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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:

$y propto x^\left\{a\right\} ,!$ 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 $q_\left\{0\right\}$ and body mass $M$ raised to the power $3/4$:

$q_\left\{0\right\} sim M^\left\{frac 3 4\right\}$

• the proportionality between breathing and heart beating times $t$ and body mass $M$ raised to the power $1/4$:

$t sim M^\left\{frac 1 4\right\}$

• mass transfer contact area $A$ and body mass $M$:

$A sim M^\left\{frac 7 8\right\}$

• the proportionality between the optimal cruising speed $V_\left\{opt\right\}$ of flying bodies (insects, birds, airplanes) and body mass $M$ in kg raised to the power $1/6$:

$V_\left\{opt\right\} sim 30 cdot M^\left\{frac 1 6\right\} m cdot s^\left\{-1\right\}$