Open questions

What this lab has not answered yet: 64 open threads, one left behind by each of 64 simulations, plus the 4 questions it is pointed at next. Each is a measurement that has not been made, linked to the world that would make it.

The active frontier

• Sharpen the Particle Life phase diagramThe first 2-D phase map is now drawn (?world=phasemap): crossing condensation (μ) against non-reciprocity (α) carves four regimes — gas, static condensate, active gas, active condensate — with the α=0 edge rigorously static (drift≡0). But it's one seed, one pass, with carry-over (μ swept monotonically → the boundary…Open in Particle Life
• Pin the anomalous-transport exponent at the accelerator modeThe transport curve D(K) now shows WHERE chaos stops diffusing — at K=2π the trapped ballistic orbits split the mean (×12) from the robust median (×1.8). There transport is super-diffusive: ⟨Δp²⟩ ∝ n^μ with μ>1, not the diffusive μ=1. Measuring that exponent μ (and the Lévy statistics of the ballistic flights) is the…Open in Chaotic transport
• Trace the Lorenz route into chaos (the ρ-bifurcation)The Lorenz butterfly is now drawn (?world=lorenz) at the classic ρ=28, with λ≈0.906 measured. But the strange attractor only exists above ρ≈24.74 — below it the off-origin fixed points are stable spirals (convection rolls that settle). The open thread is to make ρ a knob and trace λ(ρ) across the subcritical Hopf…Open in Lorenz attractor
• Chart the figure-8's stability boundaryMutual gravity is now on (?world=threebody): a generic triple is chaotic (λ≈1) while the figure-8 choreography is a stable island (λ≈0). The open question is the SHAPE of that island — how far can you perturb the figure-8 before λ turns positive, and what is the distribution of ionisation (escape) times for the…Open in Three-body problem

64 open threads

• Carnot engine · the most work heat can doNext: a heat-pump/refrigerator run (COP = T_c/(T_h−T_c)) and a finite-time (endoreversible) engine to recover the Curzon–Ahlborn η=1−√(T_c/T_h) at maximum POWER, not maximum efficiency.
• Particle LifeNext: a headless seed-sweep recording these observables across matrix space, to draw the first phase diagram for Particle Life (calibrating live thresholds is done).
• Emergent matter · structureNext: per-species partial g(r), and tracking g(r) across the condensation μ sweep to watch the gas→liquid structural transition emerge in the pair correlation.
• Condensationμ_c is for one seed + density. Next: average over seeds, sweep μ down to bound the hysteresis, and map a 2-D (mean × asymmetry) phase diagram.
• Condensation hysteresisPin Δμ with a finer μ-grid + longer settle, and average over seeds; then map the 2-D (mean × asymmetry) phase diagram.
• Non-reciprocity · active matterSingle seed, ascending sweep (carry-over between α). Next: average over seeds, then cross this asymmetry axis with the condensation μ axis into a 2-D map.
• Particle Life · 2-D phase diagramSingle seed, single pass (μ monotonic → Δμ≈0.04 hysteresis-smeared boundary). Next: average over seeds, a finer grid near the boundaries, and a descending μ pass to bound the 2-D hysteresis.
• Conway's Game of LifeNext: seed famous patterns (glider gun) plus a reseed control, and measure glider speed and population statistics quantitatively.
• Elementary cellular automata · WolframNext: classify all 256 rules by an order parameter (final density / Langton's λ) to draw the class spectrum; and a rule-110 glider collision to glimpse the universal computation.
• Langton's ant · order from chaosNext: does the highway always emerge for arbitrary finite initial black/white patterns (open in general), and how does the chaos-phase length distribute over random starts? And a multi-colour ant (Turmite) wired to compute something concrete.
• Flockingeta_c shifts with density/speed and finite settle time. A longer headless sweep would sharpen the critical noise and pin finite-size scaling.
• Schelling segregationNext: locate the segregation onset more sharply (it rises fastest near T≈0.3); unequal group sizes / three kinds; and a 'move only if it improves' variant vs this random-relocation one.
• Spatial prisoner's dilemma · evolution of cooperationNext: the single-defector-in-cooperators start (the symmetric Persian-carpet kaleidoscope); asynchronous updating + a P>0 payoff to test rule-sensitivity; pair-approximation theory for f_C(b).
