Geodesic
Relationship between mathematical and physical constants
Teoretičeskaâ i matematičeskaâ fizika, Tome 18 (1974) no. 3, pp. 427-428

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Euler's formula ($e^{\pi i}=-1$) is used to obtain a relationship between mathematical ($\pi$, $e$, $i=\sqrt{-1}$) and physical ($\varepsilon$, $h$, $c$) constants: $$ \alpha^{-1}=\frac{hc}{\varepsilon^2}= \frac{e^{2\pi}+2e^{-\pi}+4\pi}{4}=137,0361134938\dots. $$ 
@article{TMF_1974_18_3_a13,
     author = {A. K. Lorents},
     title = {Relationship between mathematical and physical constants},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {427--428},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1974_18_3_a13/}
}
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A. K. Lorents. Relationship between mathematical and physical constants. Teoretičeskaâ i matematičeskaâ fizika, Tome 18 (1974) no. 3, pp. 427-428. http://geodesic.mathdoc.fr/item/TMF_1974_18_3_a13/