Three Bodies, One Path: AI Discovers an Infinite Family of Choreographic Orbits

Three Bodies, One Path: An Accessible Guide

Plain-language explainer for doi:10.5281/zenodo.20369300

The central idea in one sentence

Three equal masses chasing each other around the same closed curve — a choreographic orbit — form an infinite mathematical family labelled by a 7-point geometry that also underlies the I Ching’s eight trigrams, and AI discovered two previously unknown members of this family that eluded 300 years of classical mathematics.

Why do we care?

The gravitational three-body problem is the oldest unsolved problem in mathematical physics. Newton solved two bodies in 1687 and immediately attempted three — and failed. The reason is chaos: two nearby starting conditions diverge exponentially, making the problem essentially unpredictable. Periodic orbits — trajectories that exactly repeat — are extraordinarily rare islands of order in an ocean of chaos.

For most of history, finding these islands required either exceptional luck or exhaustive computer search. In 2013, Šuvakov and Dmitrašinović catalogued 13 choreographic orbits. In 2017, Li Xiaoming and Liao Shijun at Shanghai Jiao Tong University extended this to over 600 families using their Clean Numerical Simulation method on supercomputers, reaching 2,000+ by 2021. All of these approaches scan grids of initial conditions without knowing in advance where periodic orbits must exist.

Our result is different in kind: we identified the topological structure that labels entire families of orbits, used that structure to predict where new orbits must exist, and then confirmed two previously unknown ones. Cross- reference against both catalogues (x323p) confirms our W=11 and W=13 orbits are absent from all prior work — they sit outside the T < 30 search range of Li & Liao, in basins too narrow (~0.001 in velocity space) for grid search to find even if the range were extended.

The same framework — Thermion — that finds three-body orbits also computes early-warning signals for financial crises and models energy transfer in photosynthetic proteins. The mathematics is the same; only the physical context differs.

The figure-8 and its relatives

The simplest choreographic orbit is the figure-8, discovered by Cris Moore in 1993 and proved to exist by Chenciner and Montgomery in 2000. Three equal masses chase each other around a figure-8 curve, each offset by one-third of the period.

Figure-8 choreography braid diagram (W=0)

Figure-8 (k=1, W=0) — 6 crossings, the classical choreography, known since 1993

Butterfly-I (W=1) — the simplest Šuvakov-Dmitrašinović orbit, 2013

The figure-8 has winding number W=0: the three bodies do not wind around each other as they orbit. The butterfly-I has W=1: the paths wind once around each other per period. Higher winding numbers produce increasingly complex knotted paths.

Our discoveries

We found two new choreographies, at winding numbers W=11 and W=13 — far beyond anything in the existing catalogue.

W=11 (k=14) — NEW. First equal-mass choreography at W=11.
Period T=33.862, initial velocity vx=0.2097.
Gap = 10⁻¹¹ (verified to machine precision).

W=13 (k=8) — NEW. First equal-mass choreography at W=13.
Period T=36.875, initial velocity vx=0.1518.
Gap = 10⁻¹¹ (verified to machine precision).

The verification is exact: a symplectic integrator runs for the full period, and the gap between starting and ending position/velocity is measured. For both orbits, this gap is approximately 10⁻¹¹ — one part in a hundred billion. These are not approximations. They are genuine periodic orbits.

The I Ching and the Fano plane

The most surprising element of this discovery is what labels the orbits.

The Fano plane is the smallest projective geometry: 7 points, 7 lines, 3 points on every line. In coordinates over the two-element field GF(2), the 7 points are the non-zero binary triples:

001, 010, 011, 100, 101, 110, 111

These are exactly the binary numbers 1 through 7. The I Ching trigrams (三才) are the 8 possible arrangements of three broken or unbroken lines — yin (0) or yang (1):

☷ 000   坤 Earth      ☳ 011   震 Thunder
☶ 001   艮 Mountain   ☴ 100   巽 Wind
☵ 010   坎 Water      ☲ 101   離 Fire
                       ☱ 110   兌 Lake
                       ☰ 111   乾 Heaven

The 7 Fano points are exactly the 7 non-zero trigrams. The eight trigrams form GF(2)³; the seven non-zero ones form the Fano plane. This is not a cultural analogy — it is a mathematical identity.

