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Analytic continuation of natural logarithm (imaginary part) Analytic continuation is a technique to extend the domain of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
Complex exponential function: The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at : () = (+) + ⏟ = + = ( + ) ⏟ The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.
Barnsley's work has been a source of inspiration to graphic artists attempting to imitate nature with mathematical models. The fern code developed by Barnsley is an example of an iterated function system (IFS) to create a fractal. This follows from the collage theorem. He has used fractals to model a diverse range of phenomena in science and ...
The honeycomb is a well-known example of tessellation in nature with its hexagonal cells. [82] In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the fritillary, [83] and some species of Colchicum, are characteristically tessellate. [84]
The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. [1] Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature.
For example, one might want to find the nearest hospital or the most similar object in a database. A large application is vector quantization , commonly used in data compression . In geometry , Voronoi diagrams can be used to find the largest empty circle amid a set of points, and in an enclosing polygon; e.g. to build a new supermarket as far ...
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 ]
Using a mass of examples, Thompson pointed out correlations between biological forms and mechanical phenomena. He showed the similarity in the forms of jellyfish and the forms of drops of liquid falling into viscous fluid, and between the internal supporting structures in the hollow bones of birds and well-known engineering truss designs.