# Logarithmic spiral

File:Logarithmic spiral.png
Logarithmic spiral (pitch 10°)
File:NautilusCutawayLogarithmicSpiral.jpg
Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it Spira mirabilis, "the marvelous spiral", and wanted one engraved on his headstone. Unfortunately, an Archimedean spiral was placed there instead.

## Definition

In polar coordinates (r, θ) the curve can be written as

$\displaystyle r = a b^\theta\ \mbox{or}\ \theta = \log_{b} (r/a)$ , hence the name "logarithmic"

and in parametric form as

$\displaystyle x(\theta) = a b^\theta \cos(\theta)\,$
$\displaystyle y(\theta) = a b^\theta \sin(\theta)\,$

with positive real numbers a and b. a is a scale factor which determines the size of the spiral, while b controls how tightly and in which direction it is wrapped. For b >1 the spiral expands with increasing θ, and for b <1 it contracts.

The spiral is nicely parametrised in the complex plane: zt, given a z with Im(z)≠0 and |z|≠1.

In differential geometric terms the spiral can be defined as a curve c(t) having a constant angle α between the radius or path vector and the tangential vector

$\displaystyle \arccos \frac{\langle \mathbf{c}(t), \mathbf{c}'(t) \rangle}{\|\mathbf{c}(t)\|\|\mathbf{c}'(t)\|} = \alpha$

If α = 0 the logarithmic spiral degenerates into a straight line. If α = ± π / 2 the logarithmic spiral degenerates into a circle.