# Logarithmic spiral Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral A low pressure area over Iceland shows an approximately logarithmic spiral pattern. The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy.

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".

## Definition

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

$r=ae^{b\theta }\,$ or

$\theta ={\frac {1}{b}}\ln(r/a),$ hence the name "logarithmic". In parametric form, the curve is

$x(t)=r\cos(t)=ae^{bt}\cos(t)\,$ $y(t)=r\sin(t)=ae^{bt}\sin(t)\,$ with real numbers a and b.

The spiral has the property that the angle ɸ between the tangent and radial line at the point (r,θ) is constant. This property can be expressed in differential geometric terms as

$\arccos {\frac {\langle \mathbf {r} (\theta ),\mathbf {r} '(\theta )\rangle }{\|\mathbf {r} (\theta )\|\|\mathbf {r} '(\theta )\|}}=\arctan {\frac {1}{b}}=\phi ,$ The derivative r'(θ) is proportional to the parameter b. In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that b = 0 (ɸ = π/2) the spiral becomes a circle of radius a. Conversely, in the limit that b approaches infinity (ɸ → 0) the spiral tends toward a straight line. The complement of ɸ is called the pitch.

## Spira mirabilis and Jakob Bernoulli

Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. While this curve had already been named by other mathematicians, the name "miraculous" or "marvelous" spiral was given to this curve by Jakob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases, but the shape stays the same with each added curve. Maybe because of this property, the spira mirabilis has evolved in nature, seen in some living beings, such as nautilus shells and sunflower heads. Jakob Bernoulli wanted the shape on his headstone, but, by error, an Archimedean spiral was placed there instead.