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Complex-valued Wallpaper Functions

Wallpaper function for cmm' with negating isometries that invert the color.

In the image at the right, the negating symmetry is portrayed in a special way: every color changes to its complementary color when you apply the isometry. We developed this way of picturing wallpaper functions when it became interesting to use complex-valued functions instead of real ones.

It has been said frequently that one cannot picture a complex-valued function on the plane without resorting to a 4-dimensional image. Our response to this has been to develop the concept of a domain coloring.

We think of the complex plane as being colored like a traditional color wheel: we put red at the complex number 1, with green and blue at the other two cube roots of unity as shown. Hues are interpolated between these positions, giving secondary and tertiary colors. A continuous blending would be possible, but here we show just twelve hues. Then we blend toward white at the center, toward black going outwards. Thus, each complex number has a color associated to it. We think of this as having colored the range of whatever complex-valued function we wish to picture.

To visualize a complex-valued function in the plane, color the plane as follows: for each pixel in the domain of the function compute the color associated with the result of the function for that input value and color that pixel with that color.

For more detail, and other applications of this idea, see our materials on complex function visualization developed in conjunction with a review [Fa] of Tristan Needham's interesting new book, Visual Complex Analysis .

When we use domain colorings of wallpaper functions with negating symmetries, we observe the phenomenon mentioned above: when negating symmetries are applied, intensities are preserved but every hue is replaced by its complement. Of course, round-off errors, along with the finite number of colors used, cause the various pixels colored white and black to look identical, when theoretically they may have subtly different colors assigned.

We can use this method to illustrate the effectiveness of the eigenfunctions of the Laplacian that we have used throughout this paper .

Completeness of the Eigenfunctions of the Laplacian Why there are 63 Types Ways to Visualize Wallpaper Functions Real-valued Wallpaper Functions