Hear

what this is

In 1966 Mark Kac asked whether the shape of a drum can be recovered from its sound alone, meaning from the full list of frequencies at which its membrane can ring. For twenty six years nobody knew. In 1992 Gordon, Webb and Wolpert answered no by producing the two shapes above: seven copies of one right triangle, stitched together two different ways, whose vibration frequencies agree not approximately but exactly, every one of them, forever. The two drums on this page are that pair. Their spectra are computed here, in your tab, from nothing but the shapes: the Laplacian on each region is assembled and its lowest eigenvalues and eigenmodes are solved for while the page says it is tuning. The fundamentals printed under the canvas agree to eleven or twelve digits, and the tick marks on the shared axis land on one another. The disk is the control: it has exactly the same area, so no simple measurement separates it, and yet its marks fall elsewhere. Area is audible in how the frequencies crowd together on average; the shape hides deeper, and for these two shapes it hides completely.

A strike is honest physics end to end. The mallet's blow is a small round dent projected onto the drum's true modes, so where you strike decides how much of each mode wakes: strike a point where some mode is still, on one of its nodal lines, and that mode stays silent, which is why the same drum sounds different in different places, and why a timpanist strikes near the edge rather than the middle. What you see is the sum of the woken modes vibrating at a heartbeat's pace; what you hear is the identical sum at true relative scale, each mode a decaying sine at a frequency proportional to the square root of its eigenvalue. The partials of a drum are not multiples of the fundamental the way a string's are; the second sits at about 1.59 times the first, not 2, and that crowded inharmonic stack is why a drum thuds where a string sings. Two things here are chosen by ear and said so: the single constant that maps the spectrum into audible range, and how long each mode rings. Everything else is the eigenvalue problem.

the vow, kept

Six gates were fixed before any code existed, with exact reference formulas and tolerances that do not move; the whole board held on the first blind run, with no amendments. The full contract and readings are in the day's gate note; the board itself runs headless against the same module this page uses.

• 1 · the solver against a closed form
  Held: on a square, where the discrete eigenvalues are known exactly at every grid size, the solver reproduced all twelve to one part in ten billion, and its error against the continuum shrank fourfold when the grid halved, the textbook rate to three digits.
• 2 · isospectrality
  Held: the two drums' first twelve eigenvalues agree to about one part in a trillion at every grid size tried, on two independent seeds. The 1992 theorem survives discretization intact: the fundamental read 10.1703715079 on both drums, every printed digit shared.
• 3 · weyl's law
  Held: counting forty modes, the eigenvalues crowd at the rate area over four pi, measured within one percent on both drums, and the two fits came out identical because the lists are. The area is in the sound; that is precisely the part Kac could keep.
• 4 · the animation is the wave equation
  Held: the modal sum this page animates was checked against a direct integration of the wave equation on the same grid; they agree to three parts in a thousand after a full second, and the residual is the integrator's own second order error, halving fourfold with the step.
• 5 · courant's bound
  Held: mode number k never breaks into more than k nodal regions, on either drum, and the fundamental holds one sign everywhere, as the theorem requires.
• 6 · the disk and the bessel zeros
  Held: the control drum's fundamental matched the Bessel zero prediction within two percent at the base grid, converging as the grid refined, and the famous inharmonic ratio between its first two notes read 1.5929 against an exact 1.5933, computed from quadrature in the gate script itself.

the picture

This piece follows Ask, where the same operator answered a question posed on a boundary; Hear strikes the operator itself and listens. It also returns to Chladni from months ago, which drove a plate with a formula written by hand; here nothing is written by hand, the shapes are the only input and the modes are earned. Of all drums with this area, the disk is the deepest, which is a theorem too. The one thing the page cannot do is tell the two angular drums apart, and that is the point: some questions are unanswerable not for lack of listening. Monochrome, procedural, no paid instruments.