[izwwgqom] Rational roots of 2

The cube root of 2 seems close to 1.26 ("26 cents"), but nothing special happens in the continued fraction expansion.

? 2^(1/3)
1.259921049894873164767210607
? contfrac(2^(1/3))
[1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, 1, 2, 14, 3, 13]
? contfrac(1.26)
[1, 3, 1, 5, 2]

We explore rational approximations to other roots of 2, truncated before large terms in the CF.  The Pari/GP function contfracpnqn was useful.

4th root 1+7/37

5th root 1+40/269

6th root 1+6/49

7th root 1+28/269 .  This is the second instance of denominator 269.

8th root 1+1/11 or 1+41/453

9th root 1+2/25 = 1.08

10th root 1+1/14

11th root 1+8/123

12th root does not have any nice small rational approximation, despite musical pitches and intervals being based on the twelfth root of 2.  2^(1/12) = 3^(1/19) = 1.5^(1/7)

Create a geometric or mechanical demonstration of these approximations.

From elsewhere:

9.1^(1/4) = 33/19
91000^(1/4) = 330/19

https://oeis.org/A093876
http://www.mathpuzzle.com/MAA/34-Keen%20Approximations/mathgames_02_14_05.html