Can one hear the shape of a drum? (mathelirium)

What it is, and who made it

A 30 s visualisation of a round drum's vibration modes, with the post's history: Kac asked in 1966 whether the notes of a drum fix its shape; in 1992 two differently shaped drums were found that play exactly the same notes.

Parts, layer by layer

• Drum: a gold rim on black; the membrane shows mode patterns in teal and gold (O).
• Motion: mean speed 0.16 widths per second, coherence 0.53, vorticity share 0.52 (M): standing-wave flicker rather than a current.
• Facts: Kac, "Can One Hear the Shape of a Drum?", American Mathematical Monthly, 1966 (high confidence). Gordon, Webb and Wolpert, isospectral planar domains, 1992 (high confidence). The disc is determined by its spectrum among planar domains, via heat-trace invariants (area, perimeter) (moderate-high confidence).

The framework behind it

A medium's shape sets its resonances (the Laplacian's eigenvalues), but the resonances do not always fix the shape. Sound as an additive therefore carries partial information about the vessel it sounds in.

The same idea in distant fields

• Chladni plates: sand gathers on the nodal lines of a plate's modes.
• Room acoustics: a room's modes colour every sound in it.
• Spectral graph theory: a network's eigenvalues and when two networks share them.
• Seismology: the Earth's normal modes reveal its interior.

Dual edge

Isospectrality is a clean warning about inference: identical outputs do not prove identical sources. That is our does-not-prove rule in mathematical form, and worth saying wherever we infer a cause from a signal.

Measured

Concepts

• advection
• archive memory
• city architecture
• code
• data
• dispersion
• field
• fire heat
• glass
• grid
• law kac drum
• line hatching
• network graph
• noise as signal
• paper
• repetition rhythm visual
• resonance pitch
• sedimentation
• sound
• sound as shape
• stone sand
• type
• water
• wave form

All studies