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EQUIANGULAR OR LOGARITHMIC SPIRAL

An equiangular spiral, also called growth or logarithmic spiral is a curve which is often seen as occurring in nature and in various natural phenomena .  Examples of naturally occurring spiral resembling the equiangular spiral include spiral galaxies, nerves in cornea, chambers in nautilus shells, arms of low pressure areas cyclones, and hurricanes.  

Rene Descartes (1596-1650), a French philosopher and mathematician, first described the equiangular spiral mathematically in polar coordinates (r,q) as:

           r = a ebq

where a and b are positive constants, and e is the base of the natural logarithm.

The spiral has the property that the angle of the tangent to the curve and the radial line at any point is constant.  It can be noted that the distances between turnings of this curve increases in geometric progression.

"Spira mirabilis" is another name for equiangular spiral.  It is a Latin word meaning miraculous spiral.  This name was originated by Jakob Bernoulli, who further studied this spiral and its properties extensively.

Equiangular or logarithmic spirals exhibit properties of self-similarity in that they are self-similar under similarity transformation.  They look similar under different scaling, and under various rotations.

One question often asked is from a given point on the curve, how many turns does it take to go around the origin of the curve before reaching the origin.  The answer is that although the distance along the curve from the given point to the origin is finite, the number of turns required to reach its origin is infinite.

A golden spiral is a special kind of equiangular spiral, which grows outward by a factor of golden ratio for every quarter of a revolution.

 

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