&quot;it is _the_ number that is hardest to approximate by any rational number&quot;<p>I&#x27;m not sure what you mean by this. Any irrational number can be approximated to an infinite number of different levels of precision by an infinite number of different rational numbers. And rational approximations to phi can be trivially generated by any Lucas sequence, starting from an infinite number of different possible seeds (not just the &#x27;1, 1&#x27; seeds of Fibonacci).<p>Approximating e, pi and the square root of 2 by a rational number is equally &#x27;difficult&#x27;. Phi is <i>exactly</i> (1 + 5^1&#x2F;2)&#x2F;2 - or, to put it another way, a half, plus half the square root of five. Are you saying that the square root of five is &#x27;uniquely&#x27; hard to approximate with a rational number?<p>If &#x27;anything that evolves to reproduce periodicity&#x27; tries to approximate this (&#x27;hardest to approximate&#x27;) number, then you surely have many specific examples of places in nature where close approximations to phi can be reliably found.<p>And that doesn&#x27;t mean &#x27;spirals that sort of look a bit fibonacci-ish even though the center in no way divides its diameters in the golden ratio&#x27;. That means, like, you can point to a plant and say &#x27;the ratio of successive buds on the stem of <i>Fooii Bariensis</i> are always in a ratio of precisely 1.62&#x27;.<p>But then to further privilege Fibonacci, not just the golden ratio, you&#x27;d have to further show that that 1.62 ratio wasn&#x27;t just a real approximation of the golden ratio but is actually 1.6181818..., a rational derived from the specific 89&#x2F;55 approximation to phi produced by the Fibonacci sequence. And then show a mechanism whereby the plant actually uses the fibonacci recursion in its growth patterns somewhere to generate this precise ratio rather than some other ratio.<p>And then you&#x27;d need to find several such examples to back up your claim that this kind of pattern is a common attractor in evolutionary space.<p>It&#x27;s just not there, sorry. There&#x27;s just no reason for growth patterns to favor phi, or Fibonacci numbers.