Logarithmic spiral Totally Explained

Everything about Logarithmic Spiral totally explained

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 by Jakob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

Definition

In polar coordinates (r, θ) the curve can be written as ť

r = ae^

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.
   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 before calculus 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 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.
   The function

x mapsto x^k

, where the constant k is a complex number with non-zero imaginary part, maps the real 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 the golden ratio for every 90 degrees of rotation (pitch about 17.03239 degrees), or approximate it using Fibonacci numbers.

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.

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 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.Further Information

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