How do fundamental constants determine the size of material bodies? The size of an object depends on the size of the molecules and atoms it is made of and these ultimately depend on the value of the fundamental physical constants. The simplest example could be the size of the hydrogen atom which is Bohr's radius $a_{0}=\frac{\hbar}{m_{e}c \, \alpha}$. Let's consider the fine structure constant (since it is dimensionless). Then the size of an H-atom is inversely dependent from $\alpha$. If it would be different than what it is, say twice as much its value, the size of the H-atom would be half as much as it is, and so on. The question is if this linear dependence would still hold for other atoms, molecules and much more complex structures? Would a doubling of $\alpha$ lead to half the length of a complex objects, such for example a table, be also inversely proprtional as $1/\alpha$ ? I tend to say that it is not obvious since the molecular orbitals are complicated to calculate and it seems to me not so straightforward that a linear scaling of $\alpha$, and therefore also of Bohr's atom, leads automatically to a linear scaling of the size of more complex structures as well. I suspect that could be determined only by complicated numerical solutions of Schrödinger's equation? But I'm not sure about that. I'm just trying to find out if there is an easier logical reason to argue how the  change of one constant leads to a change in size. Any thought about that?
 A: For a size change to make physical sense there has to be some measurable difference before and after. It is commonly assumed that the Bohr radius scales the size of molecules and everything else based on it. 
However, it turns out that the scaling is not entirely linear as a function of $\alpha$. King et al. (2010) has shown that the bond angle of water decreases in molecular simulations if $\alpha$ changes, in turn reducing the dipole
moment and hydrogen-bonding ability of water. This would at the very least make ice change in size, for the same mass. A larger $\alpha$ also increases the bond length in the water molecule despite the Bohr radius declining. Generally covalent bonds between light atoms become weaker, and this would no doubt affect the maximum size of many objects like trees while perhaps not affecting metals as much. 
Even without these local changes we can compare the size of molecules to the size of gravitationally bound structures (regulated by $\alpha_G$ in addition to $\alpha$), or structures strongly affected by the proton-electron mass $m_p/m_e=\beta$. The mass of a person has for example been estimated to scale as $\propto 10^2\alpha^{3/4}m_p^{-1/2}$ and would hence scale very differently from stellar masses $\propto m_p^{-2}$ and asteroid masses $\propto \alpha^{3/2}\beta^{3/2}m_p^{-1/2}$. (Carr & Rees 1979) is a nice overview of how the basic dimensionless constants affect the relative sizes of various objects (above estimates are from there); see also Barrow & Tipler's The Anthropic Cosmological Principle and (Tegmark, et al. 2006) for more review.
