Update (adapted from a comment below): I came across this image after launching into my now familiar goldenratio-bollocks [1] rant over at Moon's interesting and eclectic blog.

The post was, innocently enough, about the Fibonacci sequence a remarkable mathemical series that crops up with suprisingly regularity in many living systems. Its properties ensure that seeds are packed with maximum efficiency, it also appears in the number of petals of a flower and the placement of leaves around a stem. The latter distribution ensures that all leaves receive the maximum amount of sunlight. It is a formula that connects the notions of beauty and efficiency.


Well, that's of course all very well but all it really took to set me off was the inevitable appearance of the much beloved shell of the Nautilus pompilius. No discussion of the relationship between the Fibonacci series and the natural world ever seems complete without making reference to this beautiful creature. The problem is that it's all just bollocks.


Take for example this subdividing of a Golden Rectangle overlaid on top of this image of the shell. The idea is that you can produce a spiral and that matches the shell's one and this is supposed to say something deep about shell spirals and the Golden Ratio (and therefore the Fibonacci sequence which is intimately connected with it).


But, as a close examination will show, the spiral on the shell does not actually correspond to the Golden spiral. In fact, it's not even close. The spirals look similar because they are both examples of a class of mathemtical curves known as equiangular or logarithmic spirals. These curves are very common in nature and especially in seashells because their constant angle permits the creature living inside it to grow at a constant rate. My point here is that while the Nautilus and the Golden spiral are both equiangular spirals, they curve at a different rates and so it is incorrect to say that the Golden ratio (or the Fibonacci sequence) is involved [2].

Given that the Nautilus is an endangered species, I thought that at least there's a small hope that if more people appreciated this fact then perhaps a few fewer would be sawn open and sold over the internet for $15.95. Of course the reality is that they are very pretty shells anyway, and logarithmic spirals are mathematically appealing in their own right. Nautilus shells even have a fractal dimension so I don't really hold out all that much hope.

More articles on the Golden Ratio linked here.

[1] bollocks - the *-bollocks meme was, I believe, first started by Daniel Davies (d-squared) over at Crooked Timber when he coined the term Globollocks as a method for rating breathless and/or mendacious “Globalisation” pieces from neo-liberal commentators. He later extended the formulation so as to state that, to paraphase Arthur C Clarke, "any sufficiently advanced punditry is indistinguishable from bollocks".

The meme has since be put to good use by Jamie K at Blood & Treasure who has coined the term Chinabollocks to describe breathless media commentry mystifying the ancient wisdom of the Chinese nation. Classic quote: "Has anyone ever fought an actual military campaign in accordance with the doctrines of Sun Tzu? ... [I]t's in occasional vogue with MBA students, and [gets] eagerly thumbed through by middle management meatballs. So, then, officially and by definition, it's bollocks."

[2] D'Arcy Thompson is sometimes given the blame for connecting the Nautilus with the Golden Ratio. In reality he said nothing of the kind. In his famous book On Growth and Form he merely noted that its shell had a logarithmic spiral which he then explained in biological terms. My Canto edition, however, does show the spiral of the Nautilus shell on its front cover.