Rational Number Interface
The purpose of this window is to allow you to enter
exact rational coordinates into McBilliards. The
top right display shows a pair of rational numbers.
This pair may or not coincide with the current
selected in parameter space. There are 2 modes
for selecting rational coordinates.
- Dyadic mode You can use the arrow keys
to change the integer parameter n. Given the current
value of n, the Dyadic mode lets you select coordinates
of the form k/2^n, where N=0,...,2^n.
- Farey mode in this mode, you can select
any fraction p/q. This time the integer n controls the
maximum allowed sum of the coefficients in the
continued fraction expansion of the rational number.
More concretely, a rational number between 0 and 1
can be produced by iterated applications of Farey
addition. Sometimes this is called the
Stern-Brocot process though presumably it
goes by many names. With the parameter set at n you
can select any rational numbers which are produced by
at most n iterations of this process.
Once you decide on a mode to use, you have 3
ways to select a rational point in the parameter space.
- The blue cluster: You can
use the 9 keys in the 3x3
blue cluster. If you click on the top left key, you will
move one "click" to the top and to the left. And so on
for the other buttons. Here "click" means that you move
from the nearest accessible fraction to the next one.
When you do this, the rational number display at the top
left will show the selected point in the parameter space.
The middle button in the blue cluster brings you to
accessible rational number nearest the current point
in parameter space. Once again, accessible
depends on the choice of n.
- Track button If you turn on the
track button then any McBilliards interprets and
(middle mouse button) click in the parameter space
as being automatically followed by a click on the
middle blue button. In this way, a click in the
parameter space automatically produces coordinates
with rational coordinates. If the value of n is
fairly small this mode can be confusing, because the
point which is actually selected might be far from
where you actually clicked.
- Keyboard entry If you click on the rational
number display at top left you can manually enter
in rational numbers from the keyboard. McBilliards
will let you enter rational numbers which do not
lie between 0 and 1, but such coordinates have
no significance. You should choose numbers between
0 and 1.
There is one other feature of the rational number interface.
The grid button toggles the presence of a
grid whose lines have rational equations. There is a
pop-up window called the grids which allows
you to make much fancier background grids. These
grids are very useful for studying infinite families
of billiard paths.