100 Degree Theorem: Vertex Display
This is the vertex display component.
Recall from the main documentation (for the Deg100 Theorem) that each
of the little data boxes in the left two
columns contains a word W and a rational polygon P(W).
With 10 exceptions, P(W) has the property that every point
in the polygon has a periodic billiard path whose
combinatorics is described by W
When you click on one of the data
boxes this window will show the coordinates of the vertices.
If you click on one of these displayed coordinates,
the cursor in the McBilliards plotting window
will move to that point. You can use this
option most effectively if you perform the following
sequence of actions:
- select the fit option on the main McBilliards console
- select the plot tile mode on the 100 Degree
window
- turn off the verify mode on the 100 Degree window.
Here are some facts about our polyons:
- Each polygon is convex and has at most 8 vertices.
- With 42 exceptions, all the vertices have dyadic
rational coordinates. The maximum numerator is 2^17.
As for the exceptions: 2 of the exceptions correspond
to the Pi/5 point. The other 40 exceptions are the
starts of the 4 infinite families shown at the bottom
right corner of the 100 Degree window. These words
are all treated analytically in Rich's papers.
-
From amongst the 42 non-dyadic polygons,
10 polygons do not have rational vertices.
These 10 exceptions are the same ones mentioned above.
In each of these 10 cases we plotted O(W) numerically
and let P(W) be the convex hull of the vertices of O(W).
The tile O(W) is slightly nonconvex and so it does
not quite contain P(W). However, the error is
vanishingly small. Also, as we already mentioned,
Rich treats all these tiles analytically in the
papers. The plot here is just for the purposes
of illustration. The actual vertices of O(W)
in these cases are irrational multiples of Pi.
The exceptional tiles do not overlap at the
right angled line, and so it is impossible to
take a rational approximation. In the proof we
just have to deal with these irrational vertices.