# Fractal, Definition Of

Robert P. Munafo, 2000 Feb 21

Several definitions have been created over the years as mathematicians struggled with the complex properties of fractals. Here they are, roughly in chronological order:

divergent measure: Any shape that has the unusual property that when you measure its length, area, surface area or volume in discrete finite units (as in the box-counting method), the measured value increases without finite limit as the size of the discrete unit decreases to zero.

The oldest standard example is a coastline ("How long is the coast of Britain?"), which when measured one kilometer at a time might turn out to be 5000 kilometers long, but when measured one meter at a time comes out to be, say, 12000 kilometers.

self-similarity: Any object that is self-similar in a non-trivial manner. An example of trivial self-similarity is a straight line: any line segment looks the same as the whole line when magnified. Non-trivial examples include such things as the Sierpinski gasket and the Koch snowflake curve.

Hausdorff definition: Any geometric form with a non-integral Hausdorff dimension.

natural definition: A geometric figure or natural object that combines the following characteristics: a) its parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed; b) its form is extremely irregular or fragmented, and remains so, whatever the scale of examination; c) it contains "distinct elements" whose scales are very varied and cover a large range.

Mandelbrot refers to the boundary of the Mandelbrot Set as a fractal (The Fractal Geometry of Nature, p. 15). The Mandelbrot Set itself (boundary plus interior) is not a fractal if you go by the Hausdorff definition.

See also box-counting dimension, delta Hausdorff dimension, self-similarity.

Related to the study of fractals is the study of chaos; see the chaos page for a description of why there is a relation.

Acknowledgments

Translation from French of natural definition: Gerry Middleton

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.

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