BOX ANALYZER CANVAS
This window allows the user to trace through the calculations
made by our plotter when it runs the Elimination Algorithm.
This algorithm is used to decide if a given dyadic square
Q is contained in the orbit tile O(W), where W is a given word.
You can read about this algorithm in the documentation for the
plotter. In brief, the elimination algorithm starts with two
lists of vertices, TOP and BOT. Here TOP is the list of
vertices which run across the top of the unfolding and BOT
is the list of vertices which run along the button.
The elimination algorithm then considers the defining
functions for these vertices. There are 3 stages to the
algorithm
- TOP tournament This stage considers pairs of
TOP vertices and finds those which have the smallest y coordinate
for unfoldings associated to points within the given dyadic
square.
- BOT tournament This stage does the same thing
for the BOT vertices.
-
Playoffs This stage considers all pairs of the form
(t,b) where t is a TOP winner (of the tournament) and b is
a BOT winner. If McBilliards can show that the defining
functions associated to all such pairs are positive then
the given square is contained in the given orbit tile.
Here is an explanation of the display:
- Dark blue dominoes The dark blue dominoes
show the essential matches from the BOT tournament. Let's
say that the pair of numbers (7,2) shows up in a
domino, with 7 on top. This means that 7 defeats 2
in some match. In other words, vertex 7 provably
lies above vertex 2 for
all the unfoldings associated to points in the
given square. Each bottom number is a loser in
the tournament.
The BOT numbers which never appear on the bottom
are the winners of the BOT tournament. As another
way to remember things, the losers are drawn in
grey and the winners are drawn in white.
-
Light blue dominoes this is the same scheme for
the TOP tournament, with the roles of top and
bottom being completely reversed. The winners of the
TOP tournament are the numbers which never appear
on top of a light blue domino.
-
Winner lists The row of light blue squares
displays the list of TOP winners. The row of dark
blue squares shows the BOT winners.
DISPLAYING MATCHES
If you click on one of the dominoes you can see
the corresponding match. This means that
- The unfolding window displays the pair of
vertices, connected by a spine, corresponding
to the numbers on the domino. To read about
spines and their significance you
should read the documentation for the
unfolding window.
-
The function canvas shows the function associated
to the pair of vertices, as well as various
quantities associated to that function. McBilliards
plugs these quantities into its quadrant algorithm
to verify that the defining function is
positive (or negative) throughout the dyadic
square.
So far we have explained how to display the
tournament matches by clicking on the dominoes.
You can display a playoff match by clicking
on a light blue square and a dark blue
square. This pairs off 2 tournament winners.
All the remarks above now apply in this case.
There is one variant which needs to be discussed.
You can also select a chopped dyadic
square which intersects the boundary of the
orbit tile. In this case, what happens for the
playoff matches is different: The quadrant algorithm
is used to show that all the defining functions,
associated to the playoffs, are of mixed
sign on the dyadic square. This is discussed
in detail in the documentation for the
Fill mode of the plotter.
There are 3 additional buttons in this window. Here
is their function:
- Bark blue button This
button cyclically reorders the list of BOT tournament matches.
You will appreciate this feature if you are using
a very long word. Click it repeatedly and watch the
dark blue dominoes reshuffle.
-
Light blue button Same thing for the
light blue dominoes.
- Green button
swaps the placement of the two collections of dominoes.
Again, this is useful for long words.