I apologize for the vagueness. &quot;Hardest to approximate&quot; does require certain definitions first.<p>The reasoning behind this is simple continued fractions, ie. fractions like a_0 + 1&#x2F;(a_1 + 1&#x2F;(a_2 + ...)) = [a_0; a_1, a_2, ...], with a_i\in N. Every irrational number corresponds uniquely to an infinite continued fraction, and the finite &quot;steps&quot; of the SCF are the fractions that best* approximates the irrational numbers.
* a&#x2F;b is &quot;best&quot; at approximating x, if b|x-a&#x2F;b|&lt;=d|x-c&#x2F;d| for all c&#x2F;d\in Q such that d &lt;= b. The name &quot;best&quot; is to distinguish these from the &quot;good&quot; (without the b&amp;d- weights), and I think its also known as &quot;best approximation of of second kind&quot;.<p>The convergents, the finite &quot;steps&quot; of the SCF, are exactly these &quot;best&quot; approximations[1]. Such convergents are include 355&#x2F;113 for pi, and are used for many things, like pianos and most of the different systems of leap years. Fascinating stuff, really.
[1] IIRC, <a href="http:&#x2F;&#x2F;www.math.hawaii.edu&#x2F;~pavel&#x2F;contfrac.pdf" rel="nofollow">http:&#x2F;&#x2F;www.math.hawaii.edu&#x2F;~pavel&#x2F;contfrac.pdf</a> contains a full proof.<p>The size of the a_is determines when there&#x27;s going to be a jump in denominator size. The 355&#x2F;113 approx of pi is right before an 292, which is fairly large, and the next convergent is 103993&#x2F;33102. Phi, being [1;1,1,...], never reaches any such jump in denominator size, and its sequence of convergents (its best approximations, and for phi it&#x27;s the ratio of fibs) converge slower than any irrational not having a trail of ones at the end. From this, one may consider it the number &quot;least like a rational&quot;, or even &quot;the most irrational number&quot;.<p>Its properties are not directly related to the square root of five, as far as I can tell, but it is in this way the uniquely (at least as an infinite tail of a SCF) hardest irrational to approximate.<p>That being said, my initial comment was intended to point out that phi and the fibonacci numbers is quite special, and its special enough that it &quot;should&quot; occur frequently in nature. I never actually meant to comment too deep on the spiral-parts, because I know fairly little of them. My &quot;reduce periodicity&quot;-argument is only based on the thought that pi with its fourth convergent 355&#x2F;113 would almost have a period of 113 (off by ~10^{-7}), while phi with its 11th-ish convergent 233&#x2F;114 would have a not-very-almost-period 114 (off by 0.5). Phi&#x27;s ~10^{-7}-almost-period would be 1597 from its 16th-ish convergent. While one could just take any number, say 123012&#x2F;153281=[0; 1, 4, 15, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 6] and claim that has a longer period, if I calculated correctly, it has a ~10^{-7}-almost-period of 871 from its seventh convergent 699&#x2F;871, which is fairly less than phis. Note: one should probably even multiply the error 10^{-7} with the period for a more correct result, but as I hinted to, this is not my strongest subject.<p>Now, to your example: 1.62 has SCF [1; 1, 1, 1, 1, 1, 2, 2], all quite low, so it should have a fair amount of this &quot;irrationality&quot; that the fibs-ratios have.
It doesn&#x27;t have to be the fibonacci-ratios exactly (though these would be the best choice), but most* numbers trying to optimize on this property will be close to them.
* I won&#x27;t say all, because [x,y,1,1,1,1...] could possibly inherit some properties, but the number itself, 1&#x2F;(x+1&#x2F;(y+1&#x2F;phi)), could be far from phi.<p>EDIT: I only realized now that your plant was hypothetical. Anyway, a quick search yielded this <a href="https:&#x2F;&#x2F;www.mathsisfun.com&#x2F;numbers&#x2F;nature-golden-ratio-fibonacci.html" rel="nofollow">https:&#x2F;&#x2F;www.mathsisfun.com&#x2F;numbers&#x2F;nature-golden-ratio-fibon...</a>
which (if you ignore the &quot;for-kids&quot; language) has fairly good display for one of the properties, and actually mentions the continued fraction.
EDIT2: For a more serious article, see 
<a href="https:&#x2F;&#x2F;plus.maths.org&#x2F;content&#x2F;chaos-numberland-secret-life-continued-fractions" rel="nofollow">https:&#x2F;&#x2F;plus.maths.org&#x2F;content&#x2F;chaos-numberland-secret-life-...</a>
which, all the way down at the bottom, explains that the numbers of SCF ending in [1,1,...] are &quot;noble&quot;, and occur frequently as they &quot;are least susceptible to being perturbed into chaotic instability.&quot;