OBSERVATION CHAMBER / ACTIVE
Spherical Harmonic Surface Observatory
Change degree, order, amplitude, and phase to watch a real spherical harmonic deform a closed surface, with sign colour and nodal lines kept visible.
ROUTE 65 / GEOMETRY / HARMONIC SURFACE
01 / SHAPE THE SURFACE
Real spherical harmonic on a closed surface
Drag to rotate · arrows rotate · Space play/pause
Observation contract H = Re(eiαYlm) deforms a unit sphere as r = 1 + A · H / max|H|. Colour keeps the sign; bright contours mark H = 0.
Canonical renderer: depth-sorted Canvas 2D. The same path is the fallback, with no external library or CDN.
03 / READ THE INVARIANTS
Surface readout
- Current harmonic
- Y₂¹
- Current phase
- 0.00 rad
- Symmetry cue
- C₁ / odd parity
- Expected nodes
- 1 latitude + 2 meridional
- Radius range
- 0.78 … 1.22
- Mesh
- 48 × 96 cells
The field is normalized for display. The eigenvalue remains λ = l(l + 1).
Advanced diagnostics and acceptance checks
Acceptance checks use the analytic harmonic, not the shaded pixels. Display uses m = 0…l; negative m is phase-equivalent.
Observation report
Formula note
H(θ,φ) = Re(eiαYlm(θ,φ)), with −ΔSYlm = l(l + 1)Ylm. The display uses r = 1 + A · H / max|H|.
OBSERVATION GUIDE
Touchpoints for observation
Read a real spherical harmonic as a closed surface: colour shows sign, radial distance shows amplitude, and bright contours mark zeros.
- Change l, m, amplitude, and phase. Use NODES and SYMMETRY, then PLAY or STEP to watch the m-fold phase rotation.
- For m=0, the pattern has l latitude node circles and phase has no effect.
- For m>0, the visible node structure has l−m latitude circles and 2m meridional branches.
This is a finite visual model of spherical harmonics, not a material shell, an acoustic measurement, or a full PDE solver.
Runs inside the browser with no upload and no registration.
OBSERVATION POLICY
This observatory runs a browser-local real spherical harmonic model. Parameters, calculations, reports, and images are not uploaded. It does not measure a visitor, a physical surface, sound, or material properties.
Route
This chamber belongs to Shape & Space.
GEOMETRYBack to Shape & Space Route.