Fractal dimension Coastline paradox Covering and packing Minkowski-Bouligand dimension More than one, less than two Self-similarity Commentary

Felix Hausdorff, Benoit Mandelbrot mathematics

Fractal dimension

A measure of roughness A statistical index of complexity A ratio of detail to scale A measure of how completely a pattern fills a space A calculation of an object’s fractional dimension

Coastline paradox

Lewis Fry Richardson observed that the measured length of a coastline or other natural border depends on the length of the measuring stick.

Covering and packing

Construct a figure’s Hausdorff dimension by counting open balls needed to cover a figure. Construct a figure’s packing dimension by counting open balls needed to fit inside a figure. For each construction, use smaller and smaller balls.

Minkowski-Bouligand dimension

Count how many boxes of a grid are needed to cover a figure and calculate the limit as the size of the box approaches zero. A.K.A. the box-counting dimension.

More than one, less than two

How do you measure the dimension of straight line? It has a length but no width. But draw a bent line in a plane, and define the path between any two points on the line to have an infinite length. How can one say it has no width?

Self-similarity

If you have self-similarity, when we zoom in the same patterns appear. This might not be a good thing.