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Invariance under Half-Turns

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

cos(nX) cos(mY),   cos(nX) sin(mY),
sin(nX) cos(mY),   sin(nX) sin(mY),

where K and L are the lengths of the two translations and m and n are integers [Back] . 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

anm cos(nX) cos(mY)   or   bnm sin(nX) sin(mY)

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.


[Next] Invariance under a Reflection
[Skip] Lattice Waves -- Translational Invariance
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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: 12 Aug 1998 --- Last modified: Sep 30, 2003 9:26:41 AM
Comments to: CVM@maa.org