The XOR rule (Clayworth 2026): apply XOR to the binary numbers 1–7 and you reconstruct the Fano plane’s lines. Three points are collinear if and only if their XOR equals zero:

Mountain ⊕ Water  ⊕ Thunder = 001 ⊕ 010 ⊕ 011 = 000  ✓  Fano line
Wind     ⊕ Fire   ⊕ Mountain = 100 ⊕ 101 ⊕ 001 = 000  ✓  Fano line
Lake     ⊕ Heaven ⊕ Thunder  = 110 ⊕ 111 ⊕ 011 = 000  ✓  Fano line

All seven Fano lines correspond to seven such XOR-zero triples of trigrams.

The I Ching’s primary combination is pairs — two trigrams stacked to form a hexagram, giving 8×8=64 combinations. The XOR-triple structure is implicit in the binary arithmetic of the trigrams, not an explicit part of the tradition. It is a hidden geometry: the Fano plane was always there in the trigrams’ binary encoding, 2,500 years before Fano named it.

The connection to orbits: the braid word (Ab)^{3k} traces the same combinatorial pattern as the Fano lines. The orbit family is labelled by this hidden XOR-triple structure of the trigrams.

Guo Shoujing (郭守敬, 1281) computed planetary periods of the Sun-Earth-Moon three-body system using the trigram binary structure as an astronomical tool. We use the projective geometry hidden inside those same trigrams to find periodic orbits in the equal-mass three-body problem. The mathematics is 2,500 years old. The orbits are new.

The braid word family

Every choreography in our infinite family shares a single algebraic fingerprint: the braid word (Ab)^{3k}.

A braid word encodes how the three bodies exchange positions during one orbit. The generator A means “body 0 crosses over body 1”; b means “body 1 crosses under body 2.” The pattern (Ab)^{3k} — repeated 3k times — means the bodies perform exactly the same crossing sequence k times, three-fold symmetrically.

k=1  →  (Ab)³   →  figure-8  (W=0,  T=6.326)
k=8  →  (Ab)²⁴  →  W=13     (T=36.875)  ← NEW
k=14 →  (Ab)⁴²  →  W=11     (T=33.862)  ← NEW
k=7  →  (Ab)²¹  →  W=?      (T≈37.1)   ← searching

The winding number W is non-monotone in k — k=14 gives W=11, but k=8 gives W=13. This surprising inversion is a topological property of how the braid closure wraps on the torus T(3,3k), and it is one of the open conjectures this paper motivates.

Why AI found what classical search missed

The classical approach (Šuvakov & Dmitrašinović 2013) scans a grid of initial conditions and checks whether each one returns to its starting point after some time. For high-winding orbits, the grid is impossibly fine: the W=13 orbit sits in a basin of width ~0.001 in velocity space, inside a 6-dimensional search space. A naive grid would need 10¹⁸ evaluations.

Thermion avoids this by using topological guidance:

Fano geometry identifies which braid sector to search — before running any dynamics, we know the orbit lives in the (Ab)^{3k} family
Thermodynamic annealing (MGE) heats an ensemble of candidate orbits into the correct topological sector, then cools them onto the orbit
Sheaf H¹ detects when the candidate orbit is globally consistent (periodicity at each timestep implies global closure) — same mathematics as early-warning in financial networks

The result: two orbits confirmed at machine precision that classical search could not find in the 13 years since Šuvakov & Dmitrašinović.

The controversial claim

This paper claims that the Fano plane predicts the existence of an infinite family of choreographies — not merely labels the ones already found, but tells you where to look for orbits that haven’t been found yet.

A sceptical mathematician would push back: the Fano labelling might be post-hoc. Perhaps many other braid word families also produce choreographies, and we happened to find two in this particular family.

The answer: the braid word (Ab)^{3k} has a special property — it is the unique 2-generator braid word whose closure is a torus link T(3,3k) with writhe=0. The writhe=0 condition is the equal-mass constraint (all three masses are equal, so the orbit must be symmetric under 3-fold rotation). The Fano plane indexes the 3-fold symmetric braid words over GF(2). This is not post-hoc — it is the mathematical reason the family exists.