Conwayโs Big Picture consists of all pairs of rational numbers with and with . Recall from last time that stands for the lattice
and we associate to it the rational matrix
If is a natural number we write and call the corresponding lattice number-like, if we drop the zero and write .
The Big Picture carries a wealth of structures. Today, we will see that it can be factored as the product of Bruhat-Tits buildings for , over all prime numbers .
Hereโs the factor-building for , which is a -valent tree:

To see this, define the distance between lattices to be
where is the smallest strictly positive rational number such that .
We turn the Big Picture into a (coloured) graph by drawing an edge (of colour , for a prime number) between any two lattices distanced by .
The -coloured subgraph is -valent.
The -neighbours of the lattice are precisely these lattices:
And, multiplying the corresponding matrices with tells us that the -neighbours of are then these lattices:
Here's part of the -coloured neighbourhood of

To check that the -coloured subgraph is indeed the Bruhat-Tits building of it remains to see that it is a tree.
For this it is best to introduce operators on lattices
defined by left-multiplying by the matrices
The lattice lies closer to than (unless is a number) whereas the lattices lie further, so it suffices to show that the operators
form a free non-commutative monoid.
This follows from the fact that the operator
is given by left-multiplication with the matrix
which determines the order in which the occur.
A lattice at distance from can be uniquely written as
which gives us the unique path to it from .
The Big Picture itself is then the product of these Bruhat-Tits trees over all prime numbers . Decomposing the distance from to as
will then allow us to find minimal paths from to .
But we should be careful in drawing -dimensional cells (or higher dimensional ones) in this โproductโ of trees as the operators
for different primes and do not commute, in general. The composition
has as numerator in the upper-right corner and this number can be uniquely(!) written as
That is, there are unique operators and such that
which determine the -cells
These give us the commutation relations between the free monoids of operators corresponding to different primes.
For the primes and , relevant in the description of the Moonshine Picture, the commutation relations are