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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 .