An almost eternal theme in Mathematics is **the approximation of the Continuum by the Discrete**. This core idea goes back at least to Archimedes, and remains active to these very days (and quite likely for the next thousand years) .

You have a continuous structure, say a sphere, and you approximate it as the limit of a series of discrete objects, say polyhedra.

I do not need to provide more examples, I am sure you all have plenty.

But, there is also, though by no means so prominent, a reverse direction.

I am not able to date precisely when it made its debut in the history of mathematics, but it most certainly appears in the revolutionary work of Fourier: take a discrete function, say the Heaviside step function, and approximate it via series of trigonometric functions:

So, here a discrete object is realized as a limit of continuous ones (in this case smooth functions).

I am especially intrigued by this possibility, which, pushed to the extreme, will depict a mathematical world where the **Discrete is an emergent phenomenon, out of a Continuum**

Thus I ask to everybody:

can you list active research in the way of approximating discrete structures via smooth ones?

For instance, a polyhedron via a series of smooth manifolds, or examples in analytic number theory, or patterns in finite combinatorics out of .....(fill the dots) .

Any thoughtful and possibly well documented answer will get my vote, regardless the domain chosen (in fact, the more examples I will harvest from heterogeneous disciplines, the happier I shall be).

On the other hand, to get the GREEN the stakes are higher: rather than single examples, a sketch of a general perspective on the Discrete as emerging from the Continuum

**ADDENDUM**: As pointed out by Andreas Blass, I have implicitly conflated two themes here:

**discrete as limit of the continuum****emergence of the discrete from some background**

Point 2) does not seem to necessarily imply point 1) and probably the same applies the other way around. Which one I am interested in? Easy: BOTH OF THEM.

But now that Andreas has already marked this point, the GREEN ANSWER would be, perhaps, a clarification on the relationship between 1 and 2 (inter alia)