ned Productions - List of Power Rules

Here follow a variety of empirically determined power rules, also known as power laws, allometric rules or laws. I made this page because nowhere on the web seems to have a definitive list of all of them, and while this page doesn't contain them all either, it's better than anywhere else I know of.

If you know of a power rule not on this list, email me at webmaster2 <at symbol> nedprod. You'll need to provide me with at least one academic paper detailing the rule even if that paper isn't actually the original paper but rather a subsequent paper.

Where do these rules come from?

They come from the underlying fractal nature of the universe, which emerges from the chaotic attractors generated by the system of interconnections that is the universe no matter at what level you are looking at it. Depending on the form of the underlying fractal, you may get a certain guaranteed structure right across it that holds no matter what.

From Biology:

Description	Rule	Notes	Reference
Group Size for Primates	N = 10^0.093 x CR^3.389	N is group number, CR is the neocortex ratio (as against rest of brain size), = 4.1 for humans	Dunbar, R.I.M. (1998). The Social Brain Hypothesis
Whole Organism Metabolic Rate	B = B₀ x M^3/4	M is mass, B₀ is a scaling constant, = 5.66 for higher primates	Peters, R.H. (1983). The Ecological Implications of Body Size. Cambridge University Press, Cambridge. Moses, M.E. and Brown, J.H. (2003). Allometry of human fertility and energy use
Heart Rate, Reproduction Rate, Cellular Metabolic Rate	R ~ M^-1/4 R ~ B^-1/3	M is mass, B is whole organism metabolic rate
Plant Growth Rate	G ~ V ^p	G is annual increment, V is living tissue tree volume, p is less than one

Applications of Fractals to Biology

From Linguistics:

Description	Rule	Notes	Reference
The Growth of Vocabulary against Text Size	V = K x N ^b	V is the portion of vocabulary, K is a scaling factor between 10 and 100, N is the number of words in the text and b for English language is between 0.4 and 0.6	Heap's Law

From Physics:

There are loads of these in physics, but here are an example two:

Description	Rule	Notes	Reference
Gravitation	F = -6.67 x 10^-11 x M x R^-2	F is force, M the combined mass, R the distance between masses
Electrostatic	F = 8,987,742,438 x Q x R^-2	F is force, Q the combined charge, R the distance between charges

Links:

http://www.nslij-genetics.org/wli/zipf/index.html has a small list of power rules