# 1918,1974 Fractal dimension

## The book of science

Tom Sharp

Tom Sharp

Felix Hausdorff, Benoit Mandelbrot mathematics |

- 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

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

- 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.

- 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.

- 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?

- 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 on wikipedia: