# The Master Picture Theorem

Monograph guide
Billiard King homepage

```The Master Picture Theorem gives a
kind of formula for the arithmetic
graph.  The applet
at right illustrates the result.
Press the go button to activate
the applet.

The Lattice
Let A be the kite parameter
and let 2e be the offset.
We consider the lattice G
generated by the vectors

(1+A,0,0)    (1-A,1+A,0)   (-1,-1,1)

This lattice acts on 3 dimensional
space.  Any translation of the
solid body

X=[0,1+A] x [0,1+A] x [0,1]

is a fundamental domain for the
action of G on R^3.  The bottom
half of the applet shows slices
of 2 such fundamental domains.
By dragging the mouse over the
long blue strip in the middle of
the applet, you can change the
slice.

Each fundamental domain,
which we call L(eft) and R(ight),
is partitioned into polyhedra.
The slices of L and R are all
the same.  (They are all squares.
However, the slices of
the smaller polyhedra in the
partition change.  When you
change slices, you are changing
the z-coordinate.

Classifying Map
For each lattice point
(m,n) we set

t=Am+n+e

and consider the full
orbit G(t,t,t) in R^3.
This is a 3D grid of
points.  G(t,t,t) intersects
L in some polyhedron
L(m,n) and G(t,t,t) intersects
R in some polyhedron
R(m,n).

You can see this in action
by clicking on different
points of the arithmetic
graph, shown on the top
half of the applet, and
then looking at the grid
of white points plotted
in the bottom half.  The
grid you see is the
intersection of G(t,t,t)
with the  plane of
intersection .

The plane of intersection
is a plane of constant
height that contains the
point where G(t,t,t)
intersects L.  This plane
also contains the point
where G(t,t,t) intersects R.
Both points lie in the
plane of intersection,
and an entire 2D subgrid
of G(t,t,t) lies in this
plane.

You select points on the
top half of the applet
using the middle mouse
button.  If you don't have
a 3 button mouse, then
you can use the applet's
3D mouse emulator.  This
is the control panel in
the middle of the applet.

The Main Result

The two polygons L(m,n)
and R(m,n) have colors
attached to them.
We can locate these two
colors on the little
colored checkerboard in
the middle of the applet.
We draw a segment from the
center (grey square) of
the checkerboard to each
of the colors.  This
gives us the local picture
of the arithmetic graph.
This is the content of the
Master Picture Theorem.

In other words, the local
structure of the arithmetic
graph is determined by which
polyhedra in the partitions
of L and R contain points
of G(t,t,t).

Using the mouse/keyboard,
you can change the parameter
of the arithmetic graph.
This lets you see the
Master Picture Theorem
in action for other
parameters.  The applet
is set so that you can
only choose fractions p/q
with q<30.

The Domains
To really describe
the result, we need to
explain the placement
of L and R in the plane,
and also the structure
of the partitions. They
are given by the formulas

X=[0,1+A] x [0,1+A] x [0,1].
L=X+(0,-1,0)
R=X+(2+A,0,0)

The partitions
The partition of R is a
mirror image of the
partition of L.  There is
an isometry from L to R,
carrying the partition
of L to the partition of R.
The linear part of this
isometry is given by
the antipodal map p-->-p.
For this reason, it suffices
to explain the partition
on L.

The partition on L is
a translation of the
partition of X.  For this
reason, it suffices to
explain the partition
on X.  Let us write
X_A to denote the
dependence of X on the
parameter A.

There is a 4 dimensional
convex polytope X whose
A-slice is exactly X_A.
Likewise, there is a
partition of X into
4D polytopes whose slice
at A gives the partition
of X_A at the parameter A.
The polytopes in question
all have vertices with
coordinates either 0,1 or 2.
Thus, the picture you
see in the bottom half of
the applet is really a
2D slice of a 4D partition
by convex integral polytopes.

My  monograph
list the coordinates of these
convex integral polytopes,
and also gives a proof
of the Master Picture Theorem.

```