Unfolding Window

This is the unfolding window. Given a word W and a triangle T we produce the finite union of triangles U(W,T) by reflecting T iteratively, according to the digits of W. We call U(W,T) the unfolding. To see this in action, you first need to search for and find a periodic billiard path in a triangle. If you click a point in the parameter space and then do a search (using the seek button, then you will get a list of periodic billiard paths, appearing as little icons in a separate window.) If you click on one of these icons you will see the unfolding.

McBilliards mainly works with stable words . These are words for which the first and last sides of the unfolding are parallel with respect to any triangle. You can see this phenomenon by dragging around the point in parameter space and watching the unfolding change. We always rotate the picture so that the translation which matches up the first and last sides is a horizontal translation. Here we itemize some of the essential objects comprising the unfolding

CONTROLS

You can interact with the unfolding in a number of ways.

TURNING COORDINATES

Edges: Each edge has associated to it a pair of integers (A,B) called its turning pair. Assuming that we are at the point (x,y) in parameter space, the line through the first edge must be rotated through an angle of

(Ax+By) times Pi/2

to become parallel to the edge in question. Here we measure the angle mod Pi. When you select and edge on the unfolding the turning angles are displayed, as we mentioned above. If you look at the formulas for the functions displayed in the Unfold Function Window below, you will see that they are built out of the turning pairs for the spines. To read more about this, you should click on the documentation for the Unfold Function Window.

Triangles: Each triangle has associated to it a triple of integers. The triple of integers describes the location in the hexagonal grid of the word path associated to the word. The turning angle of an edge is computed from the triple of integers associated to either of the two triangles which contains the edge The formula is easier to see if we multiply the turning angles by 3.) When you click on a triangle, the triple of integers is displayed.