Shape and Order in Organic Nature: The Nautilus Pompilius

The Nautilus pompilius is an exquisite example of a planar logarithmic spiral in organic nature, yet many artists and designers claim that its shape is ordained by the Golden Section, symbolically represented by φ, where ϕ = ( 5 + 1 ) / 2 = 1 . 6180

In a logarithmic spiral, shape is conventionally analyzed by examining the growth-ratios of radius vectors. These ratios are constant for a particular curve and form the basis for calculating the acute angle β between a tangent at any point on the curve and its polar radius vector—each spiral has a unique β. The magnitude of β controls the sweep of the curve, and the closer this angle β is to 90° the tighter the spiral. For the Nautilus pompilius illustrated in this paper, the average growth-ratio of radius vectors, for θ=90°, is in the order of 1:1.3158, which yields a constant angle β of about 80°.