A catalogue of orbits that repeat.

How the orbits were computed

Every orbit starts from the initial conditions its discoverers published. Those are re-converged, with Newton’s method in 32-digit arithmetic, until the loop closes to the limit of the arithmetic, then integrated once over a full period with a high-order Taylor method and stored as a compact spline. The atlas records the period, energy, angular momentum, closest approach and the Floquet multipliers that decide whether an orbit is linearly stable. Your browser only plays the stored solution back; nothing is simulated live.

Brightness along a trace follows a long-exposure rule: the slower a body moves, the more light it leaves behind.

Nudging an orbit

Every orbit page has a kick slider. It changes the velocities and masses by a small amount in a direction that is fixed for that orbit, as far as the slider is moved, and hands the result to the personal generator’s live integrator for a few periods. Stable orbits shrug the kick off; unstable ones unravel. “New random kick” tries another direction.

Where the orbits come from

The first 2,386 sets of initial conditions were compiled by Ricky Reusser in his Periodic three-body initial conditions notebook (MIT licence); the rest were taken from the papers and their supplementary data directly, several of them at a hundred digits or more. The sources:

• Šuvakov & Dmitrašinović (2013). Three classes of Newtonian three-body planar periodic orbits.Physical Review Letters 110, 114301 (doi)
• Broucke (1975). On relative periodic solutions of the planar general three-body problem.Celestial Mechanics 12, 439–462 (doi)
• Li & Liao (2017). More than six hundred new families of Newtonian periodic planar collisionless three-body orbits.Science China Physics, Mechanics & Astronomy 60, 129511 (doi)
• Li, Jing & Liao (2018). Over a thousand new periodic orbits of a planar three-body system with unequal masses.Publications of the Astronomical Society of Japan 70, 64 (doi)
• Li & Liao (2019). Collisionless periodic orbits in the free-fall three-body problem.New Astronomy 70, 22–26 (doi)
• Hristov, Hristova & Tanikawa (2026). An extensive search for stable periodic orbits of the equal-mass zero angular momentum three-body problem.New Astronomy 125 (link)
• Hristov, Hristova, Puzynin, Puzynina, Sharipov & Tukhliev (2023). A database of high precision trivial choreographies for the planar three-body problem.Lecture Notes in Computer Science 13858, 171–180 (link)
• Hristov, Hristova, Puzynin, Puzynina, Sharipov & Tukhliev (2022). New families of periodic orbits for the planar three-body problem computed with high precision.CEUR Workshop Proceedings 3191, 45–50 (link)
• Li, Li & Liao (2021). One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems.Science China Physics, Mechanics & Astronomy 64, 219511 (doi)
• Rose (2016). Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum.PhD thesis, University of Sydney (link)
• Hristov, Hristova, Puzynin, Puzynina, Sharipov & Tukhliev (2024). Nontrivial choreographies of the planar three-body problem.Physics of Particles and Nuclei 55, 495–497 (doi)
• Simó (2001). Dynamical properties of the figure eight solution of the three-body problem.Contemporary Mathematics 292 (Celestial Mechanics, dedicated to Donald Saari) (doi)
• Broucke & Boggs (1975). Periodic orbits in the planar general three-body problem.Celestial Mechanics 11, 13–38 (doi)
• Dmitrašinović, Hudomal, Shibayama & Sugita (2018). Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler’s third law.Journal of Physics A 51, 315101 (link)
• Titov (2012). Periodic solutions of the three-body problem with unequal masses in the vicinity of the figure-eight.Russian Journal of Nonlinear Dynamics 8, 377–389 (doi)
• Three-body Moth-I¹³ computer-assisted proof (2026). A computer-assisted proof of a periodic orbit near moth I at masses 1, 1, 1.001.GitHub, Rexyysilent/three-body-moth13-cap (link)
• Hénon (1976). A family of periodic solutions of the planar three-body problem, and their stability.Celestial Mechanics 13, 267–285 (doi)
• Moore & Nauenberg (2006). New periodic orbits for the n-body problem.Journal of Computational and Nonlinear Dynamics 1, 307–311 (doi)
• Sheen (2016). Periodic three-body solutions found with a root finder, example code for Cornell’s Intermediate Dynamics course.GitHub, mws262/MAE5730_examples; six of them in the Belgrade three-body gallery (link)
• Three Body Orbits (2026). Two previously uncatalogued Broucke-region orbits.Research note, threebodyorbits.com, 8 September 2026
• Three Body Orbits (2026). Eight-hour Broucke-like orbit search.Research note, threebodyorbits.com, 9 September 2026
• Vasiljević, Raonić & Dmitrašinović (2023). An island of linearly stable non-hierarchical unequal mass periodic three-body orbits.New Astronomy 100, 101969 (doi)

