Logarithmic spiral

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Logarithmic spiral (pitch 10deg) Logarithmic Spiral Pylab.svg
Logarithmic spiral (pitch 10°)
A section of the Mandelbrot set following a logarithmic spiral Mandel zoom 04 seehorse tail.jpg
A section of the Mandelbrot set following a logarithmic spiral

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 lini"). [1] 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".

Contents

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.

Definition

In polar coordinates the logarithmic spiral can be written as [2]

or

with being the base of natural logarithms, and being real constants.

In Cartesian coordinates

The logarithmic spiral with the polar equation

can be represented in Cartesian coordinates by

In the complex plane :

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. [3] [4]

Properties

Definition of slope angle and sector Spiral-log-st-se.svg
Definition of slope angle and sector

The logarithmic spiral has the following properties (see Spiral):

with polar slope angle (see diagram).
(In case of angle would be 0 and the curve a circle with radius .)
Especially: , if .
This property was first realized by Evangelista Torricelli even before calculus had been invented. [5]
Examples for
a
=
1
,
2
,
3
,
4
,
5
{\displaystyle a=1,2,3,4,5} Spiral-log-a-1-5.svg
Examples for
Scaling by gives the same curve.
A scaled logarithmic spiral is congruent (by rotation) to the original curve.
Example: The diagram shows spirals with slope angle and . Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles resp.. All spirals have no points in common (see property on complex exponential function).
The polar slope angle of the logarithmic spiral is the angle between the line and the imaginary axis.

Special cases and approximations

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (polar slope 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 nature

Low pressure system over Iceland.jpg
An extratropical cyclone over Iceland shows an approximately logarithmic spiral pattern
Messier51 sRGB.jpg
The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy
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
{\displaystyle b=0.1759}
, resulting in a pitch of
arctan
[?]
b
[?]
10
[?]
{\displaystyle \arctan b\approx 10^{\circ }}
. Nautilus Cutaway with Logarithmic Spiral.png
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 .

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:

In engineering applications

A Kerf Canceling Mechanism (bearing).gif
A kerf-canceling mechanism leverage the self similarity of the logarithmic spiral to lock in place under rotation, independent of the kerf of the cut. [14]
ILA Berlin 2012 PD 128.JPG
A logarithmic spiral antenna

See also

Related Research Articles

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References

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  3. Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number . New York: Broadway Books. ISBN   978-0-7679-0815-3.
  4. Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes". p. 206.
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  16. Roumen, Thijs; Apel, Ingo; Shigeyama, Jotaro; Muhammad, Abdullah; Baudisch, Patrick (2020-10-20). "Kerf-Canceling Mechanisms: Making Laser-Cut Mechanisms Operate across Different Laser Cutters". Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology. Virtual Event USA: ACM: 293–303. doi:10.1145/3379337.3415895. ISBN   978-1-4503-7514-6.
  17. Jiang, Jianfeng; Luo, Qingsheng; Wang, Liting; Qiao, Lijun; Li, Minghao (2020). "Review on logarithmic spiral bevel gear". Journal of the Brazilian Society of Mechanical Sciences and Engineering. 42 (8): 400. doi: 10.1007/s40430-020-02488-y . ISSN   1678-5878.