Logarithmic Spirals

You see logarithmic spirals every day. They are the natural growth curves of plants and seashells, the celebrated golden curve of ancient Greek mathematics and architecture, the optimal curve for highway turns. Peer into a flower or look down at a cactus and you will see a pattern of logarithmic spirals criss-crossing each other like so:

This elegant spiral pattern is called phyllotaxis and it has a mathematics that is equally lovely. One reason why the logarithmic spiral appears in nature is that it is the result of very simple growth programs such as

grow 1 unit, bend 1 unit
grow 2 units, bend 1 unit
grow 3 units, bend 1 unit
and so on...

Any process which turns or twists at a constant rate but grows or moves with constant acceleration will generate a single logarithmic spiral. An equally similar cellular automata program will generate phyllotaxis.

Logarithmic curves in the artificial artist

The Artificial Artist sculpts with curves that are piecewise logarithmic, meaning that over small neighborhoods a contour will have curvature like a logarithmic spiral. This is partly motivated by ease of computation: Three equidistant points on the contour have a geometric progression of local curvatures: (dr₂/da₂)² = (dr₁/da₁) (dr₃/da₃), r=radius, a=angle from which it is possible to derive their polar parametric form: r = f(a) = e^{(p a)} This is related to the property if you draw a line from the origin to infinity, everywhere it crosses the spiral it will cross at the same angle.

Dynamic curves

Piecewise logarithmic curves are, of course, also an aesthetic choice: Logarithmic spirals are visually dynamic curves. Dynamic curves have varying curvature that suggest energy, as if there were sources of tension that keep the contour from relaxing into a simple equilibrium shape. This is especially appropriate for animals, whose skin and skeletons are held in dynamic tension by muscles, producing non-equilibrium curves:

Here is an example pointed out by the visual psychologist Rudolph Arnheim. The Syndey Opera House was originally designed with a roof of parabolic shells, meant by the architect Jorn Utzon to suggest the sails of ships coming into the harbor. Construction costs prompted a change to circular shells, but these dulled the building, because circles have constant curvature:

d²r/da² = dr/da = 0, r=radius, a=angle whereas parabolas are everywhere changing: d²r/da²=C and logarithmic spirals have infinitely dynamic curvature dⁿr/daⁿ = e^{(n p)} f(a)

Sydney opera house

Human vision and logarithmic spirals

Incidentally, logarithmic spirals are, like fractals, self-similar at all scales (f(ka) = e^k f(a)). This may be one of the reasons why they are striking to human vision: Your brain performs early visual computations at several scales (demagnifications of the image) and compares the results. A logarithmic spiral will self-correlate across all scales. In a neural network implementation of the artificial artist's coupled potential fields, this observation was exploited to produce piecewise logarithmic curves as a side-effect of curvature detection.

Where to find them

Other places you will find logarithmic spirals:

Many buildings, from Greek temples to modern skyscrapers, are proportioned in accordance with the golden mean, a constant which is the ratio of the sides of a rectangle circumscribed about a logarithmic spiral.
The silhouette of the base of a Greek column has a number of log-spiral sections.
Leaf edges in some plants (e.g., begonias) roughly follow logarithmic spirals.
Euler proposed that tracks curved in a logarithmic spiral were optimal for slowing and turning trains in railyards.
Many artists have consciously and unconscously incorporated it into their work.
The horns of many bovids such as gazelles have growth programs are similar to that of seashells.

If you know of other good examples or pictures, please send me mail.