The Hexagrid Theorem

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 the  door 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 the
 floor lines .

2. The red lines of positive
slope are parallel to the 
sides of R(p/q). These are
called the  wall lines 
or  walls  for 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 Theorem  
The 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 the  rooms 
made 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 the  arithmetic 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.