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It seems pretty well established that organisms grow according to a 3/4-power law. For example, Niklas and Enquist, in their paper "Invariant scaling relationships for interspecific plant biomass production rates and body size," PNAS 2001, 98(5):2922 -2927, say:

Annualized rates of growth $G$ scale as the 3/4-power of body mass $M$ over 20 orders of magnitude of $M$ (i.e., $G \propto M^{\frac{3}{4}}$).

Does anyone know if there is some geometric reason to expect such a growth-rate law?

$\frac{d M}{d t} \sim M^{\frac{3}{4}}$

Apparently attempts to derive this growth-rate law from Kleiber's Law, which claims that metabolic rate scales as $M^{\frac{3}{4}}$, are controversial. So I was wondering if there might be some geometric viewpoint that makes growth proportional to $M^{\frac{3}{4}}$ not unexpected.

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    The literature of allometric growth rates is littered with geometric descriptions of rates that ended up not being supported by data. It looks like it's easy to come up with an explanation but hard to know that your explanation is right.2012-02-28

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h=height, M=mass, A=cross-sectional area of stem d=density of stem V=volume of plant

The total cross-sectional area of a plant remains roughly the same at the top as at the bottom. (Da Vinci's rule).

Plants collect light for photosynthesis along their cross-sectional area. The thickness of stem A needed to support a plant is proportional to $h^3$.

$A\propto h^3$

$M = Ahd$

$A \propto M^{3/4}$

growth rate $\propto$ area exposed to the sun for photosynthesis $\propto \frac{V}h \propto A$

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    Why is the thickness of stem $A$ needed to support a plant proportional to $h^3$? That seems to be the key step, but I am not seeing the logic behind it...2012-02-29