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Logarithmic spiral


Logarithmic spiral (pitch 10°)

A logarithmic or equiangular spiral is a curve which in polar coordinates (r, θ) can be written as

<math>r = a b^\theta</math> or <math>\theta = \log_{b} \frac{r}{a}</math>, hence the name "logarithmic"

and in parametric form as

<math>x(t) = a b^t \cos(t)</math>
<math>y(t) = a b^t \sin(t)</math>

with positive real numbers a and b. Modifying a will rotate the spiral while b controls how tightly and in which direction it is wrapped. The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distance between the arms of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

The spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it Spiralis mirabilis and wanted one engraved on his tombstone. Unfortunately, an Archimedean spiral was placed there instead.

Any straight line through the origin will intersect a logarithmic spiral at the same angle; if θ is measured in radians, then this angle can be computed 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; a hypothetical logarithmic spiral with pitch 90 degrees (b = 0 or b = ∞) is a straight line starting at the origin.

Logarithmic spirals are self-similar in the following sense: enlarging a logarithmic spiral by a constant factor results in the same spiral, possibly rotated. The evolute[?] and pedal[?] of a logarithmic spirals are again logarithmic spirals.

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 only finite. This was first realized by Torricelli[?] even before calculus had been invented. The total distance covered is r/sin(α) where α is the pitch angle of the spiral and r is the straight-line distance from P to the origin.

One can construct approximate logarithmic spirals with pitch about 17.03239 degrees using Fibonacci numbers or the golden mean as is explained in those articles. One can also start with an arbitrary non-real, non-completely-imaginary complex number z and connect the points zn (n integer) in the complex plane to get an approximate logarithmic spiral. 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 plane.

Hawks approach their prey in the form of a logarithmic spiral: their sharpest view is at an angle to their flight direction; this angle is the same as the spiral's pitch.

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 a logarithmic spiral with pitch about 12 degrees.

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. Note that both the outer boundary and the inner boundary of this shape are approximate logarithmic spirals, but their pitch angles will in general be different.



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Logarithmic spiral

... distance covered is r/sin(α) where α is the pitch angle of the spiral and r is the straight-line distance from P to the origin. One can construct approximate ...