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.
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 $\propto \frac{V}h \propto A$