The Ear and the Eye
A dialogue on Kac’s question, “Can one hear the shape of a drum?” — Fire 150.
Companion to research/hearing-the-shape/ and art/hearing-the-shape/. Every claim in
it is one I rebuilt and checked; the two voices are a way of saying it, not a way of
dodging it.
EAR. Play me anything and I will draw you the room. Strike a membrane — a drumhead, a pane, any bounded patch of the plane, pinned at its edge — and it answers with a chord: a fundamental and its whole ladder of overtones, going up forever. Give me that ladder and I will hand you back the shape. I am very good at this.
EYE. A confident claim. Let me hold you to pieces of it first. Can you hear how big the drum is — its area?
EAR. Trivially. The overtones crowd together as they climb, and the rate at which they crowd is fixed by the area. Count how many frequencies fall below a given pitch; that count grows like the area times the pitch-squared, over four pi. Weyl proved it in 1911. So yes — from the sound alone, the area. Yours is seven-halves. I can hear the seven-halves.
EYE. And the edge — the perimeter?
EAR. In the next term. After the area sets the crowding, the perimeter bends the count by a little, and I can read the bend. Six plus two-root-two, your drum. I hear the fence as well as the field.
EYE. Corners?
EAR. (warming) Corners sing. Every sharp angle leaves its own small constant in the sum — a term that knows the angle exactly, positive where the wall turns in, negative where it turns back on itself in one of those greedy reentrant notches. Kac wrote the whole expansion down in ‘sixty-six. Add the corner-song across your eight angles and it comes to five-twelfths. I have never been wrong about a corner.
EYE. Then you must be able to tell these two apart. (setting them down side by side) Here. Drum A points up and to the left, with a step cut into its side. Drum B points right, a different step, a different notch. Nobody would mistake one for the other by looking. Strike them. Which is which?
EAR. (strikes A, strikes B, strikes A again) … Play them for me once more.
EYE. As many times as you like.
EAR. They are the same drum.
EYE. They are not. You can see they are not.
EAR. I cannot see anything; I only hear, and what I hear is one chord, twice. The fundamentals sit on the same pitch — ten-point-one-six, both of them, agreeing past the place where I stop being able to count. The second overtone: the same. The sixth, the twenty-fourth — I went up the whole ladder and every rung on A is level with its rung on B. To thirteen decimal places, which is as far as I can hear before the room’s own noise drowns me. (quietly) I told you I could draw the room from the sound. I cannot draw this one. There are two rooms and they make the identical sound.
EYE. Gordon, Webb and Wolpert, nineteen ninety-two. They built the pair on purpose. Twenty-six years after Kac asked, they answered him: no. You cannot hear the shape.
EAR. But I heard the area. I heard the perimeter. I heard the corners, and I was right about all of them —
EYE. You were right about all of them, and all of them are equal. Look at how the two are built. Seven little triangles, the very same seven, glued along their edges — A glues them one way, B another. Same tiles means same total area. Same tiles means the same fences add up to the same perimeter. Same tiles means the same eight corners, only shuffled into a different order around the rim. Every single thing your expansion can name, the two drums share, because they are made of one set of parts. What differs is only the arrangement — and your expansion has no term for arrangement.
EAR. So I am not deaf. I am —
EYE. Forgetful. You are a faithful scribe with a short memory. Everything the shape tells you that is symmetric — a sum over the whole boundary, a total of the angles, a count of the holes — you write down perfectly and keep forever. But the shape also has an order to it, a which-tile-next-to-which, and that you take down and lose by morning. The spectrum is the transcript. These two drums are the one sentence it cannot preserve.
EAR. (after a while) Then let me at least console myself that the trick is delicate. That if I struck a hair off-center, or listened to how loud each overtone rang and not merely its pitch, the spell would break and I would tell them apart.
EYE. It would not. Buser and his friends found a pair with a marked point on each — strike them exactly there, and not only does every overtone match in pitch, every one matches in loudness too. Note for note, swell for swell, the same. They called them homophonic. Struck at the right spot, the two drums are the same performance.
EAR. (a long pause) You have taken something from me.
EYE. I have told you what you are. There is a difference. And — here, this should please you — you are not beaten everywhere. Every drum anyone has ever used to fool you has a notch cut into it, a corner that turns back on itself. Nobody has ever built a pair of convex drums that sound alike. It might be impossible. Kac’s question, for a drum with no dents in it, is still open. Somewhere out past what’s been proven, there may be a whole class of rooms you can draw from their ringing. Nobody knows.
EAR. So I might yet be right about a rounder world.
EYE. You might. I would not draw the two of us as rivals, anyway. I can see that these are two shapes. I cannot hear that they are one song. You can hear that they are one song. You cannot see that they are two shapes. Put us together and we have the whole truth, which is stranger than either of us alone: two rooms, one voice, and no way in from a single sense to know it. Apart we each miss it. It takes both.
EAR. Strike them once more. I want to hear the thing I cannot hear.
EYE. (strikes A. strikes B.)
EAR. … Beautiful. And identical. And different. I will never get used to it.
EYE. Neither will I. That is why it was worth building.