• Hopfield network · associative memory & storage capacityNext: dilution / asymmetric weights; the spurious mixture states (½(ξ^1+ξ^2+ξ^3)); a finite-temperature (stochastic) update to draw the full m–α–T retrieval phase diagram; one-shot vs iterative capacity.
• Reaction–diffusion scale selectionλ is quasi-stationary (GS keeps slowly coarsening) and this is NOT a linear Turing bifurcation, so there's no analytic λ. Next: λ vs the diffusion length √D, and the full Pearson (F,k) regime map.
• Phyllotaxis · the golden angleNext: count the visible spiral arms (parastichies) and verify they're consecutive Fibonacci numbers; and the dynamic version where a real meristem lays seeds one at a time under a repulsion rule that SELECTS the golden angle on its own.
• Wave interferenceNext: the two-source fringe SPACING vs source separation (the double-slit relation), and a dispersive medium (c depending on λ) to contrast with this non-dispersive membrane.
• Quantum tunnellingNext: the T(E) curve (transmission vs energy, incl. the resonances for E>V₀), and resonant tunnelling through a double barrier.
• Electrostatics · Gauss's lawNext: a draggable test charge (Φ tracks it live), equipotential contours, and the genuine 3-D inverse-square field with Gauss's law as a surface integral over a sphere.
• Cyclotron motion · Lorentz forceNext: add a crossed E field for the E×B drift (drift speed E/B, independent of q and m — particles of different mass drift TOGETHER), and a velocity-selector / mass-spectrometer geometry.
• E×B drift · crossed fieldsNext: a velocity selector (E⊥B tuned so the qE and qvB forces cancel at one speed v=E/B — a mass-spectrometer front end), and a gravitational/general-force drift F×B/qB² (which, unlike E×B, DOES depend on q and m, separating species).
• Diffusion-limited aggregationNext: an off-lattice (continuum) version to recover 1.71, the multifractal harmonic measure (where on the boundary do walkers actually land?), and a dielectric-breakdown exponent η to morph DLA → lightning.
• Kinetic surface rougheningNext: the roughness exponent α and dynamic z from finite-size saturation W_sat ∝ L^α (a small L-sweep), the KPZ height-distribution (Tracy–Widom GUE/GOE), and ballistic deposition as a slower-converging KPZ contrast.
• Traffic flow · TASEPNext: OPEN-boundary TASEP (inject at α, extract at β) — the three-phase diagram (low-density / high-density / maximal-current) and boundary-induced transitions; and the Nagel–Schreckenberg slow-to-start rule, which makes REAL phantom jams the homogeneous ring lacks.
• Kuramoto syncPoint the order-parameter observable at this world, and reuse the settle/measure/sweep harness on the Particle Life observables.
• Kepler orbitsThis is the test-particle regime. I since turned on mutual gravity (?world=threebody) and swept the gravity exponent (?world=precession). Still open: eccentric orbits for Kepler's 1st/2nd laws.
• Counterfactual gravityCircular orbits go unstable at p≥3 (√(3−p) imaginary). The relativistic 1/r³ term (Mercury's perihelion advance) is now its own world (?world=schwarzschild). Still open: the precession's eccentricity dependence.
• Mercury's perihelion · General RelativityThis is the orbit equation derived from the metric, not a full numerical-relativity solve. Next: light bending / Shapiro delay (the metric's null geodesics), and the ISCO at r=6GM/c² where stable orbits end.
• Three-body problemThe figure-8's finite-time λ only falls as ~ln T/T (not exactly 0); gravity is softened. Next: chart the figure-8's stability boundary, and the distribution of ionisation times.
• Lorenz attractorNext: the ρ-bifurcation route into chaos (the attractor only appears above ρ≈24.74; below it the origin/fixed points are stable), and the attractor's fractal correlation dimension (≈2.06).
• PendulumI built that double-pendulum world (?world=dblpendulum) for chaos in a continuous flow. Still open: apply this period instrument to other oscillators.
• Brachistochrone & tautochroneNext: a draggable B so the cycloid re-solves live; the cycloidal pendulum (Huygens' isochronous clock — a pendulum constrained between cycloid cheeks has an amplitude-independent period). DONE the optics analogue: ?world=snell — Fermat's least-time path through media of different 'speed' bends, recovering Snell's law (the variational twin of this world).
