Indexing the archive…
Your Universe of Digital Possibilities
Weisskopf, 1975: from the fine-structure constant and gravity’s weakness, the size of an atom, the height of a mountain, the largest planet and the smallest star — each to an order of magnitude. Grow a body from a stone to a white dwarf and watch the tallest mountain it can hold: below a few hundred kilometres it is a potato; above, it is round, and every mountain on it is The Bone’s square–cube law written on a planet. Then turn the constants and watch everything move.
A mountain’s base flows when the weight of one molecule’s column, A mpgh, matches the energy that melts it — a small fraction ξ of a Rydberg. Weight ∝ h³ on a base ∝ h²: the square–cube law of a bone, written on a planet. On a rocky body g ∝ M1/3; where hmax reaches the body’s own radius, the potato radius, gravity stops shaping it.
Atomic matter keeps its density, so a ball of N proton masses grows as N1/3; squeezed by its own gravity, its electrons become degenerate and it shrinks as N−1/3. The two meet near the mass of Jupiter — no cold ball can be wider than about that.
Relativistic electrons push back only as hard as gravity pulls, at every radius alike: above this mass a cold star has no size at which it can stand. With the full prefactor and two nucleons per electron, ≈ 1.44 solar masses.
The hottest a ball of N protons can get before degeneracy stops the squeeze is kT ≈ αG²N4/3mec²; hydrogen burns near α²mpc². Lighter balls stay brown dwarfs. As an order of magnitude ≈ 0.3 M☉, against a measured ≈ 0.08.
Bohr’s ladder, re-derived exactly by Schrödinger. The energy depends on n alone — so 2s and 2p share a rung, a whole shell of shapes at one energy. That accidental degeneracy is the scaffold the periodic table hangs on.
Clocks deeper in the well tick slower (the −(1−rs/r) term). The event horizon is at rs = 2GM/c², where the escape speed reaches light.
Scale every length by L and a bone’s cross-section grows as L² while the weight it carries grows as L³. The stress — load per area — rises in proportion to size: a photocopied giant is a broken giant.
This is the edition’s last size, and it closes the ring. Weisskopf’s recipe needs three numbers — α, αG and me/mp — and gives everything else to an order of magnitude: the atom from α, its density and the energy that melts it from the same α, and from gravity’s weakness the largest planet, the smallest star and the heaviest white dwarf. Walk the cycle back from here. The Chip ran out of atoms of exactly this size. The City laid its roads on a planet this round. The Tree buckled under the g this mass gives; The Wing flew through the air this gravity holds down; The Drop’s 2.7 mm was surface tension against that same g; The Swim’s syrup was water made of these atoms; The Heart filled bodies with matter of this density; The Fall raced weight against air in this pull. And the mountain is The Bone again: weight ∝ h³ on a base ∝ h², stress ∝ ρgh — Galileo’s giant of 1638, written on a planet. The bone’s stress climbed as M1/3; the mountain falls as M−1/3. The ring closes on the same third.