Mountains float. Like an iceberg on the sea, a peak is held up not by the strength of the rock but by buoyancy: every metre of height above the plain is balanced by many metres of hidden root pushed down into the mantle. At the compensation depth every column weighs exactly the same. The blue team builds Airy's floating blocks; the red team tries to break the balance.
source Airy, On the computation of the effect of the attraction of mountain-masses…, Phil. Trans. R. Soc. 145 (1855) 101–104 — doi:10.1098/rstl.1855.0003. Rendered, not quoted.
Airy's picture: the crust (density ρc) is a raft of light rock floating on the denser mantle (ρm). A block that stands higher must displace more mantle, so it sinks a matching root below.
Archimedes for the crust gives the exact law: a peak of extra height h floats on a root
r = h · ρc / (ρm − ρc)
Live for the current block, distributed above vs. hidden below sea level:
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Mountains float — Airy, 1855. A peak of height h rides a root h ρc/(ρm−ρc); every column carries equal pressure at the compensation depth; it is Archimedes applied to the crust.
The same buoyancy that seats a mountain also floats the whole plate — the density contrast that makes a root is the load that feeds the-plate-tectonics. Each sphere is the next one’s premise.
The blue team’s live check: recompute the 5 km root, the equal-pressure balance, and root-grows-with-height from scratch and confirm them. If red tampers with the densities, this badge is where it shows.
Four numbers feed the engine: the surface height h of the block above the plain, the crust density ρc, the mantle density ρm, and the reference crust thickness. Typical crust is near 2.8 g/cm³, mantle near 3.3.
Everything below is computed live from these — the root, the hidden fraction, and the pressure at the compensation depth for the mountain column vs. a flat reference column. That is what you feed the panel.
Change any control — the root and the pressure balance are computed from Airy’s law on the spot, never looked up.
What the machine produces, proven: a 5 km mountain (ρc 2.8, ρm 3.3) floats on a 28 km root — about 85% of the block hidden below, exactly the iceberg ratio ρc/ρm. At the compensation depth the mountain column and a flat column weigh the same to 1e−6. A taller peak always sinks a deeper root.
The blue team’s witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
Real roots are inferred from seismology and gravity, never measured with a ruler; the density figures are averages. Airy is the first quantitative buoyancy model, not the last word.
“A mountain is held up by the strength of the rock.” Cut. Over geologic time the crust cannot support kilometres of unbalanced load — it is floated, not propped. Buoyancy, not strength.
“The root is about as deep as the mountain is tall.” Cut. The root is several times the height — r/h = ρc/(ρm−ρc) ≈ 5.6, so 5 km of peak needs 28 km of root.
“Airy and Pratt cannot both be right, so one is wrong.” Kept, corrected. Both satisfy isostasy; they differ in how compensation is arranged (root depth vs. density), and real crust mixes the two.
The red team’s move: swap the densities in the root law to r = h ρm/ρc — the wrong ratio. The root goes wrong and the columns no longer balance. The blue team’s witness (window 7) is watching.
Use the wrong density ratio and a 5 km peak “floats” on a ˜5.9 km root — too shallow to balance. The witness recomputes, the equal-pressure check fails, and it turns red. Nothing is faked; the attack is real and it is caught.