McBilliards II

What McBilliards II does:

Running McBilliards II:

• You need to download the file mcb2.jar.
The file can be run in various ways, commented on below. We always run the program on the command line from the directory containing the file mcb2.jar.

• Search Mode: The program can be used to list periodic billiard paths in a given convex polygon with vertices whose coordinates are rational. The program lists all periodic billiard paths up to a given combinatorial length.

  The following line will search for all periodic billiard paths of combinatorial length less than 20 in the convex quadrilateral with rational vertices (0,0), (1,0), (1,1), and (-1/3, 1/2):

  java -cp mcb2.jar Search -v 0 0 -v 1 0 -v 1 1 -v -1/3 1/2 -d 20

  The output should be similar to:

  The program will print out all combinatorial types of periodic billiard paths of even combinatorial length less than 20 in the polygon with vertices {0+ i*0, 1+ i*0, 1+ i*1, -1/3+ i*1/2}.
  012102
  0120121020201202
  012021023103201202
  0120210202
  012020120210231032
  01202021321232120323
  01230132
  01231321023132
  123123

  These sequences of numbers represent the orbit type of a periodic billiard path, i.e. the sequence of edges hit by the billiard path. The numbers of edges are assigned increase by one as you move in the counterclockwise around the shape. The edge with label "0" is the edge joining the first two vertices listed on the command line.

  In general, the program will do this calculation for any convex polygon with rational vertices. The vertices must be listed in counterclockwise cyclic order.

• Stable Search Mode: This mode is similar to the previous search mode, except that in this mode only orbit types of stable periodic billiard paths are listed. A periodic billiard path in an n-gon P is stable if there is an open set U of polygons containing P so that every polygon in U has a periodic billiard path with the same orbit type. This search should be faster than the standard search mode.

  The directions for running in this mode are identical, except that we replace "Search" with "StableSearch" in the command line:

  java -cp mcb2.jar StableSearch -v 0 0 -v 1 0 -v 1 1 -v -1/3 1/2 -d 20

  The output should be:

  The program will print out all stable periodic billiard paths of length less than 20.
  01202021321232120323
  123123

• Unfolding Mode: In this mode, the program opens a window and displays an unfolding of a periodic billiard path. The command is called like this:

  java -cp mcb2.jar Unfolding -v 0 0 -v 1 0 -v 1 1 -v -1/3 1/2 -d 20 -ot 01202021321232120323
  Here, I have copied the orbit type from the search results above. A window will pop up which displays the following image:
  
  Pressing the "M" key opens a menu for zooming in and out the image. By typing "M" then selecting "Zoom In" and drawing a rectangle, we obtain an image such as:
  
  Lines contained in the interior of the translucent yellow strip fold up to form billiard paths in the given quadrilateral.

Source code:

Users of this program, and new mathematical results:

Prospects for future development: