This little illustration is explained thusly:
Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the eigenfrequencies are all equal, so the timbral spectra would contain the same overtones. This example was constructed by Gordon, Webb and Wolpert. Notice that both polygons have the same area and perimeter.And then I wandered over to an article about the vibrations of a circular membrane (a drum in other words) and saw this very weird thing:
That, by the way, is one way that a circular drumhead can vibrate. Maybe if you hit it really hard in just the right spot? That's known as Mode (5d) with .
And that led me to things that vibrating strings do. Here are the first five overtones shown as standing waves on a string:
And then my head started to hurt and I had to lay down...