The logarithmic or equiangular spiral was first defined symbollically by Descartes in 1638 by the equation

r=ka

which relates the polar coordinates of any point in the spiral and expresess an exponential growth pattern. In nature it is usually associated with the growth of dead tissues forming structures, like shells of many invertebrates, corals and several horns found in mammalian species, by an accretionary process.

Shells

One of the best ways to find logarithmic spirals in nature is watching a nautilus shell.

Logarithmic spiral in a typical nautilus shell. L system: LogSpiral

The shape of many other invertebrates shells also describe logarithmic spirals, but they're not always bounded to a plane. Instead they usually raise in space forming helicospirals.

Helicospiral curve used to generate seashells. L system: Helicospiral

Helicospirals can be used to construct shells by sweeping a cross-section, usually a circle or ellipse, along their path. This procedure has been described as "growing-tube model" or "heliconical-tube model".

Two different views of a mollusc shell. L system: seashell

However, these mechanisms only simulate the geometric form of the shell but not the growth process itself that produces it. It is an incremental process that generates the expanding shell by a constant deposition of new fabric at the existing shell's rim. Thus, the shape remains the same, but the scale grows at a constant rate. Any initial slight imbalance in the speed of growth will be maintained proportionately as the shell gets bigger, leading to spiralling. In the absence of such inbalance, the shell would have instead a conical shape, as can be found in many species of molluscs.

Horns

But spirals can also be found in other organic forms like the coiling of some worms, or horns. In the latter case curvings are often more gentle than in shells. In addition, pairs of horns are specular: one rotates clockwise while the other rotates counterclockwise.

Two different pairs of horns similar to those found in antelopes and markhors. L system: horns

References

Theoretical morphology: the concept and its applications
George R. McGhee Jr.
Columbia University Press, 1998

Spiral symmetry
István Hargittai, Clifford A. Pickover (eds.)
World Scientific, 1992

The self-made tapestry
Philip Ball
Oxford University Press, 1988