- Logarithmic spiral
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of
spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated byJakob Bernoulli , who called it "Spira mirabilis", "the marvelous spiral".Definition
In
polar coordinates ("r", θ) the curve can be written as [cite book | title = Divine Proportion: Φ Phi in Art, Nature, and Science | author = Priya Hemenway | isbn = 1402735227 | publisher = Sterling Publishing Co | year = 2005]:
or
:
with "e" being the base of natural logarithms, and "a" and "b" being arbitrary positive real constants.
In parametric form, the curve is
::
with
real number s "a" and "b".The spiral has the property that the angle ɸ between the
tangent andradial line at the point ("r",θ) is constant. This property can be expressed in differential geometric terms as:
The
derivative r'(θ) is proportional to the parameter "b". In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that "b" = 0 ("ɸ" = π/2) the spiral becomes acircle of radius "a". Conversely, in the limit that "b" approaches infinity ("ɸ" → 0) the spiral tends toward a straight line. The complement of ɸ is called the "pitch"."Spira mirabilis" and Jakob Bernoulli
"Spira mirabilis",
Latin for "miraculous spiral", is another name for the logarithmic spiral. Although thiscurve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve byJakob Bernoulli , because he was fascinated by one of its unique mathematical properties: the size of thespiral increases but its shape is unaltered with each successive curve. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such asnautilus shells andsunflower heads. Jakob Bernoulli wanted such a spiral engraved on hisheadstone , but, by error, anArchimedean spiral was placed there instead.Properties
The logarithmic spiral can be distinguished from the
Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase ingeometric progression , while in an Archimedean spiral these distances are constant.Logarithmic spirals are self-similar in that they are self-
congruent under allsimilarity transformation s (scaling them gives the same result as rotating them). Scaling by a factor gives the same as the original, without rotation. They are also congruent to their owninvolute s,evolute s, and thepedal curve s based on their centers.Starting at a point "P" and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as θ goes toward -∞ is finite. This property was first realized by
Evangelista Torricelli even beforecalculus had been invented. The total distance covered is "r"/cos(ɸ), where "r" is the straight-line distance from "P" to the origin.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 0. (Up to adding integer multiples of 2π"i" to the lines, the mapping of all lines to all logarithmic spirals isonto .) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.The function , where the constant "k" is a
complex number with non-zeroimaginary part , maps thereal line to a logarithmic spiral in the complex plane.One can construct a
golden spiral , a logarithmic spiral that grows outward by a factor of thegolden ratio for every 90 degrees of rotation (pitch about 17.03239 degrees), or approximate it usingFibonacci number s.Logarithmic spirals in nature
In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:
*The approach of a
hawk to its prey. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch. [cite web|title=Organismal Biology: Flying Along a Logarithmic Spiral|url=http://www.sciencemag.org/cgi/content/short/290/5498/1857c]*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 , is believed to have four major spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees, an unusually small pitch angle for a galaxy such as the Milky Way. In general, arms in spiral galaxies have pitch angles ranging from about 10 to 40 degrees.*The nerves of the
cornea .*The arms of
tropical cyclones , such as hurricanes.*Many biological structures including the shells of mollusks. In these cases, the reason is the following: Start with any irregularly shaped two-dimensional figure "F"0. Expand "F"0 by a certain factor to get "F"1, and place "F"1 next to "F"0, so that two sides touch. Now expand "F"1 by the same factor to get "F"2, and place it next to "F"1 as before. Repeating this will produce an approximate logarithmic spiral whose pitch is determined by the expansion factor and the angle with which the figures were placed next to each other. This is shown for
polygon al figures in the accompanying graphic.See also
*
Golden spiral References
*
* Jim Wilson, [http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/related%20curves/related%20curves.html Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves] , University of Georgia (1999)
* Alexander Bogomolny, [http://www.cut-the-knot.org/Curriculum/Geometry/Mirabilis.shtml Spira Mirabilis - Wonderful Spiral] , atcut-the-knot External links
* [http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html Spira mirabilis] history and math
* [http://antwrp.gsfc.nasa.gov/apod/ap030925.html "Astronomy Picture of the Day"] ,Hurricane Isabel vs. theWhirlpool Galaxy
* [http://antwrp.gsfc.nasa.gov/apod/ap080517.html "Astronomy Picture of the Day"] ,Typhoon Rammasun vs. thePinwheel Galaxy
* [http://SpiralZoom.com "SpiralZoom.com"] , an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
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