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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").
According to Stephen Skinner, the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein. [5] Many forms observed in nature can be related to geometry; for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape.
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]
Halved shell of Nautilus showing the chambers (camerae) in a logarithmic spiral (1st p. 493 – 2nd p. 748 – Bonner p. 172) Thompson observes that there are many spirals in nature, from the horns of ruminants to the shells of molluscs; other spirals are found among the florets of the sunflower. He notes that the mathematics of these are ...
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]
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 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.
The logarithmic spiral of the nautilus shell is a classical image used to depict the growth and change related to calculus. Calculus is used in every branch of the physical sciences, [ 53 ] : 1 actuarial science , computer science , statistics , engineering , economics , business , medicine , demography , and in other fields wherever a problem ...