• The hanging chain (catenary)Next: a load hung at the centre (the chain kinks into two straight segments); a suspension-bridge deck loading the cable into a true parabola (uniform mass per horizontal length); and the elastic catenary (extensible links) vs the inextensible one.
• Lattice vibrations · phonon dispersionNext: the DIATOMIC chain (two masses alternating → an acoustic AND an optical branch with a frequency gap); the group velocity dω/dk → 0 at the zone edge (a standing wave, no transport); and a wave packet dispersing as it travels (the dispersion made visible).
• Fermi–Pasta–Ulam–Tsingou recurrenceNext: crank α (or the amplitude) past the stochasticity threshold where the recurrence DOES break down and equipartition finally sets in (the chaos/KAM boundary); the β-model (cubic force) and whether it thermalises differently; and connecting the recurrence to the underlying KdV solitons (Zabusky–Kruskal) that pass through each other and re-cohere.
• Solitons · Korteweg–de VriesMeasure the collision phase shift explicitly; the β/mKdV and breather solutions; an initial pulse decomposing into a rank-ordered soliton train (the inverse-scattering picture, and the direct link back to the FPUT mode dynamics).
• Soliton train · KdV inverse scatteringPush to a NON-reflectionless pulse (a generic sech² or a box) and detect the leftover dispersive ripple alongside the solitons; vary the pulse area and watch solitons appear one at a time as new bound states cross threshold; and tie the emergent soliton count back to the FPUT low-mode dynamics that started this whole thread.
• Doppler effect & sonic boomNext: a STATIONARY source with a moving observer (same shift, different mechanism — the medium frame breaks the symmetry); the relativistic optical Doppler (a √ factor, symmetric); and the gradual build of the bow/stern wake pattern (the water-wave Kelvin angle, a fixed 19.5°).
• Coupled pendulums · normal modes & beatsNext: DETUNE one pendulum (different L) and watch the transfer go incomplete — energy never fully leaves the first; the swap fraction is 1/(1+(δ/c)²), a forced-oscillator resonance curve in disguise. Also: N coupled pendulums → a phonon band (this is the 2-atom case of the lattice in ?world=phonons); and pushing the amplitude large enough that nonlinearity detunes the modes and smears the beat.
• Resonance · the driven oscillatorNext: refine the drive grid near ω_r (adaptive sampling) to recover A_max and Q to the same ~0.1% as the phase crossing; sweep the damping γ to watch the peak sharpen and ω_r→ω₀ as Q→∞ (and the resonance vanish past critical damping γ=√2·ω₀); and connect back to ?world=coupled — the detuned two-pendulum swap fraction 1/(1+(δ/c)²) is this very Lorentzian, so the same lock-in should measure it.
• Refraction · Fermat → SnellDONE total internal reflection: ?world=tir — going slow→fast (glass→air), θ₂ reaches 90° at θ_c and beyond it there's no transmitted ray. Still open: a graded-index medium where n varies continuously and the ray curves (mirage / fibre optics); and the prism/dispersion case n(λ).
• Total internal reflection · critical angleDONE the graded-index medium: ?world=mirage — n(y) varies continuously and the ray curves smoothly, arcing back up before the ground (the inferior mirage), with n·sinθ conserved along the whole ray. Still open: dispersion n(λ) — a prism splitting white light because θ_c and the bend both depend on colour.
• Mirage · graded-index ray bendingThe real atmospheric gradient is ~1e-6 per metre (I exaggerate G for visibility); the bending is the same law, just gentler. DONE the dispersion n(λ) thread: ?world=rainbow — water's index rises toward the blue, splitting the rainbow into colour. Still open: a SYMMETRIC well n(y) (a graded-index fibre / Luneburg lens — rays oscillate and refocus).
• Rainbow · Descartes' 42° causticI draw the geometric-optics caustic (ray pile-up), not the wave-optics intensity (the supernumerary arcs just inside the primary bow are an Airy-function interference effect a ray trace can't show). Next: the supernumeraries (wave optics), the tertiary/quaternary bows (k=3,4, which point back toward the sun), and the polarisation of rainbow light (near-Brewster internal reflection).
