A fractal is an object or quantity that displays self-similarity , in a somewhat technical sense, on all scales. The object need not exhibit exactly
the same structure at all scales, but the same "type" of structures must
appear on all scales. A plot of the quantity on a log-log graph versus scale then
gives a straight line, whose slope is said to be the fractal
dimension . The prototypical example for a fractal is the length of a coastline
measured with different length rulers . The shorter the
ruler , the longer the length measured, a paradox
known as the coastline paradox .
Illustrated above are the fractals known as the Gosper island , Koch snowflake , box
fractal , Sierpiński sieve , Barnsley's
fern , and Mandelbrot set .
See also Attractor ,
Backtracking ,
Barnsley's Fern ,
Box
Fractal ,
Cactus Fractal ,
Cantor
Dust ,
Cantor Set ,
Cantor
Square Fractal ,
Carotid-Kundalini Fractal ,
Cesàro Fractal ,
Chaos
Game ,
Circles-and-Squares Fractal ,
Coastline Paradox ,
Dendrite
Fractal ,
Dragon Curve ,
Fat
Fractal ,
Fatou Set ,
Fractal
Dimension ,
Gosper Island ,
H-Fractal ,
Hénon Map ,
Iterated
Function System ,
Julia Set ,
Kaplan-Yorke
Map ,
Koch Antisnowflake ,
Koch
Snowflake ,
Lévy Fractal ,
Lévy
Tapestry ,
Lindenmayer System ,
Lorenz
Attractor ,
Mandelbrot Set ,
Mandelbrot
Tree ,
Menger Sponge ,
Minkowski
Sausage ,
Mira Fractal ,
Newton's
Method ,
Pentaflake ,
Peano
Curve ,
Peano-Gosper Curve ,
Pythagoras
Tree ,
Rabinovich-Fabrikant Equation ,
Rep-Tile ,
San Marco
Fractal ,
Self-Similarity ,
Siegel
Disk Fractal ,
Sierpiński Carpet ,
Sierpiński Curve ,
Sierpiński
Sieve ,
Star Fractal ,
Strange
Attractor ,
Zaslavskii Map Explore this topic in the MathWorld classroom
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Cite this as:
Weisstein, Eric W. "Fractal." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Fractal.html
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