Robert P. Munafo, 2000 Feb 21

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.

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.

Translation from French of natural definition: Gerry Middleton