The Box Theorem
Billiard King homepage
The demo explains the Box Theorem, a
result about the arithmetic graph G(p/q)
associated to a rational parameter A=p/q
The applet on the right draws these graphs.
Press the GO button to activate the applet.
The Basic Result
Let p/q be a rational with pq odd.
Define V=(q,-p) and
Let R(p/q) be the parallelogram
with vertices 0; V; W; V+W;
Let L(p/q) denote the line segment
joining the midpoint of the bottom
segment of R(p/q) to the center of R(p/q).
The Box Theorem
One period of G(p/q) is contained
in R(p/q), and disjoint from L(p/q).
The Box Theorem implies that G(p/q) rises
at least q/4 units from the bottom of R(p/q).
This implies that the orbit of (1/q,1) has
diameter at least q/2.
The Box Theorem is a consequence of the
Embedding Theorem and the Hexagrid Theorem .
The Embedding Theorem says that the
arithmetic graph is an embedded curve.
Similar results hold when pq is even,
though we haven't given a proof.
In this case, we would let R(p/q) be
the parallelogram with vertices
-V, V, -V+2W, V+2W
In the odd case, there is a finer decomposition,
which we call the Father-Son (FS) decomposition.
The large R(p/q) contains the union FS(p/q) of two
smaller parallelograms F(p/q) and S(p/q)
such that one period of G(p/q) is contained in FS(p/q).
I think of S as standing for "son's room"
and F as standing for "father's room".
You can toggle the display on the
applet to show this decomposition.
The dividing line between F(p/q) and S(p/q)
has arithmetic significance. There is a
unique fraction b/a such that a lies in (0,q)
Our dividing line is parallel to the sides of
R(p/q) and contains the lattice point (a,-b).
The ceiling of S(p/q) is halfway between
the top and bottom of R(p/q).
The portion of G(p/q) in each of S(p/q)
and F(p/q) seems to have approximate
bilateral symmetry. I don't have a proof
We deduce the father-son decomposition
from 4 ingredients
1. The Hexagrid Theorem .
2. The rotational symmetry of the
extended arithmetic graph.
3. The Period Copying Theorem .
4. induction on the complexity
of the rational number.