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shell showing the chambers arranged in an approximately logarithmic spiral]]
over Iceland shows an approximately logarithmic spiral pattern]]
often have the shape of a logarithmic spiral, here the Whirlpool Galaxy ]]
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", and wanted one engraved on his Headstone . Unfortunately, an Archimedean Spiral was placed there instead.


DEFINITION


In Polar Coordinates (''r'', θ) the curve can be written as

:r = ae^{b heta}\,

or

: heta = rac{1}{b} ln(r/a),

hence the name " Logarithm ic". In parametric form, the curve is

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

with positive Real Number s ''a'' and ''b''. ''a'' is a scale factor which determines the size of the spiral, while ''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; the only difference between positive and negative ''b'' being that one spirals to the left and the other to the right.






In al figures in the accompanying graphic.


REFERENCES