Ernst Chladni (1787): bow a metal plate along its edge while holding a nodal point fixed. Sand migrates to nodal lines — the zeros of the plate's eigenmode. The pattern is governed by the Kirchhoff–Love biharmonic eigenvalue problem:
Each eigenvalue λnm corresponds to a distinct nodal geometry. The plate selects the mode whose frequency matches the driving — all other modes destructively cancel. The zero-set {unm = 0} is the visible Chladni figure.
The triskelion mode (bottom-right) appears at intermediate
plate geometries and is the 2D analogue of the dm³ three-fold orbit.
See ch-triskelion.html in AXLE/LAW3M/.
The physical content of every Chladni experiment is a selection principle: of all possible oscillatory states, only those satisfying Δ²u = λ⁴u are sustained. Non-eigenmodes are dissipated. The system locks to the zero-set of its resonant eigenfunction.
The Chladni figure is the attractor of the driven plate under dissipation. It is a zero-dimensional attractor (a curve) of the wave equation, not of a flow. The dm³ attractor Γ = {r = 1} is its dynamical counterpart.
The dm³ g-series exponentiates the operator chain: gn indexes the n-th application of G = U∘F∘K∘C. Each rung is a physical scale. Resonant locking occurs at every rung — the mechanism is identical, only the medium changes.
★ Saturn's north polar hexagon sits at rung g⁶⁴ (planetary scale ≈ 10⁷ m, rotation period ≈ 10·7 h). See §4.
Saturn's north polar hexagon is a stationary Rossby wave of azimuthal wavenumber m = 6, frequency-locked to the planet's internal rotation. It has persisted for at least 40 years of continuous observation (Voyager 1980, Cassini 2004–2017, JWST 2023). Its geometry is:
The hexagon is a Rossby wave locked at mode m = 6. The linearised shallow-water equations on a rotating sphere admit wave solutions ψ ∝ ei(mθ − ωt); the m = 6 mode phase-locks to zero in Saturn's co-rotating frame, producing a stationary geometric pattern — a planetary Chladni figure.
The dm³ ODE on the contact manifold (ℝ³, α = dz − r²dθ):
admits a unique helical attractor Γ = {r = 1}. Theorem 2.1 (proved; see gravity scales preprint §3): every orbit with r₀ > r★ ≈ 0.77594059 converges exponentially to Γ at rate μ = −2.
Chladni: {unm = 0} — nodal curve on plate.
Saturn: {ψ₆ = const} — hexagonal isoline in the jet stream.
dm³: Γ = {r = 1} — Reeb orbit of the contact structure.
All three are zeros of a differential operator acting on a resonant eigenfunction, stabilised by a dissipative / Lyapunov trapping mechanism.
The dm³ framework names six phases of attractor formation as the Cajueiro cycle (seed → overshoot → resistance → lock → branch → new seed). The same six phases appear in each resonant system:
| Phase | Chladni (g⁶) | Saturn (g⁶⁴) | dm³ |
|---|---|---|---|
| Seed | initial strike / bow | convective perturbation | r(0) ≠ 1 |
| Overshoot | transient multi-mode ring | jet meanders chaotically | trajectory spirals in |
| Resistance | dissipation kills off-mode | β-plane selects m=6 | ε₀ = 1/3 Lyapunov bound |
| Lock | sand settles on nodal line | hexagon stationary | r(t) → 1, rate e^{−2t} |
| Branch | higher harmonics persist | nested vortices form | g-series next rung |
| New seed | re-bow at new frequency | seasonal forcing | g^n → g^{n+1} |
Conjecture (Cymatics Universality). Any dissipative system governed by a self-adjoint elliptic operator on a domain with a preferred symmetry group will exhibit Cajueiro-cycle dynamics converging to a resonant attractor in the zero-set of its principal eigenfunction.
The triskelion — three interlocked spirals meeting at a common centre — appears as a Chladni mode on triangular plates and as an intermediate orbit in dm³ simulations near the saddle point (rs, zs). Its three-fold symmetry corresponds to:
The tribonacci constant η ≈ 1.839 (operator η in the dm³ ladder)
governs the three-fold recurrence and appears as the growth rate of
the saddle manifold near (rs, zs). See chEta-tribonacci.html.
Is there a contact-geometric proof that the Rossby wave locking condition ∂tq = 0 at m = 6 can be derived as a dm³ resonance condition on a suitable contact 3-manifold modelling the rotating shallow-water equations?
Construct an explicit map from Chladni eigendata (λnm, unm) on a disk D² to dm³ initial data (r₀, z₀) such that the limit cycle Γ corresponds to the nodal circle of the (0,1) mode. The map should intertwine the Kirchhoff–Love bilaplacian with the dm³ contact Laplacian Δα.
Note on det(J). The saddle Jacobian eigenvalue satisfies det(J)|rₛ = rs²(−1−rs−2rs²) ≈ −0.364 where rs = 2cos(3π/7) ≈ 0.445 is the saddle coordinate. This is distinct from ε₀ = 1/3, which is the Lyapunov stability radius from V = (r−1)²/2. See TOGT Theorem B.3 and gravity scales preprint §4.
Chladni (1787) Entdeckungen über die Theorie des Klanges · Baines & Szmytkowski (2007) Saturn hexagon dynamics · Sayanagi et al. (2010) Rossby wave m = 6 locking · Grossi (2026) Contact-Geometric Theory of Generative Transitions doi:10.5281/zenodo.20682934 · Grossi (2026) Gravity Across Scales doi:10.5281/zenodo.20747481 · AXLE/Lean 4: github.com/TOTOGT/AXLE