VW - CVM 1.1

# Wallpaper Drums

Let G be one of the 17 wallpaper groups. Imagine that an infinite, flexible membrane is deformed according to a pattern invariant under G and released from rest. Call u(x,y,t) its vertical displacement over the point x = (x,y) at the time t.

We model the progress of this membrane by the linear wave equation:

Du =
 ¶2u ¶x2
+
 ¶2u ¶y2
=
 ¶2u ¶t2

The left hand side is called the (spatial) Laplacian of u, a key quantity in our a nalysis.

This equation gives a good approximation, for small displacements, of the membrane's movement over time. Since the wave equation is itself invariant under isometries of the plane , and since the initial data is invariant under G, the membrane will have the same symmetry at every future instant. To make a well-posed problem in PDEs, however, we let the symmetry property play the role of a more traditional boundary condition, arriving at the following problem:

 ¶2u ¶x2
+
 ¶2u ¶y2
=
 ¶2u ¶t2
 u(x,0) = f(x) ut(x,0) = 0 u(gx,t) = u(x,t).

The standard technique of separation of variables yields an infinite series expansion of the solution to this problem, as follows:

The first standard step is to find all possible eigenfunctions of the Laplacian invariant under the group G. These form a basis for the space of all square-integrable wallpaper functions with group G, and each one has a characteristic way to vibrate into the future. To make a given initial pattern vibrate, we must decompose it into its eigenfunction expansion and advance each piece into the future according to its own frequency of vibration. If we return to our original concern, making static wallpaper designs, the part that we use is this: the most general wallpaper function is a superposition of multiples of the eigenfunctions of the Laplacian.

We begin with an example where the group is pmg , but you are welcome at any time to experiment on-line by combining eigenfunctions to make your own patterns .

Communications in Visual Mathematics, vol 1, no 1, August 1998.