magical isosceles triangle and 13/15 ratio

Question

It seems eerily magical that $\dfrac {13}{15}$ corresponds within $99.926$ percent accuracy to the height of an isosceles triangle the height of isosceles $= \sqrt {.75} = 0.8660254...$ and $\dfrac {13}{15}=0.866666...$

Please explain to me the statistical reason for which this could be a simple coincidence? This is how close $\sqrt{.75}$ is to $\dfrac {13}{15}$:

it's only different by a factor of $0.000739918710263$

i.e. $7.399$ thousandths..

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Answer 1

$\sqrt{\dfrac{3}{4}} = \sqrt{\dfrac{675}{900}}$ while $\dfrac{13}{15} = \sqrt{\dfrac{676}{900}}$.

So the absolute difference between their squares is $\dfrac{1}{900}$. This is small (and smaller still after taking the square roots) but not particularly special.

Answer 2

I think this coincidence is one of many that can be attributed to the "strong law of small numbers". Although it is tempting to assume that there is some sort of mystical connection there after finding such a remarkable coincidence, it is doubtful that there is one, since these coincidences occur very often and, from a statistical standpoint, inevitably.

One of my favorite numerical coincidences is that $e^\pi - \pi$ is very close to $20$. As for some more, here's a relevant xkcd comic which lists a bunch, some of which seem surprisingly useful.