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(Index, Outline)

**Allometric law** (or power-law) describes the relationship between the body parts or processes within or among living organisms, usually expressed in power-law form:

- or in a logarithmic form:

Such **Allometric functions** ( are mathematical equations derived from the the study of allometry,

For example in Body size scaling relationships between a physiological quantity (such as the respiration rate, or the maximum reproduction rate) of organisms and their body size (frequently taken to be body weight)can be expressed mathematically in terms of a curve on a graph.

Many characteristics, ranging from brain size and heart rate to life span and population density, change consistently with body size. These relationships normally fit a simple power function. The use of logarithms makes the equation easier to visualize. The exponent becomes the slope of a straight line when the logarithm of the variable (say respiration rate) is plotted against the logarithm of body mass.

The terms **isometry**, **positive allometry**, and **negative allometry** are used in relation to the slope of the line. For example if heart rate, varies proportionally to body mass this is isometry; positive allometry is where the larger animals have proportionatly higher heart rates; while negative allometry is where larger animals have proportionately lower heart rate..

## Examples

Some examples of allometric laws:

- Kleiber's law, the proportionality between metabolic rate and body mass raised to the power :

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{0} \sim M^{\frac 3 4}}**

- the proportionality between breathing and heart beating times and body mass raised to the power :

- mass transfer contact area and body mass :

- the proportionality between the optimal cruising speed of flying bodies (insects, birds, airplanes) and body mass in
*kg*raised to the power :

## See also

- allometry
- scaling law
- power law
- constructal law
- square-cube law
- Exponentiation
- Logarithm

## 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