Families

• Figure-8: Three equal masses chase each other along a single figure-eight curve, the most famous choreography. 28 on the map
• Butterfly I: Two wings of tight loops, with the bodies crossing through the centre at high speed. 19 on the map
• Dragonfly II: Slim, elongated wings that stretch far from the centre of mass. 16 on the map
• Yin-Yang III: Two interlocking halves whose lobes wrap around each other. 20 on the map
• Moth IVa: Broad, fringed wings built from many overlapping passes. 36 on the map
• Butterfly IVb: A second butterfly sequence with a different topology from class I. 32 on the map
• Moth IVc: Moth-like shapes from a third topological class. 22 on the map
• Yarn VI: Dense skeins of thread wound around a hollow core. 6 on the map
• Moth VIIa: Symmetric moths whose wings meet along a luminous seam. 8 on the map
• Moth VIIb: Related moths with a different winding of the outer lobes. 12 on the map
• Sequence VIII: The eighth sequence of the Belgrade gallery, recovered from its archived pages; three orbits that appear in no other catalogue. 3 on the map
• Broucke A: Broucke’s 1975 family of orbits with a distant third body circling a close binary; several are linearly stable. 16 on the map
• Broucke R: Retrograde relatives of the A family, with the third body circling the other way. 13 on the map
• Broucke–Boggs: Broucke and Boggs’s 1975 survey of periodic orbits at many mass ratios, most of them reported linearly stable. 23 on the map
• Broucke-like rosettes: Zero-angular-momentum relatives of Broucke’s rosettes: two equal bodies in a close binary and a lighter third body circling them the other way, its mass tuned so that the precession closes the pattern after a whole number of turns and the inner paths cross the outer one. Fourteen representatives from the site’s own September 2026 search, all linearly stable; points on a known continuous family rather than a new one. Named by the inner and outer winding counts. 14 on the map
• I.A: Class I.A orbits found with Liao’s clean numerical simulation method, for a third body of varying mass. 513 on the map
• I.B: Class I.B orbits, for equal and unequal masses. 540 on the map
• I.C: The rare class I.C orbits. 3 on the map
• II.A: Class II.A orbits with an unequal third mass. 20 on the map
• II.B: Class II.B orbits, mostly long-period. 35 on the map
• II.C: The largest class in the atlas: hundreds of II.C orbits for seven mass ratios. 844 on the map
• II.D: Two class II.D orbits with a heavy third body. 2 on the map
• Free-fall: Three bodies released from rest fall together, swing past each other and return to rest exactly where they started. 311 on the map
• Figure-8 satellites: Stable orbits found by an exhaustive search of the figure-eight’s neighbourhood: each traces the eight with extra loops and twists, and every one is linearly stable to 40 digits. 196 on the map
• Moth region: Stable orbits from the moth-like part of the search domain, between the figure-eight and the Schubart regions. 103 on the map
• Schubart region: Stable orbits that pass extremely close to a binary collision, descended from Schubart’s rectilinear orbit. They needed 100-digit arithmetic to find. 213 on the map
• Broucke region: Stable orbits related to Broucke’s isosceles orbit, with one body swinging through the other two. 74 on the map
• Choreographies: All three bodies follow one closed curve, each a third of a period behind the next. Every one here is linearly stable. 76 on the map
• Nontrivial choreographies: Choreographies whose knot is not a power of the figure-eight’s: three bodies on one closed curve with a topology of its own. Computed to 180 digits; all unstable. 164 on the map
• Simó’s choreographies: Carles Simó’s catalogue of three-body choreographies with nonzero angular momentum, the curves that made choreographies famous. All unstable except the figure-eight. 343 on the map
• Split figure-eights: Titov’s continuations of the figure-eight to a lighter third body, where the curve splits into two lobes. 2 on the map
• Certified satellite: A satellite of moth I whose existence at masses 1, 1, 1.001 is established by a computer-assisted proof. 1 on the map
• New families: Orbits from topological families that no earlier catalogue contained; all of them unstable. 123 on the map
• Moth IVa.2.A, unequal masses: The stable moth IVa.2.A continued to unequal masses: one body a little lighter or heavier than the other two, and the orbit stays stable. 7 on the map
• Rose’s Bestiary: Orbits from Rose’s survey of the symmetry-reduced problem, which reaches parts of phase space the other searches do not. 13 on the map
• New Broucke-region orbits: Three linearly stable orbits found here in September 2026 by pushing the Broucke region past the published catalogue’s limit of T* = 800: two continue known lap sequences, one is a seventeen-fold resonant satellite of the Broucke-region orbit B1. Verified independently to 160 bits and again by this atlas’s own tool; not published elsewhere. 3 on the map
• Sheen’s orbits: A dozen orbits Matthew Sheen found in 2016 with a root finder and hand-picked starting guesses while a teaching assistant on Cornell’s dynamics course, named for what they look like; six hang in the Belgrade gallery. Most of them carry angular momentum. 12 on the map

Your personal orbit

This atlas grew out of Cosmic Signature, a generator that turns a name and a date of birth into a unique three-body orbit and lets you print it.

Ratings and battles

On the battle page two orbits play side by side and you pick one; both move by the Elo rule, and the head-to-head leaderboard ranks orbits that have fought at least five battles. Every orbit page also has a five-step rating. Both are shared by all visitors: the leaderboard shows everyone’s choices, not yours alone.

Ratings are anonymous. Your browser keeps a random identifier in its local storage (not a cookie) so you can change your mind about an orbit, and the server keeps a one-way hash of your network address with each vote so that it is counted once. Rankings use a weighted average that leans towards the overall mean until an orbit has collected a few votes. Details in the privacy policy.

Colophon

Precomputation in Rust: a double-double Taylor-series integrator and eigensolver written for this atlas, DOP853 from the ode_solvers crate for the double-precision seed, tiny-skia for the thumbnails. Playback in plain JavaScript on an HTML canvas. Typeset in Archivo and JetBrains Mono. Hosted on Netlify, with the orbit data served by Cloudflare Pages.

Licences and privacy.