Coloring a modular origami

Modular origami is a type of origami that uses many sheets of paper. Each sheet is folded into a module (or unit), and then these modules are assembled creating a more complex object.

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Is it a rational number?

I was checking "The Irrationals" by Julian Havil in a bookstore, and I found this formula:

 \lim_{m\rightarrow\infty} \lim_{n\rightarrow\infty} \cos^{2n}(m!\pi x) = \left\{ \begin{array}{l l}1: & x \text{ is rational}\\ 0: & x \text{ is irrational} \end{array} \right.

It was alone in a page at the beginning of the book, even before the table of contents. No explanation, no hint, no caption. Nothing, just the formula. At first it looked like some real mathemagic to me, and I had no time to find where this formula was explained inside the book. I was very curious for two days until I finally tried to understand it. :-)

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Babbage, Analysis of Algorithms, and Analytic Combinatorics

"As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise — by what course of calculation can these results be arrived at by the machine in the shortest time?"
— Charles Babbage, 1864.

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