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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 [Back]. Early in our work with wallpaper we tried the function

cos(X) cos(0Y) + sin(2X) sin(3Y) + cos(3X) cos(2Y) + sin(4X) sin(Y)

which does indeed conform to the pmg recipe.

[example of negating wallpaper]

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 [Link], 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.
[ negating wallpaper 
with cell]
[Image]

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 [Image] 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.


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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: Sep 30, 2003 9:27:17 AM
Comments to: CVM@maa.org