# 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:*
(dr*_{2}/da_{2})^{2} =
(dr_{1}/da_{1}) (dr_{3}/da_{3}),
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:

*d*^{2}r/da^{2} = dr/da = 0,
r=radius, a=angle whereas parabolas are everywhere changing:
*d*^{2}r/da^{2}=C and logarithmic
spirals have infinitely dynamic curvature
*d*^{n}r/da^{n} = e^{(n p)} f(a)

## 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.
More on the artificial artist:

Last revised 10jul96Matthew Brand / MIT Media Lab / brand@media.mit.edu