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Invariance under Half-Turns

We have found that the eigenfunctions of the Laplacian invariant under translation are

where K and L are the lengths of the two translations and m and n are integers . But we have not yet accomplished invariance under every element of pmg. This group also includes half-turns.

Functions invariant under half-turns through the origin must satisfy the equation f(-x,-y) = f(x,y). Clearly, only terms of the first and fourth types above qualify.

At this point we pause to note that any infinite sum of terms of the form

is formally a wallpaper function with group at least as big as p2. If the series converges pointwise everywhere, we have an actual wallpaper function periodic relative to a rectangular lattice, invariant under half-turns.

Invariance under a Reflection Lattice Waves -- Translational Invariance Example in a Rectangular Lattice Constructing Wallpaper Functions Invariance under Translations