Science Lab · Physics

Magnetic Bottles for Fusion

To get energy from fusion, you have to hold a gas hotter than 100 million degrees without letting it touch anything. One way is a bottle made of magnetic field, twisted into a doughnut. Here you can turn, slice and test two exact magnetic bottles from a paper posted in October 2026. They are rare things: shapes where every number is known exactly. Change the shape, watch the field lines wind, see what each setting does, hunt for a better bottle, and check the equations yourself.

Equilibrium

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Live numbers

Check the equations

All numbers are in the paper’s units: lengths and fields are scaled so the formulas come out clean, and μ₀ = 1.

Slices through the bottle

Cut the doughnut with a flat, vertical knife at four places around it, like the paper’s Figures 4 and 8. Each slice shows the nested magnetic surfaces as rings, colored by pressure: brightest in the middle, where the pressure is highest. The dashed blue outline is the same bottle with the symmetry-breaking setting τ at zero. With the mirror test on, the pink dashed outline is the bottle turned over.

What each setting does

Find a better bottle

Every setting helps some numbers and hurts others. These three tools let you see the trade-offs and search for the best bottle by a goal you choose. They work out hundreds of exact bottles in the background, so give them a moment.

The trade-off map

The goal finder

Near the limit the layers crowd together on one side. Staying below it keeps the bottle safely nested.

Let it change

The trade-off chart

How the finder works

First it scans: it tries a few hundred bottles spread evenly over the settings it may change (fewer on a phone), and keeps the best ones. Then it polishes them with the Nelder–Mead method: a small group of trial bottles, one more than the number of settings (a triangle for two), that flips, stretches and shrinks toward better values, step by step. Last, it nudges one setting at a time, which finds answers at the end of a slider. The pink line on the map is that path. Every bottle it tries is exact; only the search is trial and error. It can miss a better bottle hiding somewhere it never looked.

The best trade-offs on the chart are what engineers call a Pareto front: you cannot improve one of the two numbers without making the other worse.

What this leaves out. Real stellarator designers ask harder questions than these. Does the plasma stay stable as its pressure rises? How fast do heat and particles leak out? Are the fast particles made by fusion held long enough to heat the plasma? Can the magnet coils that make this field be built, and built accurately enough? Answering them takes large computer codes.

These two families are exact test problems for the codes that solve the equilibrium equations, not reactor designs. Here “best” means best by the simple measures on this page, in the paper’s scaled units.

What you’re seeing

The doughnut. The outer skin is the edge of the plasma. It is a flux surface: magnetic field lines lie flat on it and never cross it. Inside are more flux surfaces, nested like the layers of an onion. Cut it open to see them. In the very middle is a single line, the magnetic axis, drawn in white.

The colors. In “field strength” mode, bright aqua means a strong field and dark blue a weak one. In “pressure” mode, each layer has one color, because pressure is the same everywhere on a flux surface. The core glows gold. The edge is purple, where the pressure has dropped to zero.

The bright lines. These are magnetic field lines on the edge surface. Follow one around the doughnut and you will see it also winds around the short way. In Family 1, every line closes on itself after one trip around the long way, having circled the short way exactly twice. In Family 2, a line usually never closes and slowly covers its whole surface. The moving dots are like charged particles, which slide along field lines.

The mirror test. Pink shows the bottle turned over: a half turn about the dashed x-axis. When τ = 0, the pink cage sits exactly on the bottle. That is stellarator symmetry. Move τ away from zero and the two no longer match. Breaking that symmetry, while keeping the solution exact, is what this paper does.

The science

What fusion needs

In fusion, light nuclei such as deuterium and tritium collide hard enough to stick together, releasing energy. They only do that when they move very fast, which means very hot. ITER, the large fusion experiment being built in France, needs its plasma at 150 million °C. At that temperature the gas is a plasma: electrons are torn off their atoms, so it is a soup of charged particles. No material wall can touch it. In magnetic-confinement fusion, the kind this page is about, the plasma floats in a vacuum, held by a magnetic field. (The other main approach, inertial confinement, instead squeezes a tiny fuel pellet with powerful lasers for a split second.)

