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Encyclopedia > Logarithmic spiral
 Logarithmic spiral (pitch 10°) Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral A low pressure area over Iceland shows an approximately logarithmic spiral pattern The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy

## Contents

In polar coordinates (r, θ) the curve can be written as This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$r = ae^{btheta},$

or

$theta = frac{1}{b} ln(r/a),$

hence the name "logarithmic". In parametric form, the curve is Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

$x(t) = a cos(t)e^{bt},$
$y(t) = a sin(t)e^{bt},$

with real numbers a and b. The parameter b controls how tightly and in which "direction" it is wrapped. For b >0 the spiral expands with increasing θ, and for b <0 it contracts with increasing θ. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

The spiral is nicely parametrized in the complex plane: exp(zt), given a z with Im(z)≠0 and Re(z)≠0. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... The exponential function is one of the most important functions in mathematics. ...

In differential geometric terms the spiral can be defined as a curve c(t) having a constant angle α between the radius or path vector and the tangential vector In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

$arccos frac{langle mathbf{c}(t), mathbf{c}'(t) rangle}{|mathbf{c}(t)||mathbf{c}'(t)|} = alpha$

If α = 0 the logarithmic spiral degenerates into a straight line. If α = ± π / 2 the logarithmic spiral degenerates into a circle. A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...

## Notes

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 mathematics, a geometric progression (also known as a geometric sequence, and, inaccurately, as a geometric series; see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

Any straight line through the origin will intersect a logarithmic spiral at the same angle α, which can be computed (in radians) as arctan(1/ln(b)). The pitch angle of the spiral is the (constant) angle the spiral makes with circles centered at the origin. It can be computed as arctan(ln(b)). A logarithmic spiral with pitch 0 degrees (b = 1) is a circle; the limiting case of a logarithmic spiral with pitch 90 degrees (b = 0 or b = ∞) is a straight line starting at the origin. The radian (symbol: rad, or a superscript c ( half circle)) is the SI unit of plane angle. ... The natural logarithm is the logarithm to the base e, where e is equal to 2. ...

Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives the same result as rotating them). Scaling by a factor b gives the same as the original, without rotation. They are also congruent to their own involutes, evolutes, and the pedal curves based on their centers. See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... Several equivalence relations in mathematics are called similarity. ... In the differential geometry of curves, an involute of a smooth curve is another curve, obtained by attaching a string to the curve and tracing the end of the string as it is wound onto the curve. ... In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ... In the differential geometry of curves, a pedal curve is a curve derived by construction from a given curve (as is, for example, the involute). ...

Starting at a point P and moving inwards along the spiral, one has to circle the origin infinitely often before reaching it; yet, the total distance covered on this path is finite. This was first realized by Torricelli even before calculus had been invented. The total distance covered is r/cos(α), where r is the straight-line distance from P to the origin. Torricelli can refer to the following: Evangelista Torricelli- Italian physicist Robert Torricelli- former American politician Torricelli languages is a subgroup of the Papuan languages This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ...

One can construct approximate logarithmic spirals with pitch about 17.03239 degrees using Fibonacci numbers or the golden ratio as is explained in those articles. Similarly, 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 2πi to the lines, the mapping of all lines to all logarithmic spirals is onto.) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis. In mathematics, the Fibonacci numbers, named after Leonardo of Pisa, known as Fibonacci, form a sequence defined recursively by: In other words,each number is the sum of the two numbers before it. ... The golden ratio, also known as the mean and extreme ratio, golden proportion, golden mean, golden section, golden number, divine proportion or sectio divina, is an irrational number, approximately 1. ... The exponential function is one of the most important functions in mathematics. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

## Logarithmic spirals in nature

Hawks approach their prey in a logarithmic spiral: their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch. The term hawk refers to birds of prey in any of three senses: Strictly, to mean any of the species in the genera Accipiter, Micronisus, Melierax, Urotriorchis, and Megatriorchis. ...

Insects approach a light source in a logarithmic spiral because they are used to having the light source at a constant angle to their flight path. Usually the sun is the only light source and flying that way will result in a practically straight line.

The arms of spiral galaxies are roughly logarithmic spirals. Our own galaxy, the Milky Way, is believed to have four major spiral arms, each of which is 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. NGC 4414, a typical spiral galaxy in the constellation Coma Berenices, is about 56,000 light years in diameter and approximately 60 million light years distant. ... Note: This article contains special characters. ...

 This graphic of an approximate logarithmic spiral made up from polygonal subunits was created using the perl interface to the ImageMagick image manipulation software: #!/usr/bin/perl use Image::Magick; #the perl interface to the ImageMagic software use Math::Trig; # The points of the polygon. ...

In biology, structures approximately equal to the logarithmic spiral occur frequently, for instance in spider webs and in the shells of mollusks. The reason is the following: start with any irregularly shaped two-dimensional figure F0. Expand F0 by a certain factor to get F1, and place F1 next to F0, so that two sides touch. Now expand F1 by the same factor to get F2, and place it next to F1 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 polygonal figures in the accompanying graphic. Biology is the branch of science dealing with the study of life. ... Suborders Araneomorphae Mesothelae Mygalomorphae See the taxonomy section for families Spiders are invertebrate animals that produce silk, and have eight legs and no wings. ... Classes Caudofoveata Aplacophora Polyplacophora Monoplacophora Bivalvia Scaphopoda Gastropoda Cephalopoda â€  Rostroconchia The mollusks or molluscs are the large and diverse phylum Mollusca, which includes a variety of familiar creatures well-known for their decorative shells or as seafood. ... Look up Polygon in Wiktionary, the free dictionary. ...

## References

• Eric W. Weisstein, Logarithmic Spiral at MathWorld.
• Jim Wilson, Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves, University of Georgia (1999)
• Alexander Bogomolny, Spira Mirabilis - Wonderful Spiral, at cut-the-knot

Results from FactBites:

 Cams (1137 words) The logarithmic spiral is a mathematical curve which has the unique property of maintaining a constant angle between the radius and the tangent to the curve at any point on the curve (figure 1). A logarithmic spiral cam (a "constant angle cam") ensures that the line between the axle and the point of contact (the "line of force") is at a constant angle to the abutting surface, independent of how the cam is oriented. The mathematical equation for a logarithmic spiral is R=beaØ.
 Logarithmic spiral - Wikipedia, the free encyclopedia (847 words) 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. Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives the same result as rotating them). Hawks approach their prey in a logarithmic spiral: their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.
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