A catalogue of orbits that repeat.

Three equal masses chase each other along a single figure-eight curve, the most famous choreography.

Two wings of tight loops, with the bodies crossing through the centre at high speed.

Slim, elongated wings that stretch far from the centre of mass.

Two interlocking halves whose lobes wrap around each other.

Broad, fringed wings built from many overlapping passes.

A second butterfly sequence with a different topology from class I.

Moth-like shapes from a third topological class.

Dense skeins of thread wound around a hollow core.

Symmetric moths whose wings meet along a luminous seam.

Related moths with a different winding of the outer lobes.

The eighth sequence of the Belgrade gallery, recovered from its archived pages; three orbits that appear in no other catalogue.

Broucke’s 1975 retrograde family: a binary and a third body circling it, the binary turning one way and the third body’s orbit the other; several are linearly stable.

The prograde counterpart of the A family: the binary and the third body’s orbit turn the same way.

Broucke and Boggs’s 1975 survey of periodic orbits at many mass ratios, most of them reported linearly stable.

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.

Class I.A orbits found with Liao’s clean numerical simulation method, for a third body of varying mass.

Class I.B orbits, for equal and unequal masses.

The rare class I.C orbits.

Class II.A orbits with an unequal third mass.

Class II.B orbits, mostly long-period.

The largest class in the atlas: hundreds of II.C orbits for seven mass ratios.

Two class II.D orbits with a heavy third body.

Three bodies released from rest fall together, swing past each other and return to rest exactly where they started.

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.

Stable orbits from the moth-like part of the search domain, between the figure-eight and the Schubart regions.

Stable orbits that pass extremely close to a binary collision, descended from Schubart’s rectilinear orbit. They needed 100-digit arithmetic to find.

Stable orbits related to Broucke’s isosceles orbit, with one body swinging through the other two.

All three bodies follow one closed curve, each a third of a period behind the next. Every one here is linearly stable.

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.

Carles Simó’s catalogue of three-body choreographies with nonzero angular momentum, the curves that made choreographies famous. All unstable except the figure-eight.

Titov’s continuations of the figure-eight to a lighter third body, where the curve splits into two lobes.

A satellite of moth I whose existence at masses 1, 1, 1.001 is established by a computer-assisted proof.

Orbits from topological families that no earlier catalogue contained; all of them unstable.

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.

Orbits from Rose’s survey of the symmetry-reduced problem, which reaches parts of phase space the other searches do not.

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.

Forty-two periodic orbits of topologies absent from the checked catalogues, found in a September 2026 search at unequal masses: the middle body lighter or heavier than the outer two, or all three different. Each is named by its free-group word, a few by a nickname; all are unstable. Candidate new families relative to the catalogues checked, not established discoveries.

Golden pancake drama is the one linearly stable orbit of the September 2026 new-family search, with masses 1, 0.612 and 1; beside it are six samples of the closed mass loop it sits on, an isola whose middle mass runs from 0.469 to 0.635 and never reaches equal masses, with a stretch of the orbit’s own topology and a stretch shaped like a threefold figure-eight satellite. Numerically verified to 160 bits; not published elsewhere.

Thirteen equal-mass orbits with angular momentum from Milovan Šuvakov’s own searches, shared with this atlas in September 2026. Seven are the figure-eight carried to nonzero angular momentum, its 1/5 to 2/3 resonances, all linearly stable; four lie on a branch that ends, at zero angular momentum, in Broucke’s isosceles collision orbit; two branch off Broucke’s A family. Numbered as in his file.

Satellites of the Broucke–Hadjidemetriou–Hénon family at equal masses, closed by a rational rotation. A satellite loop is k laps around the binary, of the retrograde (A) or the prograde (R) branch, and the pattern turns r/s of a turn per loop, so the orbit closes after s loops. Fourteen linearly stable ones from the site’s own September 2026 bifurcation search, none in the catalogues checked. Their words are powers of the family’s one-letter word: new orbits, not new topologies. Coded by branch, k and r/s. One further orbit, A 2/3 (0.3:0.6:1), is no satellite: a stable member of the A family itself at masses 0.3, 0.6 and 1, closed at two thirds of a turn per loop, a known family at a closure the catalogues do not list, included for its shape.

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. His loop-ended triangles turned out to be Broucke and Boggs’s S4 of 1975, mirrored, and is listed under that name.