![Mandel zoom 04 seehorse tail](https://upload.wikimedia.org/wikipedia/commons/b/b5/Mandel_zoom_04_seehorse_tail.jpg)
A logarithmic spiral, equiangular spiral, or growth spiral is a
self-similar spiral
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
that often appears in nature. The first to describe a logarithmic
spiral was
Albrecht Dürer
Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an umlaut) or Due ...
(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
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
, who called it ''Spira mirabilis'', "the marvelous spiral".
The logarithmic spiral can be distinguished from the
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
by the fact that the distances between the turnings of a logarithmic spiral increase in
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
, while in an Archimedean spiral these distances are constant.
Definition
In
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
ic spiral can be written as
or
with
being the base of
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
s, 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
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
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
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
, 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
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as
nautilus
The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.
It comprises six living species in ...
shells and
sunflower
The common sunflower (''Helianthus annuus'') is a large annual forb of the genus ''Helianthus'' grown as a crop for its edible oily seeds. Apart from cooking oil production, it is also used as livestock forage (as a meal or a silage plant), as ...
heads. Jacob Bernoulli wanted such a spiral engraved on his
headstone along with the phrase "
Eadem mutata resurgo
''Eadem mutata resurgo'' is a Latin phrase that literally translates to "''Although changed, I arise the same''".
Background
The word-for-word translation of the phrase is
:"Same having-changed I-rise".
:''Eadem mutata resurgo''.
The sense is bet ...
" ("Although changed, I shall arise the same."), but, by error, an
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
was placed there instead.
Properties
![Logspiral](https://upload.wikimedia.org/wikipedia/commons/e/e0/Logspiral.gif)
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
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and work ...
even before
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
had been invented.
* Sector area:
* Inversion:
Circle inversion
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
(
) maps the logarithmic spiral
onto the logarithmic spiral
![Spiral-log-a-1-5](https://upload.wikimedia.org/wikipedia/commons/9/94/Spiral-log-a-1-5.svg)
* 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
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
: A result of the previous property: A scaled logarithmic spiral is
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
(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
involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or ...
s,
evolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...
s, and the
pedal curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to:
Computers and other equipment
* Footmouse, a foot-operated computer mouse
* In medical transcription, a pedal is used to control ...
s based on their centers.
* Complex exponential function: The
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
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 polar slope angle
of the logarithmic spiral is the angle between the line and the imaginary axis.
Special cases and approximations
The
golden spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter turn it makes.
Approximations of the golden spira ...
is a logarithmic spiral that grows outward by a factor of the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
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 number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s.
In nature
![Nautilus Cutaway with Logarithmic Spiral](https://upload.wikimedia.org/wikipedia/commons/4/48/Nautilus_Cutaway_with_Logarithmic_Spiral.png)
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
Hawks are bird of prey, birds of prey of the family Accipitridae. They are widely distributed and are found on all continents except Antarctica.
* The subfamily Accipitrinae includes goshawks, sparrowhawks, sharp-shinned hawks and others. Th ...
to its prey in
classical pursuit
Motion camouflage is camouflage which provides a degree of concealment for a moving object, given that motion makes objects easy to detect however well their coloration matches their background or Disruptive coloration, breaks up their outline ...
, 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.
*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.
*The arms of spiral
galaxies. Our own galaxy, the
Milky Way
The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...
, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees. However, although spiral galaxies have often been modeled as logarithmic spirals,
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
s, or
hyperbolic spiral
A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation
:r=\frac
of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called Reciprocal spiral, too..
Pier ...
s, 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.
*The nerves of the
cornea
The cornea is the transparent front part of the eye that covers the iris, pupil, and anterior chamber. Along with the anterior chamber and lens, the cornea refracts light, accounting for approximately two-thirds of the eye's total optical power ...
(this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).
[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.]
*The
bands of
tropical cyclone
A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depend ...
s, such as hurricanes.
*Many
biological structures including the shells of
mollusk
Mollusca is the second-largest phylum of invertebrate animals after the Arthropoda, the members of which are known as molluscs or mollusks (). Around 85,000 extant species of molluscs are recognized. The number of fossil species is e ...
s. In these cases, the reason may be construction from expanding similar shapes, as is the case for
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
al figures.
*
Logarithmic spiral beaches
A logarithmic spiral beach is a type of beach which develops in the direction under which it is sheltered by a headland, in an area called the ''shadow zone''. It is shaped like a logarithmic spiral when seen in a map, plan view, or aerial photog ...
can form as the result of wave refraction and diffraction by the coast.
Half Moon Bay (California)
Half Moon Bay is a bay of the Pacific Ocean on the coast of San Mateo County, California. The bay is approximately semi-circular, hence the name half moon, with sea access to the south. Coastal towns located there are Princeton-by-the-Sea, ...
is an example of such a type of beach.
In engineering applications
* Logarithmic
spiral antennas are frequency-independent antennas, that is, antennas whose radiation pattern, impedance and polarization remain largely unmodified over a wide bandwidth.
* When manufacturing mechanisms by subtractive fabrication machines (such as
laser cutters
Laser cutting is a technology that uses a laser to vaporize materials, resulting in a cut edge. While typically used for industrial manufacturing applications, it is now used by schools, small businesses, architecture, and hobbyists. Laser cutt ...
), 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.
*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.
See also
*
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
*
Epispiral
*
List of spirals
This list of spirals includes named spirals that have been described mathematically.
See also
* Catherine wheel (firework)
* List of spiral galaxies
* Parker spiral
* Spirangle
* Spirograph
Spirograph is a geometric drawing device that ...
*
Mice problem
In mathematics, the mice problem is a continuous pursuit–evasion problem in which a number of mice (or insects, dogs, missiles, etc.) are considered to be placed at the corners of a regular polygon. In the classic setup, each then begins to m ...
, a geometric problem asking for the path followed by mice chasing one another whose solution is a logarithmic spiral
*
Tait–Kneser theorem
In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other.
The logarithmic spiral or the pictured Archimed ...
References
*
* Jim Wilson
Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves University of Georgia (1999)
*
Alexander Bogomolny
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and M ...
Spira Mirabilis - Wonderful Spiral at cut-the-knot
External links
history and math
*
*
''SpiralZoom.com'' an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
Online exploration using JSXGraph (JavaScript)YouTube lecture on Zeno's mice problem and logarithmic spirals
{{Spirals
Spirals
Spiral
Spiral
Plane curves