$\sqrt{2} \ln \pi = 1.618892$… is approximately equal to the golden ratio $\phi = 1.618033$… . Is this just a coincidence? Could it be some kind of first-order approximation?
2026-03-25 18:56:34.1774464994
$\sqrt{2} \ln \pi \approx 1.618033…$, the golden ratio. Why?
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$$\frac{433 \pi ^2+540 \pi+156}{304 \pi ^2+457 \pi-652}$$ is in relative error of $5.9\times 10^{-18}$%
Have fun with this.