Parameter Space of Triangles
This window allows you to examine the parameter space of all triangles. It is essentially the main window in McBilliards. You interact with the parameter space with the mouse:

• Left click to zoom in.
• Right click to zoom out.
• Middle click to move the point in the parameter space. You can hold down the middle mouse button while moving the mouse to smoothly change the point in the parameter space. In particular, the unfolding window will become animated if you do this. You can perform other actions by selecting an interaction mode from the green parameter space controller canvas.

Basic Definitions: Here are a few definitions which are basic to this window, and to all of McBilliards. A word is a finite string in the digits 1,2,3. The shortest word of interest to us is 123123. A periodic billiard path in a triangle gives rise to an infinite repeating word in the following way: We label the sides of the triangle by the digits 1,2,3, and then simply record which sides are hit as the path goes about its business. In describing such infinite periodic words, we list a single period. W is called the combinatorial type of the billiard path. Each point in this window represents a triangle. Specifically, the point (x,y) represents the triangle, two of whose angles are 90x and 90y. Given a word W, the orbit tile O(W) is defined as the set of points (x,y) which correspond to triangle having a periodic billiard path of combinatorial type W.

Core Operation: Now that the basic terms have been defined, we can explain the core of what McBilliards does:

• Searching: Using the middle mouse button, you can select a point in parameter space. Once you select a point, you can use the search console (at 6:00 on the main console.) to seek periodic billiard paths in the triangle corresponding to the selected point. You specify a depth N, and then push the seek button. McBilliards then finds all combinatorial types of periodic billiard path having length at most N which work for the given point.
  
• Plotting At the end of your search, an auxilliary window pops up showing all the combinatorial types McBilliards has found. You can then click on the icons in this auxilliary window to select a word W. Once you have W, you can plot the orbit tile O(W) using the plot console (at 5:00 on the main console) You can use the color selector (at 3:00 on the main console) to choose the plot color.
  
• Modifying Tiles The little green console at 7:00 on the main console allows you to modify plotted tiles. You can
  • raise a tile in relation to other plotted tiles.
  • lower a tile in relation to other plotted tiles.
  • recolor a tile.
  • erase a tile.
  • zoom in so that a given tile fits tightly in the screen.
  In each case, you select the appropriate button on the green console and then use the middle mouse button. The standard option is the default. In this case, a click on a plotted tile merely focuses McBilliards on that tile. In particular, McBilliards switches its current word to W when you click on O(W).

Additional Features: There are many ways you can interact with this window. These other actions typically have their own documentation. However, it seems worthwhile to give an overview of some of the less technical ones.

• Unfolding Given a triangle T and a word W, the unfolding U(W,T) is a union of triangles obtained by reflecting T across its sides according to the digits of W. If you bring up the unfolding window, you can see a picture of the unfolding, and manipulate it in various ways.
  
• Rational Point Selection. Using the rational number interface at 10:00 on the main console you can select points in the parameter space which have rational coordinates.
  
• Memory At least on the stand-alone version, McBilliards has a memory system, whereby you can save words and orbit tiles in files and recall them at a later time. This system doesn't work in a web-browser because of security issues related to interaction with files.
  
• Color Scheme Using the Settings window you can change the color scheme of this (and many other) windows. One way to activate the settings window is to click on the little S icon in the top left corner of this window. A second way is to open up the more popups window and then select the settings window from the menu.
  
• Word Inspection The Word window , which can be called as a popup, gives you a much more detailed look at the words found by the McBilliards searcher.
  
• Global Positioning Using the Where am I window you can keep track, in a global way, of where you are in parameter space. In other words, if you zoom way into the parameter window, you might not know where you are in relation to other objects which are no longer on the screen. The global positioning system takes care of this problem.
  
• Grid Selection After you play with McBilliards for a while, you will undoubtedly discover strong evidence for infinite families of periodic billiard paths. The corresponding orbit tiles frequently are arrayed in a certain kind of pattern. There is a general purpose Grid window which allows you to overlay the parameter space with various kinds of grids. These grids sometimes shed light on the structure of the infinite families of orbit tiles.

The above list is still only a fraction of what McBilliards actually can do. However, the remaining operations are of a more advanced or specialized nature.