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Our human bodies pick out a length scale (let’s say 1m). How unique is this scale and why did it arise?

In other words, how much smaller could humans, or multicellular lifeforms in general, be while sticking with approximately the same architecture of life? I know that the weight to muscle ratio is one argument against scaling things up massively, but what about smaller? Would such arguments still be available we just lower the gravitational field strength? If so, might there be intelligent aliens kilometers big?

The question is vague, I’m not looking for one particular answer.

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  • $\begingroup$ The question asks about smaller sizes, but in the sentence beginning with "If so,..." you suddenly switch to larger sizes. The question also seems unfocused to me. In various places, you refer to humans, multicellular lifeforms, and intelligent lifeforms. $\endgroup$
    – user4552
    Oct 15 '19 at 19:17
  • $\begingroup$ @BenCrowell edited. $\endgroup$ Oct 15 '19 at 19:18
  • $\begingroup$ If you want to be multicellular then you can't be any smaller than a cell, so the minimum size of a cell gives you a strong lower bound. (Cells can be fairly small but they obviously can't be smaller than a DNA molecule and are always substantially bigger. In addition, all known multicellular organisms are eukaryotes, which have quite big cells, relatively speaking.) $\endgroup$
    – N. Virgo
    Oct 15 '19 at 20:41
  • $\begingroup$ On the other hand, there's no obvious upper limit, and there are single fungal organisms that are several square kilometres in size. $\endgroup$
    – N. Virgo
    Oct 15 '19 at 20:43
  • $\begingroup$ The meter was based on $1/40,000,000$ of the Earth's circumference. "Foot" might be a better example in this context. $\endgroup$ Oct 16 '19 at 3:13
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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.

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Physicist Don Page estimates

the height of the tallest running, breathing organism on a habitable planet as the Bohr radius multiplied by the three-tenths power of the ratio of the electrical to gravitational forces between two protons.

This gives 3.6 meters for The Height of a Giraffe, which is the title of his paper.

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  • $\begingroup$ It's hard to imagine that giraffes can't get taller on smaller planets (where $g < 9.81\frac{m}{s^2}$). $\endgroup$ Oct 15 '19 at 20:49
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This answer is extremely approximative:

The scale of life should be somewhere between the smallest size physically meaningfull: the Planck lenght $\ell_{\textsf{P}} \approx 1,6 \times 10^{-35}~\mathrm{m}$, and the largest size set by causality in our universe: the Hubble lenght $\ell_{\textsf{H}} \approx 2 \times 10^{26}~\mathrm{m}$. The most likely scale of life should be around the geometric average of these extreme scales: \begin{equation} \ell_{\text{Life}} \sim \sqrt{\ell_{\textsf{P}} \, \ell_{\textsf{H}}} \approx 6 \times 10^{-5}~\mathrm{m} = 0.06~\mathrm{mm}. \end{equation} This is indeed about the size of a human cell.

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