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
- The vertices There is a row of vertices which
runs along the top side of the unfolding. We call these vertices
TOP. Likewise there is a row BOT of vertices running along
the bottom of the unfolding. W describes a periodic
billiard path for T if and only iff all the TOP vertices
of U(W,T) lie above all the bottom vertices. You can
select vertices by clicking the middle mouse button
on them. We will explain this in more detail below.
- The leaders The TOP leader is the lowest
TOP vertex. The BOT leader is the lowest BOT vertex.
If you click on the compute leaders option, you
can see these vertices automatically selected. The
leading vertices will be joined together by a path of
edges which we call a spine .
- The strip In case all the TOP vertices are
above all the BOT ones, there is a maximal strip which
lies above all the BOTs and below all the TOPs. This
strip is automatically drawn, in a semitransparent
color, over the unfolding. Any horizontal line contained
in the strip is the unfolded image of a periodic billiard
path.
- spines There are three types of edges in
a triangle, and thus all the edges of the unfolding are
divided into three types. A connected path made from
a single type of edge is what we call a spine .
We use these spines to compute the functions which define,
in parameter space, the orbit tile O(W) of all triangles
having a periodic billiard path of type W. We explain
this in more detail in the documentation for the
window below this sone. If you select a pair of
vertices, you will see them connected by one of the
spines.
CONTROLS
You can interact with the unfolding in a number of ways.
- Recoloring If you bring up the Settings popup,
you can recolor virtually every aspect of the picture: the background
color, the triangle color, the strip color, the spine color, and
so on.
- Scaling and motion A left mouse click on the window
dilates the image about the click point. A right mouse click on the window
shrinks the image about the click point. You can also drag the unfolding
around using the middle mouse button.
- Vertex Selection
If you press the middle mouse button near a vertex of the unfolding, you
can select this vertex. Mainly, we are interested in selecting pairs of
vertices. You select a pair by the following procedure:
- Click
on the top little black rectangle to the left of this window.
- Select a vertex. The name and number of the vertex should appear
inside the black rectangle you have previously clicked.
- Click
on the bottom black rectangle.
- select another vertex. The name and number of this vertex will
appear in the most recently selected rectangle.
Once you select a pair of vertices, the unfolding window will
connect them by a spine and display it.
- Edge and Triangle
Selection If you press the middle mouse button near an edge or
a triangle of the unfolding you can select it. Not much happens in
this case, except that the turning coordinates for the
edge or triangle are displayed at left, near the two black
rectangles we mentioned above. We will explain more about the
turning coordinates below. These turning coordinates are vital
for our computations of the orbit tile associated to W.
- Compute Leaders toggle We already mentioned above that
turning on causes McBilliards to compute the leading vertices
automatically. Once you turn on this feature, you still need
to click a point in the parameter space (again) to activate it.
- Function Window toggle To each pair of vertices
in the unfolding, we attach a function of parameter space.
Essentially, this function computes the y-coordinate of
one of the vertices minus the y-coordinate of the other one,
when the picture is suitable scaled. These functions and
their values are displayed at the window below this one.
You can click on the open function window window
to either open or close this window.
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.