The spectrum of the surface.
Resonance uses the same midsurface, cotangent stiffness matrix K, lumped mass matrix M, and reflecting rims as Waves. It solves the generalized eigenproblem for twelve nonuniform modes:
K φₙ = λₙ M φₙ
fₙ = √λₙ / (2π) (wave speed c = 1)
u(t) = (1−b) φₐ cos(√λₐ t)
+ b φᵦ cos(√λᵦ t + δ)The uniform zero-frequency mode is omitted. Modes are ordered by increasing frequency and normalized by surface area (φᵀMφ = 1). A common color scale preserves relative mode amplitudes; a new shape sets a new display scale. Dark nodal lines stay fixed for a single mode; a mixture’s instantaneous zero contours can move. Opposite phase does not guarantee cancellation between different spatial modes.
Frequencies use model units and simulation seconds, with speed fixed at one. They are not audible pitches or measured properties of bronze. Near-equal frequencies can produce slow beating; the listed beat period is 1/|fₐ−fᵦ|. Animation plays at one simulation second per active second, with long frame gaps capped. Freeze holds the exact displayed time.
A shifted banded Cholesky solve with subspace iteration and Rayleigh–Ritz extraction computes the modes on 1,664 nodes. The mass-normalized eigenproblem residual must be below 10⁻⁶ before a result is shown. Maximum relative mode residual: —. The eigenvectors are interpolated for display; the dark contours are visual approximations. Mode order may swap and signs or bases may change near repeated eigenvalues when the form changes. This is a discrete scalar membrane model, not a shell elasticity or sound simulation.
Waves live on the geometry.
A scalar field travels on the middle surface of the sculpture. Gold and blue show positive and negative displacement; the color scale stays fixed as waves decay. Both faces show the same membrane field. This is not a model of bronze elasticity or sound.
M ü + 2γ M u̇ + c² K u = 0
E = ½ u̇ᵀ M u̇ + ½ c² uᵀ K u
A cotangent finite-element stiffness matrix K and lumped area matrix M are assembled from 1,664 vertices and 3,200 triangles. The rims use the natural zero-flux boundary condition, so waves reflect. A click launches a Gaussian velocity impulse using shortest paths along mesh edges, with its area-weighted mean removed.
Implicit midpoint steps of 1/120 simulation second conserve the discrete quadratic energy without damping, up to solver error. With damping, the step's lost energy is explicitly accounted for. The simulation runs separately from drawing; its clock shows simulated time, not a claim of real-time performance. A study pauses at 30 seconds.
Maximum relative solve residual —Relative energy-accounting error —
Shape changes rebuild the operator and restart the most recent strike pattern at time zero. Thickness only affects the displayed shell. Bands only affect its markings. Additional clicks add to the evolving field; Replay starts all retained sources together. The coarse field is interpolated onto the display mesh. Wave fronts and edge-graph distances have discretization error; energy conservation alone does not establish spatial accuracy.
Operator construction follows Keenan Crane's discrete differential geometry notes, sections 6.2 and 6.5. The time integration and checks are implemented here.
From a lattice to a complete surface.
The folded support of the supplied sculpture is the starting surface. It now has two opaque faces, a continuous inner rim, and a continuous outer rim. The photographed surfaces and separate lattice tiles are replaced by this complete shell.
26,112vertices
52,224triangles
χ = 0Euler characteristic
Let u travel around the opening and v travel from the inner rim to the outer rim. The source supplies sampled coordinates S₀(u,v). The controls apply:
r′ = r + (a − 1) rᵢₙ(u) (1 − v)²
θ′ = θ + τv
z′ = fz
S± = S′ ± ½t n̂
Here a is opening scale, τ is twist in radians, f is fold amplitude, and t is shell thickness in model units. The two offset surfaces join at both rims. Their mesh connectivity is closed and has genus one. This does not certify an intersection-free or fabrication-ready solid at every setting.
The bands mark equal steps in the radial parameter v; they are not equal-height contours or equally spaced geodesics. Fifty is the initial band count, not a claim that this version runs a cellular automaton. The original geometry was supplied in Fifty iterations of a self, credited there to Russell Foltz-Smith.