VW - CVM 1.1

Experiment Interactively with Wallpaper Design

Here you are invited to experiment with wallpaper functions. You can select which type of lattice to use and then choose coefficients and frequencies for the building blocks of a wallpaper function in that lattice. The symmetry of the resulting wallpaper function depends both on the types of building blocks you combine and the parity of the frequencies.

For example, in the case of the rectangular lattice , choosing the sum n + m to be even will reduce the lattice to a rhombic one.

It's easy to create too much symmetry by accident. For instance, in a rectangular lattice, if every coefficient of x happens to be even, the cell will be twice as small as anticipated.

Instructions: First choose a lattice type; the default is rectangular. Then enter whatever values you choose for the coefficients and frequencies. The frequencies must be integers. Push the add term button to add a term; if you change your mind you can delete the most recent term and work your way backward through the list. We recommend that you add a few terms and modify them as desired before you press the button to redraw the wallpaper. You will find that pressing this button is a bit of a time investment. There currently is no way to save your designs, unless you use your own screen capture program.

If you haven't read the section on Negating isometries , you can experiment for yourself to see which combinations of building blocks result in functions invariant under the various wallpaper groups.

This might be a good time to dip into the tables of recipes, where there are also examples made with this applet. We recommend starting with the rectangular lattice , but the applet is also useful for the square and hexagonal lattices.

Negating Isometries Constructing Wallpaper Functions Eigenfunctions with 3-Fold Rotations