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Concept
Logarithmic spiral
Summary
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). 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 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.
In polar coordinates the logarithmic spiral can be written as
or
with being the base of natural logarithms, and , being real constants.
The logarithmic spiral with the polar equation
can be represented in Cartesian coordinates by
In the complex plane :
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.
The logarithmic spiral has the following properties (see Spiral):
Polar slope: with polar slope angle (see diagram and animation).(In case of angle would be 0 and the curve a circle with radius .)
Curvature:
Arc length: Especially: , if . This property was first realized by Evangelista Torricelli even before calculus had been invented.
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