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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").
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically . Natural patterns include symmetries , trees , spirals , meanders , waves , foams , tessellations , cracks and stripes. [ 1 ]
A whorl is a single, complete 360° revolution or turn in the spiral or whorled growth of a mollusc shell. A spiral configuration of the shell is found in numerous gastropods, but it is also found in shelled cephalopods including Nautilus, Spirula and the large extinct subclass of cephalopods known as the ammonites.
Approximations of this are found in nature Fibonacci spiral: circular arcs connecting the opposite corners of squares in the Fibonacci tiling: approximation of the golden spiral golden spiral = special case of the logarithmic spiral Spiral of Theodorus (also known as Pythagorean spiral)
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 Fibonacci Spiral consists of a sequence of circle arcs. The involute of a circle looks like an Archimedean, but is not: see Involute#Examples.
The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b: [10] = or = (/), with e being the base of natural logarithms, a being the initial radius of the spiral, and b such that when θ is a right angle (a quarter turn in either direction): =.
On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
A whorl (/ w ɜːr l / or / w ɔːr l /) is an individual circle, oval, volution or equivalent in a whorled pattern, which consists of a spiral or multiple concentric objects (including circles, ovals and arcs). [1] [2]