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.