http://newenergytimes.com/library/1989fph/1989fph.htm

Fractal geometry will make you see everything differently.  There is danger in reading further.  You risk the loss of your childhood vision of clouds, forests, galaxies, leaves, feathers, flowers, rocks, mountains, torrents of water, carpets, bricks, and much else besides.  Never again will your interpretation of these things be quite the same.

Michael Barnsley
Fractals Everywhere (1988)

To what extent do models help?  It is interesting that very often models do help, and most physics teachers try to teach how to use models and to get a good physical feel for how things are going to work out.  But it always turns out that the greatest discoveries abstract away from the model and the model never does any good.

Richard P. Feynman
The Character of Physical Law (1965)

http://www.snowcrystals.com/

A mathematical definition of dimension

The Sierpinski Triangle as a model fractal

http://www.jimloy.com/fractals/sierpins.htm

The Sierpinski Triangle and the Chaos Game

http://serendip.brynmawr.edu/playground/sierpinski.html

Measuring the Fractal Dimension

d = ln m / ln r

Sierpinski gasket: when size (r) doubles, number of elements (m) triples

d = ln 3 / ln 2 = 1.585

d = ln m / ln r

Conversely, the length of the branches in a fractal tree are related to the fractal dimension and the branching number

fractal antennas for cell phones

Physical systems can generate Sierpinski Gaskets

http://classes.yale.edu/fractals/MandelSet/ComplexNewton/Basins/OpticalBasinBdry/Sweet.html

Biological systems can generate Sierpinski Gaskets

Note: many different ways to produce Sierpinski gasket

Another important type of fractal: Diffusion Limited Aggregation

http://angel.elte.hu/~vicsek/books.html

DLA simulation

DLA can model many natural fractals

A more detailed explanation of Dielectric Breakdown Models (DBM) and DLA

http://webvision.med.utah.edu/sretina.html

Bacterial growth

http://classes.yale.edu/fractals/Panorama/Biology/Bacteria/Bacteria2.html

Manganese oxide dendrites

http://appserv01.uni-duisburg.de/hands-on/files/autoren/nordm/nordm.htm

Quarter-power scaling in biological systems

Many biological systems depend upon branching fractal networks

http://library.thinkquest.org/26242/full/ap/ap11.html

Max Kleiber and Allometric scaling

Comparison of measured exponents and theoretical predictions (After West, Brown, Enquist)

 Cardiovascular Variable Predicted Exponent Measured Exponent Respiratory Variable Predicted Exponent Measured Exponent Aorta radius 3/8 = 0.375 0.36 Tracheal radius 3/8 = 0.375 0.39 Aorta pressure 0 = 0.000 0.032 Interpleural pressure 0 = 0.000 0.004 Aorta blood velocity 0 = 0.000 0.07 Air velocity in trachea 0 = 0.000 0.02 Blood volume 1 = 1.000 1.00 Lung volume 1 = 1.000 1.05 Circulation time 1/4 = 0.250 0.25 Volume flow to lung 3/4 = 0.750 0.80 Circulation distance 1/4 = 0.250 No Data Volume of alveolus 1/4 = 0.25 No Data Cardiac stroke volume 1 = 1.000 1.03 Tidal volume 1 = 1.000 1.041 Cardiac frequency -1/4 = -0.250 -0.25 Respiratory frequency -1/4 = -0.250 -0.26 Cardiac output 3/4 = 0.750 0.74 Power dissipated 3/4 = 0.750 0.78 Number of capillaries 3/4 = 0.750 No Data Number of alveoli 3/4 = 0.750 No Data Service volume radius 1/12 = 0.083 No Data Radius of alveolus 1/12 = 0.083 0.13 Womersley number 1/4 = 0.250 0.25 Area of alveolus 1/6 = 0.166 No Data Density of capillaries -1/12 = -0.083 -0.095 Area of lung 11/12 = 0.917 0.95 Oxygen affinity of blood -1/12 = -0.083 -0.089 Oxygen diffusing capacity 1 = 1.000 0.99 Total resistance -3/4 = -0.750 -0.76 Total resistance -3/4 = -0.750 -0.70 Metabolic rate 3/4 = 0.750 0.75 Oxygen consumption rate 3/4 = 0.750 0.76