Hyperchoreography catalogue

1818 records · d 2–11 · N 3–12
tile
how to read a tile
Main view — a static orthographic 3-D shadow of principal axes 1, 2, 3 of the pooled second moment (all N bodies, all samples). Depth is encoded as line opacity. The stored coordinate basis is meaningless (the action is O(d)-invariant), and a plain (1,2)-plane plot renders a large class of high-dimensional records as a featureless circle — hence 3-D. Dots are the bodies, brightest first, body 0 ringed.
Strip — one panel per principal plane (1,2), (3,4), (5,6)…, in three states: a curve = both axes exceed the display threshold; a horizontal line = exactly one does; a short grey rule = neither (or the axis does not exist). So visible axes = 2×(curve panels) + (line panels), countable by eye. The threshold is λ/λ₁ > 1e−5 and can differ from stored deff. This is not a dimension proof. Panels are normalised per axis, so shape here is qualitative only.
Meter — under each panel, two bars of height √(λk/λ1): the magnitudes the strip's per-axis normalisation throws away. Grey = dead axis; a hairline = that axis does not exist at this d.
Ribbon — the mutual distances |q0−qk|(t), k = 1…⌊N/2⌋, over one period, floor = 0. Dead-flat lines mean a relative equilibrium (rigid body rotation); wavy lines show non-rigid motion, not an existence proof or a common inertial curve. The lowest sampled minimum is minsep.
even speed — every displayed loop uses a 2π time window, so a loop whose path is 80 radii long sweeps the frame ~15× faster than one 5 radii long. On, each tile's clock is divided by its own path length so the grid reads at one pace; off, all tiles share the true time base.
dimension colour, fixed to 2…11 so it is stable across harvests:
Badges — rigid relative equilibrium (rigidity < 1e-4, a clean gap in the data — trivial however high its d); frame {…} rotating frame with its rotation-rate multiset; cover multiple cover; action cluster matching numerical action, N and ambient d (neither orbit equivalence nor a continuous family is established); residual a record whose numerical state residual exceeds 1e-9 or whose stored coefficients exceed 1e-6. Click any tile for the full record.
Evidence, rotating frames, and prior work — audit update

This gallery renders numerical Fourier approximations, not proof witnesses. Historical proof flags require recomputation after corrections to interval bounds and the action remainder. The revised prover has rechecked the figure eight and the full-dimensional 12-body example in R11 (#13 in d11_n12_g2.bin); the catalogue as a whole has not been re-proved.

For a rotating-frame record, Qj(t) = exp(Ωt)q(t + 2πj/N). Closure after 2π does not imply that every body follows the same inertial curve. Displayed dimensions, twist, stability-related indices, and action clusters are numerical diagnostics.

Prior work: Xiaoming Li and Shijun Liao report 21 equal-mass spatial choreographies within a larger numerical periodic-orbit dataset in arXiv:2508.08568v1 (2025). Their orbit identities and reported stability must be credited and compared before making novelty claims; they are not interval existence certificates.

Read the mathematical audit, scientific credits, and reference-seed workflow.

Throughout, d is the dimension the motion actually occupies. The ambient space a record was found in is a search setting, not a property of the orbit, and only bounds it: a search run in R7 returns orbits of every d from 2 upwards. The selection below is ranked over the whole catalogue; open a section for the rest.