Why a magnetic field can hold it

A magnetic field pushes on a moving charge sideways, at right angles to its motion. Across the field, the particle goes around in a small circle. Along the field, nothing stops it. Together that makes a spiral that follows the field line. The circle’s radius is

r = m v⊥ / (q B)

A deuterium nucleus moving at 1,000 km/s across a 3-tesla field circles with a radius of about 7 mm. (In a 100-million-degree plasma, the typical, root-mean-square speed of a deuterium nucleus is about 1,100 km/s.) So particles are tied to field lines, but they run freely along them.

Why a doughnut, and why the twist

Particles escape out the ends of a straight tube, so bend the tube into a ring and the field lines have no ends. But a plain ring field is not enough. Where it curves, the field is stronger on the inside than the outside. Ions drift one way and electrons the other, the charges separate, and the electric field that builds up pushes the whole plasma outward into the wall.

The cure is to twist the field lines, so each one goes over the top and under the bottom as it travels around. The drifts then cancel out on average. The amount of twist is the rotational transform:

ι = (turns the short way, around the axis) ÷ (turns the long way, around the ring)

Tokamaks and stellarators

A tokamak, like ITER, runs a strong electric current through the plasma itself. That current makes part of the field, and the sum is twisted. A stellarator gets the twist from twisted magnet coils outside the plasma, so it needs no current in the plasma and can run steadily. Lyman Spitzer invented the stellarator at Princeton in 1951, in work that led to the Princeton Plasma Physics Laboratory.

The price is complicated coils. Wendelstein 7-X in Greifswald, Germany, the world’s largest stellarator, has 50 non-planar and 20 planar superconducting coils. Its plasma ring has a radius of 5.5 m, from the center of the machine to the middle of the plasma. The plasma tube itself has a radius of about 0.52 m and holds 30 m³ of plasma in a field of up to 3 tesla. Its twisted shape could only be worked out with supercomputers. It made its first plasma on December 10, 2015.

Equilibrium: forces in balance

Hot plasma pushes outward with its pressure p. Electric currents J in the plasma, crossing the magnetic field B, push back. When the two balance everywhere, the plasma is in equilibrium:

J × B = ∇p,   J = ∇ × B / μ₀,   ∇ · B = 0

The last equation says field lines never start or stop. Dot the first with B and you get B · ∇p = 0: pressure cannot change along a field line. So the field lines lie on surfaces of constant pressure. These are the flux surfaces, labeled by a number ψ that is 0 on the axis and δ on the edge:

B · ∇ψ = 0,   p = p(ψ)

Stellarator symmetry

Many stellarator designs have a hidden symmetry: turn it upside down with a half turn about one horizontal line (here the x-axis) and it looks the same, with the field reversed. In symbols, with S(x, y, z) = (x, −y, −z):

ψ(Sx) = ψ(x),   B(Sx) = −S B(x)

The equilibrium codes that design stellarators often build this symmetry in, because it halves the number of numbers they must find. Real machines are never perfect, so it matters to know what happens without it.

Grad’s conjecture

For a perfectly round doughnut (a symmetric tokamak) equilibria are easy to find. For a fully three-dimensional shape they are hard. In 1967 the mathematician Harold Grad doubted that smooth toroidal equilibria, with pressure that changes from surface to surface, exist at all, apart from special cases such as shapes that are the same all the way around the ring. This doubt became known as Grad’s conjecture, and it stayed an open question for decades.

In September 2026, Matt Landreman (University of Maryland) wrote down exact formulas for two families of smooth, fully 3-D equilibria with nested flux surfaces, finite twist and pressure that drops from the axis to the edge. They are counterexamples to a precise mathematical form of the conjecture. Both families were stellarator symmetric.

In October 2026, Opal Issan, Jonathan Citrin, Matt Landreman and Brendan Tracey (Google DeepMind; UC San Diego; University of Maryland) added one more number to each family, τ, that breaks the stellarator symmetry and leaves every equation exactly satisfied. With τ = 0 you get Landreman’s solutions back. So these smooth 3-D equilibria are not a side effect of stellarator symmetry.

