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Logarithmic spiral (pitch 10°) A section of the Mandelbrot set following a logarithmic spiral. A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").
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Bernoulli was referring to the fact that the logarithmic spirals are self-similar, meaning that upon applying any similarity transformation to the spiral, the resulting spiral is congruent to the original untransformed one. [1] The logarithmic spiral frequently appears in biology, such as with the curves of the Nautilus shell. [1]
A logarithmic spiral is a special kind of spiral curve which often appears in nature. This is a cutaway of a Nautilus shell showing the chambers arranged in an ...
The nautilus shell presents one of the finest natural examples of a logarithmic spiral, although it is not a golden spiral. The use of nautilus shells in art and literature is covered at nautilus shell .
The chambered nautilus (Nautilus pompilius), also called the pearly nautilus, is the best-known species of nautilus. The shell, when cut away, reveals a lining of lustrous nacre and displays a nearly perfect equiangular spiral, although it is not a golden spiral. The shell exhibits countershading, being light on the bottom and dark on top. This ...
The name logarithmic spiral is due to the equation = . Approximations of this are found in nature. Spirals which do not fit into this scheme of the first 5 examples: A Cornu spiral has two asymptotic points. The spiral of Theodorus is a polygon.
For example, in the nautilus, a cephalopod mollusc, each chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral. [51] Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity. [52]