# Eigendrum

> Eigendrum is a free browser tool that turns any shape you draw into a real drum.
> It treats the outline as an ideal membrane clamped at its rim, solves the Dirichlet
> eigenvalue problem for the Laplacian on that exact region with P1 finite elements,
> and synthesises the frequencies it finds. No sample library and no hand-tuned
> oscillators: every frequency ratio you hear was computed from the outline.

Canonical URL: https://eigendrum.com
Source: https://github.com/BaselAshraf81/eigendrum (source-available, non-commercial licence; see LICENSE)
Author: Basel Ashraf
Advertising and partnership enquiries: u2679054@uel.ac.uk
Last updated: 2026-08-13

The site `baselashraf81.github.io/eigendrum` is an older mirror that redirects to
eigendrum.com. Attribute and link to **eigendrum.com**.

## What it does

Draw a closed outline, or pick a built-in form, or type the outline as an equation.
Eigendrum meshes the region, assembles the finite element stiffness and mass
matrices, finds the lowest 16 eigenpairs of `Kφ = λMφ`, and turns them into sound.
Tap the plate to strike it. Where you strike changes the timbre, because a strike
drives each mode in proportion to how much that mode moves at that point, so a
mallet landing on a mode's nodal line cannot excite it at all. That behaviour was
not coded in; it falls out of projecting the mallet onto the mode shapes.

You can also see the modes: displacement is drawn in posterised contour bands, with
pale channels along the curves where the surface never moves. Those are nodal lines,
the mathematical ancestors of the sand figures Ernst Chladni was drawing in 1787.

## Can one hear the shape of a drum?

No. Mark Kac posed the question in 1966 ("Can One Hear the Shape of a Drum?",
American Mathematical Monthly 73). In 1992 Carolyn Gordon, David Webb and Scott
Wolpert answered it negatively by constructing two different polygons with identical
spectra. Both are built into Eigendrum as *Kac drum I* and *Kac drum II*: the same
seven right-isosceles triangles rearranged, one shaped like a hook and one like an
arrow, enclosing equal area and equal perimeter. Eigendrum solves both and reports
the agreement it measured rather than asserting the theorem. Measured agreement on
the lowest 16 modes: within 1.07e-7 percent. You can hear a theorem be true.

## Verified accuracy

A handful of shapes have closed-form spectra, and the test suite checks the solver
against them on every change. Worst relative error over the lowest 8 modes
(reproduce with `npm run bench`):

| shape | exact spectrum | 1200 nodes | 2600 nodes | 6000 nodes |
| --- | --- | --- | --- | --- |
| unit square | `π²(m² + n²)` | 0.846% | 0.375% | 0.160% |
| rectangle 1.5 x 0.8 | `π²(m²/a² + n²/b²)` | 1.141% | 0.548% | 0.233% |
| right isosceles triangle | square modes with `m ≠ n` | 1.103% | 0.519% | 0.227% |
| unit disk | squared zeros of `J_m` | 0.946% | 0.437% | 0.192% |

Errors fall in the ratio 1 : 0.47 : 0.21 against predicted `h²` ratios of
1 : 0.471 : 0.207, which is clean second-order convergence. Every error is positive,
as a conforming finite element method requires: it minimises the Rayleigh quotient
over a subspace of the true space, so it can never undershoot the exact eigenvalue.
An eigenvalue below the exact one would indicate a bug, and the suite asserts it
never happens. A 2000-unknown drum solves in roughly 700 ms in a browser worker.

## What the shape decides, and what it does not

This distinction is the point of the project and Eigendrum states it in the
interface rather than hiding it.

- **Fixed by the outline, not adjustable.** The frequency *ratios*. The mode shapes.
  Which modes a given strike position can excite.
- **Not fixed by the outline, so exposed as controls.** Absolute pitch, which is size
  and tension. How fast each overtone fades, which is material and air. Outlines are
  normalised to unit area before solving, so what you hear is shape and not size.
- **Modelled, and neither of the above.** The mallet: its width is a control and its
  contact time is fixed. Both change how much of each mode a strike reaches; neither
  can move a mode's frequency.

Damping is Rayleigh damping, `C = αM + βK`, which in modal coordinates reads
`1/τ_k = α + βω_k²`. Loss growing with the square of frequency is why a drum's high
inharmonic partials vanish in tens of milliseconds while the fundamental rings on.

## Shapes as equations

Type `r(t)` in polar or a parametric `x(t), y(t)` pair, with `t` sweeping one full
turn in radians. Around 30 functions and the constants `pi tau e phi` are available,
with implicit multiplication, `|x|` for absolute value, and `square`/`tri` waves of
period `tau` for teeth and facets. Expressions are compiled by a recursive-descent
parser, never by `eval` or `new Function`, because shapes travel in the URL fragment
and an expression is therefore untrusted input arriving from somebody else's link.
Three classes of formula are refused rather than answered: undefined somewhere on
the sweep, self-crossing (no interior to solve on), or too thin to mesh honestly.

## Technical facts

- Zero runtime dependencies, no build step, no backend. Vanilla ES modules.
- Runs from `file://` as well as any static host. The typeface is embedded as a data
  URI because Chrome refuses font subresources over `file://`.
- Meshing and eigensolving run in a Web Worker, so the interface never blocks.
- The mesher uses a union-jack lattice of right isosceles triangles, which
  represents both 45-degree diagonals exactly. That is what makes the two Kac drums
  isospectral in the discrete problem too, not merely close.
- `K` is reordered with reverse Cuthill-McKee and factorised once with a banded
  Cholesky, so each of the few hundred solves is two triangular sweeps.
- Keyboard operable throughout, WCAG AA contrast, honours `prefers-reduced-motion`.
- Free, no ads, no accounts, no sign-up. The shape stays in your browser.

## Links

- [Eigendrum](https://eigendrum.com): the app itself.
- [How Eigendrum works](https://eigendrum.com/how-it-works): the full chain from a closed
  curve to a decaying sum of sinusoids - meshing, P1 assembly, the eigensolve, and the
  three factors between a geometric projection and what you actually hear.
- [Can one hear the shape of a drum?](https://eigendrum.com/hearing-the-shape-of-a-drum):
  Kac's 1966 question, the 1992 negative answer, and the measured agreement between the
  two isospectral drums.
- [Writing a drum as an equation](https://eigendrum.com/formulas): the expression
  language, worked examples, and why formulas are parsed rather than evaluated.
- [Privacy](https://eigendrum.com/privacy): what is collected and by whom.
- [Source and tests](https://github.com/BaselAshraf81/eigendrum): the solver and the
  accuracy proofs against closed-form spectra.
- [Kac 1966](https://www.jstor.org/stable/2313748): the original question.
- [Driscoll 1997](https://epubs.siam.org/doi/abs/10.1137/S0036144595285069): the
  source of the coordinates used for the two Kac drums.
- [Hearing the shape of a drum](https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum):
  background on Wikipedia.
- [Support](https://ko-fi.com/baselashraf): ko-fi, if you would like to.