What you are seeing
The square plate
u(x,y) = cos(nπx)·cos(mπy) − cos(mπx)·cos(nπy)Theorem
Proof
The circular plate, and Saturn
u(r,θ) = Jk(αk,s·r) · cos(kθ)Theorem
Proof
/-- Standing wave pattern of Saturn's polar hexagon. f(theta) = cos(6 * theta). Nodal lines {f = 0} form the six sides of the hexagon. -/ noncomputable def chladni6 (theta : ℝ) : ℝ := cos (6 * theta) /-- Invariant under rotation by π/3: six-fold symmetry. -/ theorem chladni6_sixfold_sym (theta : ℝ) : chladni6 (theta + π / 3) = chladni6 theta := by unfold chladni6 have h : 6 * (theta + π / 3) = 6 * theta + 2 * π := by ring rw [h, cos_add_two_pi] /-- Each nodal angle is a zero of chladni6. -/ theorem hexagon_nodes_are_zeros (i : Fin 6) : chladni6 (hexagon_nodal_angles i) = 0 := by unfold chladni6 hexagon_nodal_angles push_cast have h : 6 * ((i.val : ℝ) * (π / 6) + π / 12) = i.val * π + π / 2 := by ring rw [h, cos_add, cos_pi_div_two, sin_pi_div_two] simp [sin_int_mul_pi]
The globe
Ylm(θ,φ) = Plm(cos θ) · cos(mφ) | sin(mφ)Theorem
Proof
The hypersphere
Φ = cosp(η) · sinq(η) · cos(p·ξ₁ + q·ξ₂)Theorem
Proof
Sonic Starquakes — the missing math
ωℓ ∝ √((ℓ−1)(ℓ+2))| Observed | Mode | Predicted | Error |
|---|---|---|---|
| 18 Hz | ℓ = 2 | 18.0 Hz | anchor |
| 30 Hz | ℓ = 3 | 28.5 Hz | 5.1 % |
| 93 Hz | ℓ = 10 | 93.5 Hz | 0.6 % |
| 150 Hz | ℓ = 16 | 147.9 Hz | 1.4 % |
What is and is not settled
Is the mathematics proved?
| Statement | Status |
|---|---|
| Plate gradient, analytic against numerical | 9 × 10⁻¹⁰ |
| Φ harmonic in ℝ⁴, five (p,q) pairs | 4 × 10⁻⁷ |
| Hopf form against polynomial form | 6 × 10⁻¹⁶ |
| Spherical harmonic eigenvalue −l(l+1) | 5 × 10⁻⁷ |
| Six-fold symmetry, hexagon nodes are zeros | Lean ✓ |
One honest caveat