VW - CVM 1.1

A Motivating Example

In order to construct a wallpaper function with pmg symmetry, our recipe instructs us to combine terms like cos(nX)cos(mY) and sin(nX)sin(mY) with certain parity restrictions . Early in our work with wallpaper we tried the function

which does indeed conform to the pmg recipe.

One easily checks that this does have the necessary translations, half-turns, reflections, and glide reflections to qualify as a wallpaper function with group pmg , but surely the most striking thing about the picture is the presence of anti-symmetries.

The most noticeable anti-symmetries may be the negating vertical mirrors. In addition to these, a thorough search reveals negating half-turns, glide reflections, and negating translations halfway along the diagonal of the cell.

The cell diagram for the pmg wallpaper shown above. Negating isometries are shown in green.

These particular translations, which we will call diagonal half-translations, are key ingredients in the recipe for this type of wallpaper function.

In this example, by accidentally choosing the sum m + n to be odd in every term, we forced a negating diagonal half-translation. This is the source of our remark above that sometimes care is required to prevent anti-symmetries.

A three-dimensional view of the graph of this function illustrates that the blue mountaintops are exact inversions of the fiery pits. The mirror lines are clearly visible as straight lines running toward a vanishing point in the distance.

Classification and Naming The Algebra of Wallpapers Negating Isometries