The Hilbert Curve is a space filling fractal pattern with the purpose of mapping all natural numbers onto the 2d plane.
From a conventional standpoint, this feels nearly impossible as it feels as if there is no intuitive way to fill 2d space with a single dimensional line.
Why is it Useful over a Simpler Curve?
One can imagine that if a “space filling” curve were to exist, then it would entail repeating a pattern with infinite precision, approaching a state of a perfect mapping from 1 to 2 dimensional.
But why does this not work for any curve, and why is the Hilbert Curve an exception?
It is because of how any given natural number input to the curve will “approach” a certain coordinate as the Hilbert Curve value approaches :
Think of the definition of a Hilbert Curve as something like the following:
Other space filling curve options don’t have this property, and natural number mappings will jump around and never find a stable value.
It is for this reason, that we can prove that as the Hilbert Curve approaches infinite precision, there exists a mapping of all natural numbers to the 2d coordinate plane, such that every point in the plane has a natural number associated with it.