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:
-
The regular hexagon.
-
The integral hexagon. This hexagon has
the property that the set of midpoints of the edges of
the associated hexagonal grid, together with the
centers of the hexagons, forms the integral lattice
Z+Z.
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.
- hex The hexagons of the hexagonal grid.
- grid A grid of parallelograms which goes through
the midpoints of the edges of the hexagonal grid.
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.