Word Window


The Hexagonal Grid
The constructions in this window are based on a hexagon. Given a hexagon, we form a hexagonal grid in the plane by taking the edges of the tiling by translates of the hexagon. There are two nice choices of hexagon: A button at the bottom of this window lets you toggle between the two options.

The hexpath
Let W be a word in the digits 1,2,3. For instance W=123123. We associate to W a path H(W) in the hexagonal grid, as follows: We group the edges of the hexagonal grid into three types, depending on their direction. We then trace out the edges on the grid according to the digits of the word. The word W is stable if and only if H(W) is a closed path. We call H(W) the hexpath In our example, H(W) is a hexagon. In general, McBilliards only searches for billiard paths in triangles which correspond to stable words. The main purpose of the word window is to draw the hexpath and some related objects. We will explain the auxilliary objects, called spines below. On the control panel for this window (at bottom) you will see a 3 by 4 array of buttons. These buttons control what is drawn in the main window. If you turn on the button which corresponds to draw hexpath then McBilliards will draw the hexpath. This is the default setting.

Animation
The hexpath need not be embedded. For this reason, it is useful to see how the path is actually drawn in the plane. If you turn on the button corresponding to animate hexpath you will see a white marker trace around the path. Another white marker traces around the edge of the window. The two white markers are in correspondence. You can read off the digits of the word around the edge of the window.

Subword Selection
Around the edge of the window you should see a border of colored boxes. These boxes are naturally in bijection with the digits of W. If you click on these boxes, using the left and right mouse buttons, you can select a subword of W. When you have successfully selected a segment of these boxes, the segment you have selected will light up in another color. Also, a portion of the hexpath will light up. You might wonder why you would want to do this. Here is the explanation: One of McBilliard's search algorithms uses a subword as input and then only looks for words which contain this subword. This search algorthm is useful for looking for words with special properties.

Geometric Interpretation
Now we explain the geometric interpretation for the hexpath. Say that an edge of the hexagonal grid is affiliated with W if this edge is incident to a vertex of the hexpath. The affiliated edges of the hexpath are in canonical bijection with the edges of the triangles which appear in the unfolding of the word. The coordinates of the midpoints of these edges, in a natural way, measure how these unfolding edges are rotated in space. Here is the precise formula: We will work with the integral hexagonal grid. Suppose that e1 and e2 are two edges of the unfolding U(W,T), and e1 and e2 are the corresponding edges on the hexagonal grid. Let A be the counterclockwise amount by which one must rotate e1 so that it is parallel with e2. Suppose that T is obtuse and X=(x1,x2) are the two small angles of T. Then mod pi we have the equation A=X.( e1 - e2 ) We are interpreting e1 - e2 as the vector which points from the midpoint of e1 to the midpoint of e2 It turns out that the hexpath is very closely related to the Fourier transforms of the functions which define the edges of O(W) the orbit tile associated to W. We explain this point of view in our research papers in McBilliards.

Topological Interpretation
The hexpath also has a nice topological interpretation. One can consider an periodic billiard path of even length (and these are all we consider) to be a closed geodesic in the 3 punctured sphere S3 obtained by doubling the triangle. W is a stable word if and only if the corresponding closed geodesic lies in the commutator subgroup of the fundamental group of the S3. Such a loop automatically lifts to the universal abelian cover UAC of S3. It turns out that UAC has a canonical deformation retract to the (1-dimensional) hexagonal grid. Composing the retraction map with our lift to UAC, we exactly produce our hexpath.

The Spines
Letting d be one of the digits 1,2,3, the d-spine is defined to be the union of type-d edges of the hexagonal grid which are affiliated to W. You can perform all of the same functions with respect to one of the spines that you can perform with respect to the hexpath. You can try this out by clicking on various of the buttons in the 3 by 4 array on the control panel. We mentioned above that the hexpath is closely related to the Fourier transforms of the functions defining the orbit tile. To get an exact relationship, one needs to use the spines. Indeed, if you read the documentation for the unfold window, you will see that McBilliards uses these spines to compute the defining functions for the orbit tile. There is a button on the control panel of the word window which lets the word window hear the unfolding window. In this way, you can see precisely how the spines correspond to the defining functions.

The Squarepath
The squarepath is a synonym for the 3-spine, when it is rendered relative to the integral hexagonal grid. The squarepath is especially relevant for obtuse triangles, where the type-3 edge is (by convention) the longest edge. The squarepath is a closed loop contained in the 1-skeleton of the usual tiling of the plane by squares.

Background display
There are two natural backgrounds (in the sense of graphical images) for the hexpath and the spines. The hexpath naturally lives on the hex background. The spines are more closely associated to the grid background. The choice of integral hexagon makes the grid into the square grid. Thus, each of our choices of hexagon shape is maximally symmetric with respect to one background or the other.

changing colors:
Here we mention one last feature of this window. If you bring up the Settings popup window, you can select this window, and then change a number of the colors. Many windows in McBilliards use this convention: You can change their colors fairly arbitrarily using the settings popup.