An interactive demo

Hearing the shape of a drum

After Kac, 1966.
Page created by Mihkel Kree using Claude in May 2026; solver and text revised July 2026. More projects at llmphysics.com.

In 1966, Mark Kac asked: can one hear the shape of a drum? That is, can two membranes of different shapes ever produce exactly the same spectrum of natural frequencies? Sketch any drum shape below: the page computes its frequency spectrum and its vibration modes — Chladni figures for membranes — and the ♪ Hear it button strikes your drum, synthesizing its sound from the computed spectrum. A detailed walkthrough for undergrad physics students follows after the demo.

Draw a drum shape

The two GWW shapes are Gordon–Webb–Wolpert's 1992 answer to Kac's question: different shapes, identical spectrum. Load one, then the other — amber ghost bars linger so you can watch the spectra coincide.
Draw a blob or pick a preset — results update when you lift the pen.

Frequency spectrum

One numbered line per mode at x ∝ √λn, with the drum rescaled so its bounding box's longer side is 1 (a square drawn at any size lands on the textbook λ1 = 2π² ≈ 19.7). Hover a bar to read λ and highlight its mode shape below; after each recompute, amber ghost bars show the previous spectrum for comparison.

Mode shapes

Blue = negative, white = zero, red = positive. The curves where the colour crosses white are the nodal lines — the drum analogue of Chladni patterns, the curves where sand would settle. Hover a tile to highlight its line in the spectrum above.
Brush sets the pen width; Eraser and Undo (Ctrl+Z) fix mistakes. Grid N: raster resolution — higher is more accurate but slower (the eigensolver itself is exact to a verified residual, so all remaining error is discretization). Modes: how many eigenvalues are computed and plotted.
A word of caution

The following physics explainer was written by Claude Opus 4.7, a large language model by Anthropic, and later reviewed, corrected, and updated for the current solver by Claude Fable 5, with the numerics validated against closed-form spectra. It has not been reviewed by a human physics expert. Treat it as an informed starting point, not as an authoritative reference.

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