Logarithmic spiral

Logarithmic spiral (pitch 10°)

Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter b = 0.1759.

Romanesco broccoli, which grows in a logarithmic spiral

A section of the Mandelbrot set following a logarithmic spiral

An extratropical cyclone 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

Polygon spiral

A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

Definition

In polar coordinates $(r, \theta)$ the logarithmic curve can be written as^[1]

r = ae^{b\theta}\,

\theta = \frac{1}{b} \ln(r/a),

with $e$ being the base of natural logarithms, and $a$ and $b$ being arbitrary positive real constants.

In parametric form, the curve is

x(t) = r(t) \cos(t) = ae^{bt} \cos(t)\,

y(t) = r(t) \sin(t) = ae^{bt} \sin(t)\,

with real numbers $a$ and $b$ .

The spiral has the property that the angle $\phi$ between the tangent and radial line at the point $(r, \theta)$ 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 of $\mathbf{r}(\theta)$ 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 $b = 0$ ( $\textstyle\phi = \frac{\pi}{2}$ ) the spiral becomes a circle of radius $a$ . Conversely, in the limit that $b$ approaches infinity ( $\phi$ → 0) the spiral tends toward a straight half-line. The complement of $\phi$ is called the pitch.

Spira mirabilis and Jacob Bernoulli

Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiral was placed there instead.^[2]^[3]

Properties

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

Logarithmic spirals are self-similar in that the result of applying any similarity transformation to the spiral is congruent to the original untransformed spiral. Scaling by a factor $e^{2 \pi b}$ , where b is the parameter from the definition of the spiral, with the center of scaling at the origin, gives the same curve as the original; other scale factors give a curve that is rotated from the original position of the spiral. Logarithmic spirals are also congruent to their own involutes, evolutes, and the pedal curves based on their centers.

Starting at a point $P$ and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as $\theta$ goes toward $-\infty$ is finite. This property was first realized by Evangelista Torricelli even before calculus had been invented.^[4] The total distance covered is $\textstyle\frac{r}{\cos(\phi)}$ , where $r$ is the straight-line distance from $P$ to the origin.

The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. (Up to adding integer multiples of $2\pi i$ to the lines, the mapping of all lines to all logarithmic spirals is onto.) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.

The function $x \mapsto x^k$ , where the constant $k$ is a complex number with non-zero imaginary part, maps the real line to a logarithmic spiral in the complex plane.

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.