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.

http://www.snowcrystals.com/

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

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

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

Sierpinski Gasket is useful:

fractal antennas for cell phones

Physical systems can generate Sierpinski Gaskets

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

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

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

Sierpinski Gasket is useful:

fractal antennas for cell phones

Physical systems can generate Sierpinski Gaskets

Biological systems can generate Sierpinski
Gaskets

Note: many different ways to produce Sierpinski gasket

Another important type of fractal: Diffusion Limited Aggregation

DLA can model many natural fractals

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

Note: many different ways to produce Sierpinski gasket

Another important type of fractal: Diffusion Limited Aggregation

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

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

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

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 |