It is argued in the paper http://cogprints.org/3667/ (Coincidence, data compression, and Mach’s concept of “economy of thought”, by J.S. Markovitch) that most likely it is not simply a coincidence that $e^\pi-\pi$ is an almost integer. However I don't find the arguments very convincing.

It is argued in the paper http://cogprints.org/3667/ (Coincidence, data compression, and Mach’s concept of “economy of thought”, by J.S. Markovitch) that most likely it is not simply a coincidence that $e^\pi-\pi$ is an almost integer. However I don't find the arguments very convincing.

P.S. A random thought. It is known (see http://aapt.scitation.org/doi/10.1119/1.3456565 The rolling sphere, the quantum spin, and a simple view of the Landau–Zener problem, by A.G. Rojo and A.M. Bloch) that $e^{-\pi}$ is related to the rotation angle of the North pole of the unit sphere when the sphere rolls along the Cornu spiral $\varphi=\frac{1}{4}s^2$ from its one pole to the another. So, if the mentioned approximate identity is not a coincidence, there should exist some approximate description of this rolling that produces $e^{-\pi}\approx \frac{1}{20+\pi}$.