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.

When the length of our measuring stick approaches zero, the length of the coast of Great Britain approaches infinity, which makes the length of a coastline seem indeterminable; however, the Hausdorff dimension of the coastline of Great Britain is 1.25, quite a bit more determinable.

See also in

The book of science:Fractal—Benoit MandelbrotReadings in wikipedia: