Jospepp, it's the Twelfth root of two, not the square root.
Square root of two is 1.41421.....
Twelfth root of two is the magic fret spacing factor 1.0595 (ish).

it's the number you have to multiply by itself 12 times to make two.
Space your frets, or pitches by that factor and by the time you have done it twelve times and you are half way between the nut and the bridge, and/or at double the original pitch.
That's why it works for a twelve tone equal temperament. 24th root of two would give you a quarter tone scale, A scale based on the square root of two would only contain two notes in the octave!
If you are going to base your music theory on mathematics, you must get the maths right!
Personally, I find all the circle of fifths concept something of a case of creating a mnemonic which is harder to remember than the thing it is meant to help you to learn, which is the scale relationships of notes and chords in the different keys.
Cheers
Dave