Three equal point masses attract each other by Newtonian gravity in a plane. Almost every initial condition gives chaos, but a discrete set gives periodic orbits — the three bodies retrace the same choreography forever. The ones here come from the 2013 catalogue of Šuvakov & Dmitrašinović, which found thirteen new classes beyond the classical figure eight.
Three points in the plane have six degrees of freedom. Throw away the things that do not change the shape of the triangle — where it sits (2), how big it is (1), which way it points (1) — and two degrees of freedom remain. That two-dimensional space is a sphere.
Concretely, from the Jacobi vectors ρ = (x₁ − x₂)/√2 and λ = (x₁ + x₂ − 2x₃)/√6 one builds a unit vector
n = ( |λ|² − |ρ|² , −2 ρ·λ , 2 ρ∧λ ) / ( |ρ|² + |λ|² )
which is automatically of length one. Every triangle is a point of this sphere, and the whole trajectory becomes a closed curve on it.
A sphere with three points removed has fundamental group the free group on two generators, ⟨a, b⟩ — rank two, not three, because a loop around all three punctures slips off over the back of the sphere.
Cut the equator at the punctures; it falls into three arcs. Take two of them, α from 0 to 2π⁄3 and β from 4π⁄3 to 2π, as cuts. Their union is a tree joining all three punctures, and cutting the sphere along it leaves a disc — so a loop's homotopy class is decided entirely by which cuts it crosses, and in which direction. Crossings of the third arc happen inside the disc and mean nothing.
At each equator crossing, read off the longitude and the direction of travel:
a, going north → Ab, going north → BConcatenating these letters over one period gives the orbit's raw word. Two ambiguities in it are not physical, and removing them gives the labeled class — the two lines the Reduction panel shows.
aA and bB are the identity. A
cancelling pair means the curve nicked across a cut and came straight back, which is
homotopically nothing.The panel stops there deliberately, because every further reduction throws away something physical. Taking the primitive root of wq discards how many times the orbit went round; quotienting by the six relabellings of the three bodies discards which body is which; quotienting by the mirror discards the handedness. Each is standard for a catalogue label, and each merges orbits that are genuinely different — so none of them is shown as a stage of this orbit's own reduction.
They are used in exactly one place: the Agreement line, which checks this orbit against the word as the paper prints it. That comparison has to apply them, because the paper writes several of these words with a different choice of body labels than the convention used here, and prints yarn's word as an explicit cube.
One warning worth repeating: even the labeled class is an invariant, not a complete label. Distinct orbits with different periods and energies can share one. It organises a catalogue; it never certifies that two solutions are the same.
The 3D sphere shows the curve as it really is. The Mercator map unrolls it so that longitude is horizontal and the equator is a straight line: the three punctures become three vertical lines, the letters become crossings of the coloured segments of the horizontal midline, and the word can be read straight off left to right. Mercator stretches near the poles without bound, so latitude is clipped at ±85° — the dotted borders.
Integration is an adaptive Dormand–Prince 5(4) scheme at tight tolerance, with the orbit sampled on a uniform time grid over exactly one period. Syzygies are found from sign changes of the signed area (x₂ − x₁) ∧ (x₃ − x₁), which vanishes exactly on the equator, then located by bisection on a cubic Hermite interpolant. The initial conditions were Newton-refined from the published five-digit values, by shooting on the half-period symmetry the whole family shares: every one of these orbits starts in the configuration x₁ = −x₂, x₃ at the midpoint, v₁ = v₂, and returns to that same set at exactly T⁄2. Shooting over the half period rather than the full one halves the error amplification, which is what makes the refinement converge at all.
Fourteen of the seventeen orbits then close to around 10⁻¹⁰ and reproduce the published word exactly. Three do not: butterfly IV and the two yin-yang II orbits pass within about 10⁻³ of a binary collision, and each such passage multiplies any error enormously. Over a full period the amplification exceeds what double precision can absorb, so their initial conditions are not representable at all — the closure error stalls near 10⁻⁵ no matter how the refinement is done or how tight the integration tolerance is set. Those three are shown with a warning: the motion is real, the shape curve is real, but the tail of the word is not the published one. Every panel reports its closure error, so you can see this for yourself rather than take it on trust.