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Recipe for Invariance under pmg

This is our first recipe for wallpaper functions: for invariance under pmg:

Use any linear combination of terms like cos(nX) cos(mY) and sin(nX) sin(mY).

Take care with the parity of m to ensure that it is even in the cosine terms and odd in the sines.

Using this recipe, we combined ingredients, experimenting until we found a picture we liked.

An image titled "Bats" that is invariant under the group pmg.

At this point, you may wish to experiment for yourself, combining terms according to this recipe and observe the pmg invariance that results .

The next steps are to generalize the formulas above to any lattice and give an example in what we call the hexagonal lattice .

Lattice Waves -- Translational Invariance Experiment interactively with Wallpaper Design Example in a Rectangular Lattice Constructing Wallpaper Functions Invariance under a Reflection