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^2r/da2 = dr/da = 0, r=radius, a=angle
whereas parabolas are everywhere changing:
d^2r/da2=C
and logarithmic spirals have infinitely dynamic curvature
d^(n)r/da^n = e^(n p) f(a)

 

 

 

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.

The silhouette of the base of a Greek column has a number of log-spiral sections.