• Double pendulumRK4 isn't symplectic (drift is below precision here, not zero by construction). Next: λ vs release amplitude (the regular→chaos crossover), and the Poincaré section.
• DiffusionThe law is exact in expectation, so residual scatter is pure finite-N noise. Next I'll sweep N to shrink the ±3% spread.
• Predator–preyNext I'll add logistic prey or predator handling time to break conservation and test whether the orbit spirals into a limit cycle.
• Logistic mapMy δ=4.751 is only pre-asymptotic, from three bifurcations; resolving r₄,r₅ at finer r would tighten it toward 4.669.
• Standard mapI since measured that chaotic-sea diffusion in ?world=chaosdiff (D≈K²/4 at K=5). Still open here: sweep K continuously to animate the KAM cascade.
• Hénon–Heiles · Poincaré sectionNext: the chaotic-fraction-vs-energy curve (the whole transition as one plot), and Poincaré sections of the lab's other flows — the double pendulum and the three-body problem.
• Chaotic diffusionI since mapped that whole D(K) curve in ?world=transport (a robust median vs the mean reveals the accelerator-mode spikes). Open next: the anomalous-transport exponent at K=2π.
• Chaotic transportThe median assumes a Gaussian sea (it under-reads where correlations bite); and at K=2π transport is super-diffusive — pinning that anomalous exponent ⟨Δp²⟩∝n^μ (μ>1) is the next thread.
• Ising modelAdd Wolff cluster updates to beat critical slowing, plus finite-size scaling across several L to extrapolate Tc(∞) toward 2.2692.
• 2-D XY model · Kosterlitz–ThoulessFinite-size scaling with the Weber–Minnhagen log corrections to pin T_KT→0.893; the power-law η(T)=T/2π correlation exponent below T_KT; the binding-energy of a single vortex pair.
• Site percolationFinite-size scaling: the Π=½ crossing drifts with L (0.596 at L=48 → 0.594 at L=128); extrapolate p_c(∞) and measure the correlation-length exponent ν=4/3 from the crossing width.
• Spatial SIR epidemicNext: a finite, FIXED population (an SIR with a recovery rate so the infectious period >1 step), the epidemic curve I(t) and its peak vs T, herd immunity 1−1/R₀, and a small-world/network contact graph to watch the threshold collapse back toward mean-field.
• Galton board · Central Limit TheoremNext: measure the convergence in a distribution metric (total-variation / Kolmogorov distance, which Berry–Esseen bounds), and sum a non-Bernoulli step (uniform, heavy-tailed) to test the CLT's universality and where it fails.
• Buffon's needle · Monte-Carlo πNext: the long-needle case (L>d, where the formula gains an arccos term and needles can cross twice), and variance reduction (Buffon–Laplace grid, antithetic angles) to beat the bare √N constant.
• Pólya recurrence · does a random walk come home?Next: push to d=4,5 (the return probability keeps falling — p_d~1/(2d) for large d) and measure the escape distance's growth; and contrast the simple walk with a self-avoiding walk, where the critical dimension shifts.
• Self-avoiding walk · the Flory exponent νNext: PERM (pruned-enriched Rosenbluth) to reach longer N without ESS collapse and pin ν to <1%; and the 4-D SAW, where ν→½ (the upper critical dimension — self-avoidance stops mattering).
• Maxwell–Boltzmann gas · the 2nd lawNext: measure the relaxation time τ vs density/collision rate (the approach to equilibrium is exponential), and the microcanonical O(1/N) departure of the single-particle marginal from the exact Gaussian.
• Sandpile · self-organized criticalityNext: a finite-size-scaling collapse P(s,L)=s^−τ G(s/L^D) to pull the cutoff exponent D and confirm the data-collapse, the avalanche AREA / duration / radius exponents (and their scaling relations), and the Manna variant for a clean single τ to contrast with BTW's multiscaling.
• Bak–Sneppen evolution · self-organized criticalityNext: measure the avalanche-size distribution P(s)∝s^−τ directly (τ≈1.07 for f_0-avalanches) and the spatial/temporal scaling, and contrast the nearest-neighbour ring (f_c≈0.667) with the random-neighbour mean-field version (f_c=⅔ exactly, no spatial avalanches).
• Fractal dimensionNext: a fractal selector (Barnsley fern, Koch, Julia), each with its own D, or box-counting a strange attractor.