Read here about allometry and its relationship to issues of scale.


Allometry is the study of the relationship of body size to shape, anatomy, physiology and behavior. As such, its concerns are central to the proper comprehension of scale as advocated by this site. Such comprehension requires not only the ability to objectively survey a vast range of scales, but to also assess the meaning of events at such scales, particularly as the events pertain to life.

Bonner is a recent researcher into allometry, see his short biography or his five rules of allometry, below.

About Allometry

Allometry is the study of the relationship of body size to shape, anatomy, physiology and finally behaviour, first outlined by Otto Snell in 1892, D'Arcy Thompson in 1917 and Julian Huxley in 1932. The study of allometry is known for its statistical shape analysis and 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 of allometris 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. The relationship between these two measured quantities is often expressed as a power-law or logarithm.
  • in the power law expression, y is proportional to x^a, where a is the scaling exponent of the law.
  • in a logarithmic form, log y is proportional to log x

Methods for estimating allometric exponents from data are complicated because the usual method for fitting lines (least-squares regression) does not account for error variance in the independent variable (e.g., log body mass). Therefore, specialized methods can be used, including measurement error models and a particular kind of principal component analysis.

Ontogenetic Allometry: Focus on Growth

Allometry often studies shape differences in terms of ratios of the objects' dimensions. Two objects of different size but common shape will have their dimensions expressed in the same ratio and compared. Take, for example, a biological object that grows as it matures. Its size changes with age but the successive shapes assumed as it grows are highly conformant, or 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.

Static Allometry

Static allometry examines shape variation among individuals of a given age and sex. Comparisons of species are used to examine interspecific or evolutionary allometry. These studies are called "static" to contrast with ontogenetic allometry, which is "dynamic" as it focuses on growth.

Haldane's Principle

Haldane wrote an essay called "On Being the Right Size". Haldane's thesis is that sheer size very often defines what bodily equipment an animal must have:
"Insects, being so small, do not have oxygen-carrying bloodstreams. What little oxygen their cells require can be absorbed by simple diffusion of air through their bodies. But being larger means an animal must take on complicated oxygen pumping and distributing systems to reach all the cells." Many of his examples are based on the area-volume power-law relationship, although he does not use that terminology. The bigger an animal gets, the more they would have to change their physical shape, but the weaker they would become.

This link became known to others as Haldane's principle, being referred to as such by urban planning theorist Jane Jacobs. Another planning theorist, Christopher Alexander, refers to this principle in his theory of Independent Regions in "A Pattern Language": "...just as there is a best size for every animal, so the same is true for every human institution. In the Greek type of democracy all the citizens could listen to a series of orators and vote directly on questions of legislation. Hence their philosophers held that a small city was the largest possible democratic state..." The conceptual metaphor to animal body complexity has been of use in energy economics and secession ideas.

Bonner On Allometry

Bonner's book "Why Size Matters" summarizes the latest issues of allometry.

Bonner's Rules

Size difference, as indicated by their weight, affect different properties: for a given weight, an appropriate strength is required for support and movement; an appropriate surface, so that an adequate diffusion of oxygen and food substances can reach the inner tissues (entropy pumping); an appropriate division of labor, so that the body of the animal or plant can function (control); and the appropriate rate of metabolism for that functioning.

Strength Vs. Weight

RULE 1 Strength varies with size.
Weight-strength: Strength <varies as> Weight ^ 2/3

Surface Vs. Weight

RULE 2 Surfaces that permit diffusion of oxygen, of food, and of heat in and out of the body, vary with size.
Size and diffusion: Surface <varies as> Weight ^ 2/3

Complexity Vs. Weight

RULE 3 The division of labor (complexity) varies with size.
Complexity <varies as> Weight ^ a
This results from a direct trend of rules 1 and 2.

See for example a chart of evolution from RNA replication and reproduction to multi-cellular organisms.

Metabolism Vs. Weight

RULE 4 The rate of various living processes varies with size, such as metabolism, generation time, longevity, and the speed of locomotion.
Matabolic rate <varies as> Weight ^ c
Y (metabolic rate) <varies with> X (body weight) ^ a [constant]
log y <varies with> log x, mouse to elephant curve

Abundance Vs. Weight

RULE 5 The abundance of organisms in nature varies with their size.
Size-abundance: Abundance <varies as> Weight ^ -b

Chart of visualizing animal abundance: by dividing the distance between organisms (B) and their diameter (D) it can be seen that over a size span of 7 orders of magnitude the ratio is moderately constant.

Size in diameter

Distance between
1-10 micron
3 micron
44 micron
Amoeba, arthropods
10^-100 micron
40 micron
390 micron
0.1-1 mm
0.5 mm
5.8 mm
Arthropods, other invertebrates
1-10 mm
5 mm
35 mm
Intertebrates, shrews, small birds
1-10 cm
5 cm
50 cm
10^-100 cm
10 cm
300 cm
Large mammals
1-10 M
1.5 M
100 M

Alometric Graphs

Below, some graphs showing allometric relationships across a large collection of scales.

Brain size of 200 species

Allometric log/log chart of endocranial size vs. weight from Bonner

Allometric Body Mass vs. Cruising Speed in Constructal Theory--this GNU licensed image is displayed as fair use and does not constitute acceptance of the GNU license on this website.

Rensch's Rule

Rensch's rule is an allometric law concerning the relationship between the extent of sexual size dimorphism (SSD) and which sex is larger. Across species within a lineage, size dimorphism will increase with increasing body size when the male is the larger sex, and decrease with increasing average body size when the female is the larger sex.

Links and References

Click below for other pages about the human scale:

Click below to read about other core concepts involving scale: