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The here column is measured in your browser on the shape you are looking at: it is recomputed whenever you move a slider and open this tab. Force balance is tested by finite differences of B alone at 300 random points in the plasma. Nothing is solved, so nothing can be tuned to pass. The same module is held to these claims across seven configurations by field.selftest.mjs.
A fusion plasma is held in place by a magnetic field whose lines wrap around nested doughnut-shaped surfaces, with the pressure constant on each surface and highest in the middle. The balance is (∇×B)×B = ∇p: magnetic force against pressure push. Tokamaks make this easy by being symmetric: every slice around the ring is the same. Stellarators twist the field in three dimensions instead, which removes the large current a tokamak needs, but then nobody could prove the balance could hold exactly.
In 1967 Harold Grad suspected it couldn't. He conjectured that, apart from symmetric cases, smooth equilibria with nested surfaces and non-constant pressure do not exist. Close approximations were possible, but not the real thing. For sixty years stellarator design has been built around approximate equilibria on the strength of that doubt.
Earlier in 2026, Gómez-Serrano and coauthors proved that non-symmetric solutions exist, with every field line closing in the poloidal direction and B vanishing on the axis. Then Matt Landreman wrote two families down explicitly, in Cartesian coordinates, using only square roots, trig and hyperbolic functions. They are smooth, fully three-dimensional, with nested surfaces and pressure that falls from the axis outward. They involve no expansion and no small parameter.
ι = 2 is a symmetric tokamak field (Solov'ev's) stretched by √(1+ε) one way and √(1−ε) the other. That shouldn't work, because stretching usually breaks force balance. It survives because in this field a moving point obeys a harmonic oscillator in each coordinate, ẍ = −x, ÿ = −y, z̈ = −4z, and stretching x and y separately leaves those equations alone. Every field line closes after exactly one turn round the torus, and the axis rises and falls like a saddle.
Sheared ι is a different construction, built from complex analysis. Its field lines twist at a rate that changes from surface to surface, so some close and most never do. That is the property real stellarators need.
Press → tokamak: ε slides to 0 and the shape relaxes into an ordinary symmetric tokamak, then twists back out. The symmetric case turns out to be one point in a continuous family, not the only one.
Open sections. The coloured curves come from the formulas. The orange dots come from something else entirely: a field line followed step by step through B alone, marked each time it passes the slice. If the formulas were wrong, the dots would drift off the curves. They land on them.
Neither family is a reactor design. ι = 2 with every line closed is a degenerate case, and the sheared family's twist changes only a little: about 1% across the plasma at the default settings, as the sections chart shows. What they settle is existence. It is now certain, not just believed, that strongly non-symmetric equilibria with perfect surfaces exist. And because they are exact, they are ideal test cases for the numerical codes stellarator designers use.