The same magnet in
three dimensions
If you build the same magnet in three dimensions instead of two, does it still have a tipping point — and can you still find it when nobody has ever solved the problem exactly?
Everything up to this rung lived on a flat sheet. That flatness was a gift: in two dimensions, Lars Onsager solved the Ising magnet by hand in 1944, so every measurement had an exact number waiting to be checked against. Stack the same magnets into a cube and that gift is withdrawn. Nobody has an exact solution for the three-dimensional Ising model. Eighty years of effort, and the best anyone has are very good numbers from very careful computation.
The physics changes in a way you can almost guess. Each magnet now has six neighbours instead of four, so there is more peer pressure holding the crowd in line, so you have to heat it harder before it breaks. The transition moves from 2.269 up to about 4.51 — in the model's own units, which is worth saying plainly: these are not degrees of anything, they are the dimensionless temperature the model is written in.
What does not change is the method. The same coin-flip rule, the same checkerboard trick for updating half the sites at once, the same hunt for the temperature where the magnetisation fluctuates most wildly. The instrument was pointed at unfamiliar ground and asked to come back with a number.
It came back with 4.504, against a benchmark of 4.5115 — off by less than two parts in a thousand. The panel below runs that cube. Because a cube is hard to see into, the left canvas shows a single slice through it, and you can move the slice around to convince yourself the whole thing is doing the same thing.
left · one plane through the cube (ember = up, dark = down), with three further planes ghosted behind it — move the slice and the character does not change, which is the point · right · the susceptibility ladder. Ember is the 16³ cube walking 4.1 → 4.9; moss is a 32² sheet walking 1.8 → 2.8, run live on this page for contrast. Six neighbours push the peak from 2.269 to 4.51. Both demo curves are short ladders, not the published run.
The published run — 2026-06-16 — walked 21 temperatures across [4.1, 4.9] at L = 12, 16 000 sweeps each with 6 000 discarded to burn-in, seed 42, and finished on a CPU in 28 seconds. The susceptibility peak landed at 4.504 against the Monte Carlo literature benchmark of 4.5115. The specific heat, an independent observable from the same run, peaked at 4.42 — near enough to agree, far enough apart to be worth reporting as its own number rather than folded into the headline.
Three numbers the receipt also carries — β = 0.3265,
γ = 1.2372, ν = 0.6301 — are the published literature
exponents for this class, stored for reference. This run did not measure them and
this page does not claim them.