In the Fibonacci Spiral, when you divide the area of the square, which holds that specific part of the spiral by the area of that specific section of the spiral, you get the root of the Golden ratio, ($\approx 1.2730$).
Can anyone explain why this happens?
Example: $$ \frac{13^2}{\frac{(13^2\pi)}{4}}$$
The ratio $\frac{13^2}{(13^2\pi)/4}$ is simply $\frac4\pi$, which is approximately $1.27324$, while the square root of the golden ratio, $\sqrt{\frac12(1+\sqrt5)}$, is approximately $1.27202$. Close, but not the same number.
This appears to be one of many "mathematical coincidences" where a number generated by one mathematical formula is close to the number generated by a very different mathematical formula. There are a lot of ways such coincidences can be generated, because there are many operations we might choose to apply to any given number that sound simple enough.
For example, why take the square root of the golden ratio? Obviously because the golden ratio itself is much too big. If the square root were still too big, maybe we could try the cube root. But the square root turns out to come within about $0.1\%$ of the exact result.
Lest that seem too remarkable, an error of $0.1\%$ is not particularly good as mathematical coincidences go. For example, $\pi ^{4}+\pi ^{5}\approx e^{6}$ within $0.000 005\%$.
See https://en.wikipedia.org/wiki/Mathematical_coincidence for many more mathematical coincidences. The paper http://numerical.recipes/whp/NumericalCoincidences.pdf describes an algorithm for finding more of them.