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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 ]
This phyllotactic pattern creates an optical effect of criss-crossing spirals. In the botanical literature, these designs are described by the number of counter-clockwise spirals and the number of clockwise spirals. These also turn out to be Fibonacci numbers. In some cases, the numbers appear to be multiples of Fibonacci numbers because the ...
The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. [ 3 ] [ 4 ] [ 5 ] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci , who introduced the sequence to Western ...
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 science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature. In developmental biology , pattern formation refers to the generation of complex organizations of cell fates in space and time.
The nerves of the cornea (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern). [12] The bands of tropical cyclones, such as hurricanes. [13] Many biological structures including the shells of mollusks. [14]
The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. [3] The first few Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... .
The pattern of zeros in a constant-recursive sequence can also be investigated from the perspective of computability theory. To do so, the description of the sequence s n {\displaystyle s_{n}} must be given a finite description ; this can be done if the sequence is over the integers or rational numbers, or even over the algebraic numbers. [ 11 ]