  The topological properties of the quantum Hall effect in a crystalline
lattice, described by Chern numbers of the Hofstadter butterfly quantum phase
diagram, are deduced by using a geometrical method to generate the structure of
quasicrystals: the cut and projection method. Based on this, we provide a
geometric unified approach to the Hofstadter topological phase diagram at all
fluxes. Then we show that for any flux, the bands conductance follow a two
letter symbolic sequence . As a result, bands conductance at different fluxes
obey inflation/deflation rules as the ones observed to build quasicrystals. The
bands conductance symbolic sequences are given by the Sturmian coding of the
flux and can be found by considering a circle map, a billiard or trajectories
on a torus. Simple and fast techniques are thus provided to obtain Chern
numbers at any magnetic flux. This approach rationalize the previously observed
topological equivalences between the Fibonacci and Harper potentials (also
known as the almost Mathieu operator problem) or with other trigonometric
potential, as well as the relationship with Farey sequences and trees.
