Fractal Drum

A drum vibrates at characteristic frequencies, depending mainly on the size, shape, tension, and composition of its sound-generating drumhead. This spectrum of frequencies—the set of pure tones produced by a vibrating membrane stretched across a frame—gives a drum its particular voice.

Physicists and mathematicians have long recognized that the shape of the boundary enclosing a membrane plays a crucial role in determining the membrane's spectrum of vibrations. So, what happens when a drum has an intricately indented rim—one so wiggly that the boundary consists of crinkles atop crinkles?

To find out, Michel L. Lapidus and his colleagues set out some years ago to study mathematically the vibrations of a drum with a fractal boundary—a fractal known as the Koch snowflake.

Starting with a large equilateral triangle, you can add smaller triangles to the middle of each side to create a six-pointed star. Adding even smaller triangles to each of the star's twelve sides generates a crinkly shape. Continuing the process by adding increasingly small triangles produces the intricately frilled Koch snowflake.

Idealized to its mathematical essence, a drum is a flat, two-dimensional surface held fixed along its rim. Only the interior moves, which greatly restricts the surface's possible motions.

The resulting vibrations, or normal modes, represent the solutions of a mathematical expression known as the wave equation. In the drum's case, the solutions specify the vertical displacement of each point on a surface bounded by a closed curve, such as a circle, rectangle, or fractal.

In the 1990s, working with Lapidus, Cheryl A. Griffith, Robert Renka, and John W. Neuberger of the University of North Texas sought to display normal-mode vibrations of a drum with a fractal boundary on the computer screen. They computed the first 40 eigenfunctions of the Koch snowflake. Their colorful images vividly showed the dramatically frilled edges of the waveforms created on such membranes.

Entranced by the beautiful forms, Neuberger commissioned mathematician and sculptor Helaman Ferguson to create a bronze rendering of one of the modes that the team studied. Ferguson called his creation Texas Snowflake 13^th Eigenfunction.

Helaman Ferguson's sculpture depicts the thirteenth eigenfunction, or normal mode of vibration, of the Koch snowflake. This mode has seven bumps and sixfold symmetry.

At Neuberger's request, Ferguson then created another sculpture, this time depicting the sixth eigenfunction of the Koch snowflake, with so-called Neumann boundary conditions. This mode has threefold symmetry.

Reference:

Peterson, I. 1994. Beating a fractal drum. Science News 146(Sept. 17):184-185.

Photos by I. Peterson