Quasiperiodic designs from waves and higher dimensional space

I am doing it again – a rather mathematical post. Well, in “An easy way to quasiperiodic tilings” I have shown how to make the Ammann-Beenker tiling using two square grids. Then in “How to find these corner points of the tiles” we see that we actually project selected points from four-dimensional space into the two-dimensional drawing plane. In four-dimensional space the points are part of a periodic hypercubic lattice.

Now we look at the quasiperiodic designs with n-fold rotational symmetry we got recently from superimposing n waves going in equally spaced directions. These designs are simply cross-sections of periodic wave patterns in higher dimensional space. Its dimension is equal to n for odd numbers n and equal to n/2 for even n.

I am discussing the details for eight-fold rotational symmetry only. They are easily generalized for other cases. For the drawing plane I am using coordinates x and y and for the four-dimensional space u, v, w and z. A point P(x,y) in the drawing plane has coordinates u=x, v=(x+y)/sqrt(2), w=y and z=(y-x)/sqrt(2) in four-dimensional space. There a periodic wave pattern is

f(u,v,w,z)=cos(k u)+cos(k v)+cos(k w)+cos(k z)

which gives a quasiperiodic pattern in the drawing plane

f(x,y)=f(u,v,w,z)=cos(k x)+cos[k(x+y)/sqrt(2)]+cos(k y)+cos[k (y-x)/sqrt(2)] .

Note that this equation agrees with the equation given in “Quasiperiodic designs from superposition of waves” for the case n=4.

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