Allometry is the study of the relationship between size and shape, , first outlined by Otto Snell in 1892 and Julian Huxley in 1932. Allometry is a well-known study, particularly in statistical shape analysis for its theoretical developments, as well as in biology for practical applications to the differential growth rates of the parts of a living organism's body. One application is in the study of various insect species (e.g., the Hercules Beetle), where a small change in overall body size can lead to an enormous and disproportionate increase in the dimensions of appendages such as legs, antennae, or horns.
Allometry studies shape differences in terms of ratios of the objects' dimensions. Two objects of different size but common shape will have their dimensions in the same ratio. Take, for example, a biological object that grows as it matures. Its size changes with age but the shapes are similar. Studies of ontogenetic allometry often use lizards or snakes as model organisms because they lack parental care after birth or hatching and because they exhibit a large range of body size between the juvenile and adult stage. Lizards often exhibit allometric changes during their ontogeny.
Allometric scaling's key equation for all of life equates Metabolic Rate P, to body mass W, raised to a power (4μ - 1)/4μ, where μ is percentage of capture by mass W of energy from oxidation reactions, for storage or expenditure. The equation models the rate at which electrical power needs to be captured, given the limitations of organic chemistry, by a biological system, in order to sustain itself for a duration. With metabolic rate expressed in calories/second, or watts (Joules/second), an extremely high metabolic rate, like that seen for masses less than one milligram at μ less than 15%, translates to many seconds, and long life for cells if the energy is available.
This scaling equation, when graphed for a range of values from e-13 to e+8 grams, and from -22 to 141% metabolic efficiency μ, clearly models the evolution of life to the extent that it was driven by metabolic considerations, that is, terms of bioenergetics. This model clearly depicts how energy fluctuations were equilibrated by biological systems in such a way as to drive evolution. This model explains the difference between cancerous cells and stem cells in terms of proliferation and metabolic efficiency, how the former can be triggered to apoptosis, and the proliferation of the latter can be extended, through the application of electrochemistry acting upon μ.
This equation models the mass increase of fat cells, and illustrates, in terms of metabolism and bioenergetics, how fat cells are related to the evolution of mammals by making prolonged in utero development possible. This equation clearly depicts why resort to dieting to reduce fat cell mass is a bad strategy, why increased activity levels is a good strategy, and how to increase muscle mass independently of muscle use.
This equation is not well understood even by its major proponents who, on the one hand, hint it might hold the secret to the aging process, and, on the other, believe that all metabolism is mass dependent, operates at 1000% μ, is the result of respiration, and is not anaerobic, or electrochemical. These beliefs have no variable in the equation to represent them, and even conflict with its terms. The denominator of μ is expressed in amperes. P is expressed in watts. W is expressed in grams mass, and the numerator of μ is the rate at which anabolic organic reactions are triggered by available amperes. The value for W is a reflection of the size and duration of working of the numerator, appearing as either growth or division.
In contrast, one refers to isometric scaling when growth does not lead to a change in geometry of an organism.