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
"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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