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.