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# Allometric law

An allometric law describes the relationship between two attributes of living organisms, and is usually expressed as a power-law:

$y \propto x^{a} \,\!$ or in a logarithmic form: $\log y \sim a \log x \,\!$

where a is the scaling exponent of the law. Methods for estimating this exponent from data tend to involve a particular kind of principal component analysis.

## Examples

Some examples of allometric laws:

• Kleiber's law, the proportionality between metabolic rate q0 and body mass M raised to the power 3 / 4:
$q_{0} \sim M^{\frac 3 4}$
• the proportionality between breathing and heart beating times t and body mass M raised to the power 1 / 4:
$t \sim M^{\frac 1 4}$
• mass transfer contact area A and body mass M:
$A \sim M^{\frac 7 8}$
• the proportionality between the optimal cruising speed Vopt of flying bodies (insects, birds, airplanes) and body mass M in kg raised to the power 1 / 6:
$V_{opt} \sim 30.M^{\frac 1 6} m.s^{-1}$

• allometry
• constructal law
• power law
• Rensch's rule
• scaling law
• square-cube law
• Metabolic theory of ecology

## References

• A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000. ISBN 0-521-79388-2
• A. Bejan, Constructal theory of organization in nature: dendritic flows, allometric laws and flight, Design and Nature, CA Brebbia, L Sucharov & P Pascola (Editors). ISBN 1-85312-901-1