# How unique is the length scale picked out by intelligent life?

The length scale picked out by animal life may be very different from the length scale picked out by intelligent life.

For animals, the upper limit is set by bone strength considerations. As Galileo first pointed out the mass scales as $L^3$ and if bones have radius $r$ then for a given maximal axial compression $\sigma$ the animal will get into trouble when $\sigma \approx \frac{g\rho L^3}{\pi r^2}$. If one uses allometric scaling laws one ends up with (for an earth-gravity planet with Earth-bone strength animals) $\approx 140$ tons as a maximum mass of land animals (with a confidence interval from 100 to 1000 tons, which also emerges when one considers muscle strength and locomotion considerations). At about water density a 1000 ton animal is $10\times 10\times 10$ meters -- even on (terrestrial) planets with lower gravity or with less dense animals this is not going to change by orders of magnitude since we take the cube root of the mass. One can distribute the mass into something longer or taller (in the later case, look out for Euler buckling!) but the size scale will be tens of meters.

What sets this scale is the relative ratio between the strength of molecular bonds ($\sigma$) and the the gravitational forces ($g\rho$). These can be estimated from fundamental physics. In Barrow & Tipler (1986) (p.310-318) they estimate that the gravitation on a planetary body will be on the order of $g\sim Z^{2/3}\left(\frac{\alpha_G}{\alpha}\right)^{1/2}\alpha^3\left(\frac{m_e}{m_N}\right)^2m_N$ and the molecular binding energy is $\sim \epsilon \alpha^2 m_e$ (where $\epsilon\sim 10^{-3}$). Here $Z$ is the atomic mass of typical elements, $\alpha_G$ is the gravitational fine structure constant, $\alpha$ the usual fine structure constant, $m_e$ the electron mass, and $m_N$ the nucleon mass. So they get a maximal length scale of $$L\leq Z^{2/3}\epsilon^{3/4}\left(\frac{\alpha_G}{\alpha}\right)^{1/4}a_0 \sim 73 \text{ cm}$$ (they get a bit of an underestimate since bones are better than their calculation). But it is basically a size on the meter scale.

They point out that warm-blooded animals cannot be smaller than a certain size since their heat loss to the environment is proportional to surface area $\sim L^2$ while their heat generation scales with volume $\sim L^3$, so $\text{loss}/\text{generation}=1/L$: if the metabolism cannot climb arbitrarily high (due to cooking of proteins and lack of food) there will be some size limit. Which is $\sim$ 2cm for mammals and birds.

The reason *intelligence* may be different is that it might potentially be electronic "solid state life" somebody built, which could be distributed over long distances (suffering a speed penalty as the frequency of thinking goes as $c/L$ but in principle able to run on an interstellar computer network -- the ultimate limit is set by the expansion of the universe on mega- to gigaparsec scales) or made very small (presumably down to a nanocomputer level). In fact, the limit for nanocomputers is also a limit for normal life: below a certain size there are not enough atoms to sustain the required complexity and fault tolerance (statistical fluctuation size for $N$ body systems scale as $1/\sqrt{N}$).

So the uniqueness of the length scale depends a bit on what capacities life has to have (standing up, locomotion, information processing), the assumed environment (molecular structures on planetary surfaces, in oceans, low-temperature chemisty), and limits (error correction, heat retention).

If one changes the assumptions the scale will shift. Zero-gravity life (in oceans or space) avoids most of the mechanical issues and can presumably be very large, but will likely exist in an environment placing other constraints. Nuclear matter life in the core of a neutron star (assuming there are QCD states allowing ordered information processing!) would tend to be very small and fast, with a length scale presumably set by communications lags ($\text{QCD timescale}\times c \approx 2.9979\times 10^{-16}$ m).

So in the end what sets the length scale of intelligent life will be (1) where it comes from - maybe water-carbon life from planets is the typical case, and (2) whether it subsequently changes itself to exist in other environments.