Fractals in the Natural Sciences

http://www.amazon.com/exec/obidos/tg/detail/-/0691024383/qid=1110295495/sr=1-1/ref=sr_1_1/002-4121393-8405649?v=glanc e&s=books



Cold Fusion


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)



large snowflake dendritic snowflake

http://www.snowcrystals.com/

A mathematical definition of dimension

Dimensions

The Sierpinski Triangle as a model fractal
Sierpinski Triangle


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 tree



Sierpinski Gasket is useful:
fractal antennas for cell phones

Fractal antenna

Physical systems can generate Sierpinski Gaskets


Mirrored Spheres

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


Biological systems can generate Sierpinski Gaskets

Sierpinski shell


Note: many different ways to produce Sierpinski gasket




Another important type of fractal: Diffusion Limited Aggregation


DLA cluster



DLA Fractal

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


DLA simulation

http://apricot.polyu.edu.hk/~lam/dla/dla.html

DLA can model many natural fractals


Electrostatic discharge

http://205.243.100.155/frames/lichtenbergs.html

Electrostatic breakdown

lightning






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


http://classes.yale.edu/fractals/Panorama/Physics/DLA/DBM/DBM.html

Arterial network - retina

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

Bacterial growth

Bacterial growth



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