Family 1: the stretched Solov’ev bottle

With a = √(1+ε), b = √(1−ε), s = x²/a² + y²/b², Ω = √(1−τ₀²) and F = √(1 − (1−s)² − 4z² − 4τ₀sz):

Bx = [(2z + τ₀s)x − (a/b)F y] / s
By = [(2z + τ₀s)y + (b/a)F x] / s
Bz = 1 − s − 2τ₀z,   p = pa − 2ψ/Ω²

The paper also gives every field line in closed form. Each one is labeled by two fixed numbers (u, v) and the flux label is simply ψ = (u − ua)² + v², with ua = −Ω²ε/2. That gives the volume and the average β in closed form:

V = 2π² √(1−ε²) δ / Ω³    βV = 2(pb + δ/Ω²) / (1 − Ω²ε²/2 + δ/Ω²)

Here β = 2μ₀p/B² compares plasma pressure with magnetic pressure, and pb is the pressure at the edge (zero in this explorer). The surfaces are nested tori when 0 < δ < (1 − Ω²ε)²/4. At ε = 0.5 and δ = 1/64: V = 0.267 and βV = 3.51% for τ₀ = 0; V = 0.411 and βV = 4.49% for τ₀ = ±0.5.

Family 2: the complex-phase bottle

This one uses complex numbers. With ω = x + iy, K = ω̄ √(1 + ε/ω̄²) and Ξ = ωK + π/2 − S + iτ₁:

Bx + iBy = e−iλz · i sin Ξ / (2K)
Bz = Re(e−iλz cos Ξ) / λ
ψ = [sin²(λz) + λ²Bz²] / 2,   p = pa − ψ/λ²

The magnetic axis is a flat ellipse at z = 0. The surfaces are nested tori when 0 < δ < ½ sin²(min(S, π/2)). The twist ι changes from the axis to the edge, which is called magnetic shear. At the paper’s Figure 5 settings (ε = 1.2, S = 1.4, λ = 1.6, δ = 0.14) this explorer finds ι = 2.800 on the axis and 2.810 at the edge for τ₁ = 0, but 2.564 and 2.543 for τ₁ = 0.5: the shear flips sign, as the paper describes.

Why exact solutions matter

Real stellarators are designed with computer programs (VMEC, SPEC and DESC are three) that solve the equilibrium equations step by step. To trust them, you test them on problems whose answers you already know exactly. Exact answers for smooth, fully 3-D shapes like these are very rare. These families are such tests, and the new ones test the codes without stellarator symmetry.

Try this

Find the symmetry. Turn on the mirror test in Family 1 and press the τ₀ = 0 button. The pink cage hugs the bottle. Now slide τ₀ slowly to 0.5. The real bottle sinks and the turned-over one rises, until they no longer touch. Watch the “mirror gap” number.
Count the twist. In Family 1, turn off the particles and follow one bright line once around the long way. Count how many times it passes over the top: twice, because ι = 2. Change ε, τ₀ or δ: it is always 2.
Shear that flips. In Family 2, watch the ι chart while you move τ₁ from 0 to 1. At τ₁ = 0 the curve rises toward the edge. Past about τ₁ = 0.3 it falls instead. Try λ too: the chart does not change at all, because ι does not depend on λ.
The biggest bottle. Push δ to the top of its range. The inner edge of the doughnut gets close to the place where the formulas stop making sense, and the surfaces crowd together there. Then look at the slices.
Mirror twins. Set τ = 0.5 in either family and turn on the mirror test. Remember where the pink cage is. Now set τ = −0.5: the bottle moves exactly to where the cage was. Flipping the sign of τ turns the bottle over, which is why the volume and β never change when you do it.
Climb the map. In Family 2, set the trade-off map to color by βV, with S across and δ up. Bright means high: the best bottles sit along the top, where δ is near its limit, and toward a big axis. Click a bright spot to load it. Then run the goal finder with δ kept below 60% of its limit, and watch it settle for less.
No free lunch. In the trade-off chart, put the ripple across and βV up. Follow the gold front: to get a higher βV, you have to accept a more uneven field on the edge. Tap the two ends of the front and compare their shapes in the 3-D view.
Read the check box. Its numbers sit near 10⁻¹⁰. That is round-off: the computer keeps about 16 digits, and the derivatives are measured over tiny steps. Press “Test new random points” and change any setting: the numbers stay tiny. If a formula had a single wrong sign, they would be near 1.

