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Heighway dragon curve. A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently.
Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal: [1]
Koch curve: 3 Koch curves form the Koch snowflake or the anti-snowflake. 1.2619: boundary of Terdragon curve: L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle. 1.2619: 2D Cantor dust: Cantor set in 2 dimensions.
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Sierpinski triangle created using IFS (colored to illustrate self-similar structure) Colored IFS designed using Apophysis software and rendered by the Electric Sheep.. In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar.
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.
It is also an example of a fractal curve. The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. [ 1 ]
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