The Myth:
The shell of a nautilus (when cut in half lengthwise) forms a golden spiral.
The Reality:
The shell of a nautilus forms a logarithmic (or equiangular) spiral. The golden spiral is a logarithmic spiral, just not one that we see in nature. The ratio that actually seems to be a better approximation of what has been measured in nature is about 4/3, but there is a great deal of variability.
References:
[Fal05] Clement Falbo, The golden ratio–a contrary viewpoint, The College Mathematics Journal 36 (2005), no. 2, 123.
[Liv02] Mario Livio, The golden ratio, Broadway Books, New York, 2002, The story of phi, the world’s most astonishing number. MR MR1938220 (2003k:11025)