Logarithmic spiral

The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").

[1][2] More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

The logarithmic spiral is distinct from the Archimedean spiral in that the distances between the turnings of a logarithmic spiral increase in a geometric progression, whereas for an Archimedean spiral these distances are constant.

the logarithmic spiral can be written as[3]

being the base of natural logarithms, and

The logarithmic spiral with the polar equation

can be represented in Cartesian coordinates

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.

has the following properties (see Spiral): The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch angle 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.

In several natural phenomena one may find curves that are close to being logarithmic spirals.

Logarithmic spiral ( pitch 10°)
A section of the Mandelbrot set following a logarithmic spiral
Definition of slope angle and sector
Animation showing the constant angle between an intersecting circle centred at the origin and a logarithmic spiral.
Examples for
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 , resulting in a pitch of .