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").[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 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.
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.[4][5]
Properties
The logarithmic spiral has the following properties (see
Spiral):
Pitch angle: with pitch 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.[6]
Sector area:
Inversion:Circle inversion () maps the logarithmic spiral onto the logarithmic spiral
Rotating, scaling: Rotating the spiral by angle yields the spiral , which is the original spiral uniformly scaled (at the origin) by . Scaling by gives the same curve.
Self-similarity: A result of the previous property: 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).
Relation to other curves: Logarithmic spirals are congruent to their own
involutes,
evolutes, and the
pedal curves based on their centers.
Complex exponential function: 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 :
The pitch 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 (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. Here follow some examples and reasons:
The approach of a
hawk to its prey in
classical pursuit, assuming the prey travels in a straight line. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.[7]
The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the Sun (or Moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.[8]
The arms of spiral
galaxies.[9] The
Milky Way galaxy has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees.[10] However, although spiral galaxies have often been modeled as logarithmic spirals,
Archimedean spirals, or
hyperbolic spirals, their pitch angles vary with distance from the galactic center, unlike logarithmic spirals (for which this angle does not vary), and also at variance with the other mathematical spirals used to model them.[11]
The nerves of the
cornea (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).[12]
Many
biological structures including the shells of
mollusks.[14] In these cases, the reason may be construction from expanding similar shapes, as is the case for
polygonal figures.
A kerf-canceling mechanism leverages the self similarity of the logarithmic spiral to lock in place under rotation, independent of the kerf of the cut.[16]
A logarithmic spiral antenna
Logarithmic
spiral antennas are frequency-independent antennas, that is, antennas whose radiation pattern, impedance and polarization remain largely unmodified over a wide bandwidth.[17]
When manufacturing mechanisms by subtractive fabrication machines (such as
laser cutters), there can be a loss of precision when the mechanism is fabricated on a different machine due to the difference of material removed (that is, the
kerf) by each machine in the cutting process. To adjust for this variation of kerf, the self-similar property of the logarithmic spiral has been used to design a kerf cancelling mechanism for laser cutters.[18]
Logarithmic
spiral bevel gears are a type of spiral bevel gear whose gear tooth centerline is a logarithmic spiral. A logarithmic spiral has the advantage of providing equal angles between the tooth centerline and the radial lines, which gives the meshing transmission more stability.[19]
In
rock climbing,
spring-loaded camming devices are made from metal cams whose outer gripping surfaces are shaped as arcs of logarithmic spirals. When the device is inserted into a rock crack, the rotation of these cams expands their combined width to match the width of the crack, while maintaining a constant angle against the surface of the rock (relative to the center of the spiral, where force is applied). The pitch angle of the spiral is chosen to optimize the friction of the device against the rock.[20]
^C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure and regeneration,"
Invest Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.
^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.
ISSN1678-5878.
^Todesco, Gian Marco (2018). "Weird gears". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 6: Between Culture and Mathematics. Springer International Publishing. pp. 179–193.
doi:
10.1007/978-3-319-93949-0_16.
ISBN9783319939490.