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The Hexagrid Theorem is a structural result that implies the Box Theorem. The applet on the right demonstrates the Hexagrid Theorem. Press the GO button to activate the applet.## The Extended Arithmetic Graph

We explained the extended arithmetic graph at the end of our discussion about the arithmetic graph. The basic arithmetic graph is a single component of the extended arithmetic graph. Assuming that you activated the applet, the yellow curve shown at the bottom is one period of of the basic arithmetic graph. The remaining curves are part of the extended arithmetic graph. You should turn on the 'box' display (and then turn it off again) just to see how the plot we have made relates to the box R(p/q) discussed in connection with the Box Theorem. The extended arithmetic graph EG(p/q) is periodic with respect to a certain lattice of translations. A fundamental domain for this lattice is obtained by stacking (p+q)^2/4 copies of R(p/q) on top of each other. This applet lets you draw the portion of EG(p/q) contained in up to the first 8 boxes on the stack. You change the amount using the arrow key. Initially, there are 3 shown.## The Hexagrid

The hexagrid is made from 6 infinite families of parallel lines. You should click on the 'hexagrid' button on the applet display to see the hexagrid. The Hexagrid Theorem gives information about how the arithetic graph sits with respect to the hexagrid. There are 4 infinite families of blue lines. These are called thedoor lines. The reason for the name will become clear momentarily. The red lines of negative slope are parallel to the top and bottom of R(p/q). These are called thefloor lines. 2. The red lines of positive slope are parallel to the sides of R(p/q). These are called thewall linesorwallsfor short. You can turn on and off the display of these various families by toggling the grid display on the applet. Here is the main result:Hexagrid TheoremThe extended arithmetic graph never crosses a floor. Moreover, the extended arithmetic graph only crosses a wall within one unit of where a door line crosses the wall.This result only works for odd rationals. I give a proof in my monograph . There should be a similar result for even rationals, but I haven't explored it. This applet only lets you see the odd rational case. We have highlighted the portions of the extended arithmetic graph that actually cross walls. The remaining components are confined to theroomsmade by adjacent walls and doors. If you turn on both the hexagrid and the box display, and focus your attention on the drawn box, you can see that the Hexagrid Theorem immediately implies the Box Theorem. The point is that the intersection of a door line with the center wall of R(p/q) necessarily occurs more than halfway up the wall. We have currently chosen the parameter 13/29. You can change the parameter to any odd rational p/q with p>1 and q<40. The theorem also works for p=1, but we excluded this case in the applet to avoid having to a bit of extra programming.## Constructing the Hexagrid

To make the Hexagrid Theorem precise, we need to explain how the hexagrid is constructed. Let L be the line extending the bottom of R(p/q), the basic box. If you turn on the hexagrid display and the kite display, You will see a green quadrilateral Q(p/q) appear. Q is affinely equivalent to the original kite K(p/q). We call Q thearithmetic kite. The floor lines are parallel to the short diagonal of Q. The spacing between consecutive floors is the same as the spacing between the top and bottom of R(p/q), the basic box. The wall lines are parallel to the long diagonal of Q. The spacing between consecutive walls is half that of the spacing between the sides of R(p/q). Each door line is parallel to one of the sides of Q. Since Q has 4 sides, we get 4 infinite families of parallel lines. The spacing between the door lines within a family is easily deduced by the spacing between the floor and wall lines. Alternatively, as we see that the door lines only intersect the line extending the bottom of R(p.q) at points of the form k(q,-p) where k is an integer. It only remains to describe the arithmetic kite Q. The bottom vertex is (0,0) The top vertex is (x,y) where x=q y=(p+q)^2-2p^2)/(2q) The left vertex is (0,(p+q)/2) The right vertex is (x,y) where x=pq/(p+q) y=((p+q)^2-4p^2)/(2p+2q)## Rotating the Hexagrid

It is a consequence of the Master Picture Theorem that the extended arithmetic graph has a canonical extension to the entire Z+Z lattice. This extension has both translation and order 2 rotational symmetry. In particular, there is an order 2 rotational symmetry Y about the point (-a,b) where b/a is the same fraction that arises in the father-son decomposition The symmetry Y does not preserve the Hexagrid H. This means that Y(H) is a second hexagrid that has all the same properties as the first hexagrid. In other words, the Hexagrid Theorem applies to 2 different hexagrids. You can see Y(H) on the applet, using the "rotate hexagrid" control panel. Notice that one of the walls of the rotated hexagrid divides the bottom yellow component of the arithmetic graph at the same place that the father-son decomposition divides this yellow curve. This fact is the beginning of our proof that the father-son decomposition really holds.