A common mistake: “Magnets pull the plasma inward”

It is natural to picture the magnets pulling the hot gas toward the middle, like a magnet picking up a nail. That is not how it works. Plasma particles are not iron, and the magnetic force on a moving charge is always at right angles to its motion. It never speeds a particle up or slows it down. It only bends its path.

So a particle slides freely along a field line and circles tightly around it. The trap works because the field lines themselves never leave the bottle: each one lies on a closed flux surface and winds around forever. What holds back the plasma’s pressure is the push of currents in the plasma against the field, J × B, balanced at every point. That balance is exactly what the “check the equations” box measures.

How the explorer works

The model. Nothing here is simulated step by step. The field, the pressure and the flux label ψ come straight from the paper’s formulas, in its dimensionless units. Family 1’s surfaces and closed field lines come from its exact field-line formulas. For Family 2 the explorer uses elliptic coordinates: we found that ωK = σ + i(ε/2) sin 2ζ, where σ labels confocal ellipses and ζ is the angle around them. That turns ψ = δ into a formula you can solve for each point. Family 2’s field lines are followed with a fourth-order Runge–Kutta method, three times around, as in the paper.

The numbers. Family 1’s volume and β come from the closed forms. Family 2’s are computed by adding up the volume in thin slices, with the vertical part done exactly. Family 2’s ι is the ratio of two magnetic fluxes through each surface: the flux the short way around over the flux the long way around. That is the same average twist the paper’s field-line equation gives, but it can be added up on a grid over the surface, which stays fast and accurate even near the top of δ’s range. The |B| range on the edge is found at 400 spread-out points and then polished to the exact highest and lowest values. The edge pressure is set to zero, as in the paper’s Figures 3 and 7.

What’s left out. These are ideal equilibria: no resistance, no flows, and no check that they are stable. A real plasma can be in balance and still be unstable, like a pencil standing on its point. There are no coils here either. The formulas are mathematical test cases, not machine designs, and their sizes are in units of the formulas, not meters.

How it was checked. Every check below was run in Node.js before the page was built. The “check the equations” box repeats the first four live.

  • ∇·B, J×B − ∇p, (B·∇)B + ∇Q and B·∇ψ by central differences at 400 random points inside the plasma for 10 parameter sets: all below 4 × 10⁻⁹ relative. With the pressure sign flipped, the force residual jumps to about 1. Over 28 extreme slider settings: below 4 × 10⁻⁷, shrinking 16-fold when the step shrinks 4-fold, as pure difference error should.
  • Drawn surfaces: ψ off by at most 2 × 10⁻¹³ (Family 1) and 3 × 10⁻¹⁴ (Family 2) of its value.
  • Family 1’s field-line formulas: Ω dx/dζ = B to 3 × 10⁻¹⁰; Jacobian −ab/Ω³ to 5 × 10⁻¹⁰. A traced line closes after one turn with ι between 2.0000 and 2.0003, and ∮dℓ/|B| = 7.2551975 against 2π/Ω = 7.2551975.
  • Family 1 volume on a 400 × 400 × 200 grid: 0.26708 and 0.41122 against the formula’s 0.26710 and 0.41123. ⟨|B|²⟩ matched the formula to 6 digits.
  • Family 2 volume and averages against a 240 × 240 × 120 grid: V = 1.7642 vs 1.7647, βV = 0.24546 vs 0.24540.
  • Family 2’s ι equation against traced field lines: the angle agrees to 2 × 10⁻⁸ over three turns. ι is the same for λ = 0.8, 1.6 and 3, and for τ₁ = ±0.6, to 10⁻⁷.
  • Family 2’s ι as a ratio of fluxes against the paper’s field-line equation followed for 400 periods with 512 steps each, at 40 random settings including δ at 98% of its limit: within 5 × 10⁻⁶. (Following the equation with only 32 steps per period, as an earlier version of this page did, was off by up to 16% near the top of δ’s range.)
  • The words in “What each setting does” against its live table: every statement that says which way a number moves was checked at 2,000 random settings of both families, with no exceptions. The statements are based on sweeps of every setting across its whole range at five starting points per family.
  • Symmetry: with τ = 0, ψ and B pass the mirror test exactly. Changing τ to −τ gives exactly the turned-over bottle. Both families also repeat every half turn around the ring, for any τ.

Sources

@teddyrookbook GitHub