bounce with the termination point, you have "closed the loop" so to speak, and formed an orbit. Can you prove this? Did you have to adjust the number of bounces in order to hit it? If so, why?

    Fun with emitter

    PS. Increase bouncing after you hit the termination point to see the full orbit.

  14. Consider the sworl we see near a vertex in this 7-sided fluxagon:

    Interesting sworl near a vertex

    The fluxagon will appear after ~400 bounces with the initial angle at 317 degrees 1 minute 23 seconds. Move the Origination point, if necessary, to produce the sworl or vortex. The flux seems to be in an orbit, but the exact orbit may be a fraction of a arc-second in either direction. Increase bouncing past 1000.

    Can you explain why the flux lines seem to avoid this area?

Contributions to Math-Minded exercises are welcome! Send to timehavenmedia@gmail.com.


The above exercise numbers may change over time, so be careful not to reference solely by number. (The exercises themselves may change.)

Bounce your ideas off us!

Please tell us about PolygonFlux being used in high school or a university class - send your ideas or experiences to timehavenmedia@gmail.com

A Note on circular billiards and other shapes

Circles are more predictable than polygons, but can produce an interesting range of fluxagons — polygons, stars with any odd number of points, rectangles, etc. We may add circles and ovals to a future release. We'd like to add non-regular polygons such as scalene, right, and isosceles triangles.


Gardner, M. Bouncing Balls in Polygons and Polyhedrons. Ch. 4 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 29-38 and 211-214, 1984.

Knuth, D. E. Billiard Balls in an Equilateral Triangle. Recr. Math. Mag. 14, 20-23, Jan. 1964.

Tabachnikov, Serge. Geometry and Billiards, in the American Mathematical Society bookstore.


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