Search results
Results from the WOW.Com Content Network
Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the coastline of Britain for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like ...
L-Systems branching pattern having 4 new pieces scaled by 1/3. Generating the pattern using statistical instead of exact self-similarity yields the same fractal dimension. Calculated: 1.2683: Julia set z 2 − 1: Julia set of f(z) = z 2 − 1. [9] 1.3057: Apollonian gasket
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island [1] [2]) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" [3] by the Swedish mathematician Helge von Koch.
Many fractal structures or patterns could be constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Figure 6. For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion limited aggregation, L-system.
Visible patterns in nature are governed by physical laws; for example, meanders can be explained using fluid dynamics. In biology , natural selection can cause the development of patterns in living things for several reasons, including camouflage , [ 26 ] sexual selection , [ 26 ] and different kinds of signalling, including mimicry [ 27 ] and ...
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. As examples, "landscapes" revealed by microscopic views of surfaces in connection with Brownian motion , vascular networks , and shapes of polymer molecules all ...
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.
The first few steps starting, for example, from a square also tend towards a Sierpiński triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals." [2] [3] Iterating from a square. The actual fractal is what would be obtained after an infinite number of iterations.