09 September 2016

Dutch gold: the Fibonacci spiral

(Continuation of part 2)

 Any spiral that has a form parameter equal to 

φ = (√5-1)/2 = 0.61803....

or its inverse
Φ =  (√5+1)/2 = 1.61803....

could be called golden. The most popular choice is fπ/2 = Φ. After each complete turn, the radius of this golden spiral has grown by a factor Φ4=6.853..., while the Nautilus factor is merely 3. No need for a microscope here: 
the golden spiral expands way too fast to (even approximately) match Nautilus Pompilius. 
Among the countless spirals in the universe, it's not impossible that golden spirals accidentally occur, but definitely not in Nautilus Pompilius. This should be curtains for golden Nautilus mysticism, but wait! In Proportion And the Living World (here), Rachel Fletcher writes (p.45)
Golden Mean symmetries are prevalent throughout the natural world, as in the well-known example of the nautilus shell (see Figure 8). (...) Each chamber relates to the next successively larger one according to the ratio of 1:√Φ. (...) The shell as a whole fits perfectly inside a rectangle of similar 1:√Φ. (...) proportions.
Impressive, no? Well... the nautilus shell will fit exactly inside any rectangle if you allow, as Fletcher does, that the fourth side (right in the rectangle above) is not tangent to the curve. Just for fun, here is my Nautilus curve (f =3) perfectly fitting inside a square.

 On the other hand, if you want the four sides to be tangents, the rectangle is unambiguously determined, and its sides are in a ratio of 3:4 (0.75984 according to Geogebra).

Like da Vinci's Vitruvian Man (see here), Nautilus Pompilius prefers simple ratios to learned stuff: its growth factor per complete turn is (give or take a little) 3, and it fits inside a rectangle of proportions (more or less) 3:4. Nothing golden left!


This said, the golden spiral has a nice feature which is due to the mathematical properties of the golden proportion. [Update. Here I got carried away. The only thing special about the golden spiral is that squares can be used instead of rectangles. In fact there is only 1 rectangle whose spiral fits perfectly, and it's not the golden one.] In any logarithmic spiral, the portions corresponding to quarter turns are similar curves—meaning that one is an enlarged or reduced copy of the other—and each of these curves could be approximated by a quarter circle. But in a golden spiral, the successive squares containing these quarter circles fill out a golden rectangle (1 by φ)!


So let's start with a rectangle having OA=1, OB=φ. Cutting off a square (red), we are left with a rectangle similar to the first one, but scaled down from 1 to φ. Cutting off squares three more times (blue, then red, then blue again), we are left with a rectangle similar to the first one (downscaled from 1 to φ4) and in the same position. That everything fits so nicely is a result of the basic property φ+φ2=1. The corners of the successive rectangles move along the three lines OF, AB and CD. Four quarter circles, one in each of the first four squares, form a very decent approximation of one complete turn of the true golden spiral (drawn in gold, pole in Z). After that, this four-quarter-circles curve is repeated in the inner rectangle (downscaled from 1 to φ4), et cetera. The curve consisting of quarter circles is not a spiral, but it fits nicely into a golden rectangle. The true golden spiral, on the other hand, does not: its arcs XE and YF are outside (barely visible, but Geogebra detects everything). Nor are the sides of the rectangle tangent to the spiral.

Two successive squares in the above figure have sides in a proportion of 1:φ. In the so-called Fibonacci spiral successive squares have sides in the proportion of two successive Fibonacci numbers. The larger the numbers, the better the approximation to 1:φ. While easy to construct, this "spiral" (consisting of quarter circles) is but a poor substitute for the golden spiral. For one thing, there is no genuine pole, because there is no continuation beyond the two smallest squares, both of side 1. The "approximate pole" moves with every new step.

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