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Fractal fern in four states of construction. Highlighted triangles show how the half of one leaflet is transformed to half of one whole leaf or frond.. Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
Starting in the 1950s Benoit Mandelbrot and others have studied self-similarity of fractal curves, and have applied theory of fractals to modelling natural phenomena. Self-similarity occurs, and analysis of these patterns has found fractal curves in such diverse fields as economics, fluid mechanics, geomorphology, human physiology and linguistics.
Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape ...
A 4K UHD 3D Mandelbulb video A ray-marched image of the 3D Mandelbulb for the iteration v ↦ v 8 + c. The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and further developed in 2009 by Daniel White and Paul Nylander using spherical coordinates.
The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 50/log 10 = 1.6990. Images generated with Fractal Generator for ImageJ [23]. Generator for 50 Segment Fractal. 1.7227: Pinwheel fractal: Built with Conway's Pinwheel tile.
Lyapunov fractals can be calculated in more than two dimensions. The sequence string for a n-dimensional fractal has to be built from an alphabet with n characters, e.g. "ABBBCA" for a 3D fractal, which can be visualized either as 3D object or as an animation showing a "slice" in the C direction for each animation frame, like the example given ...
The Pythagoras tree is a plane fractal constructed from squares. Invented by the Dutch mathematics teacher Albert E. Bosman in 1942, [1] it is named after the ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem.
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