Periodic and quasiperiodic images from cross-sections of 4-dimensional space

In the earlier post “Quasiperiodic designs from waves and higher dimensional space” I have shown that the quasiperiodic wave pattern with 8-fold rotational symmetry is a special cross-section of a periodic pattern in  4-dimensional space. Here I will rotate the drawing plane in this space to make a transition from a periodic image with 4-fold rotational symmetry to a quasiperiodic image with 8-fold rotational symmetry.

Just think of the short novel “Flatland” by Edwin A. Abbott. We are a citizen of flatland and we can only see the two-dimensional drawing plane. Around us we have a periodic four-dimensional object, which rotates. What do we actually see ?

Using the coordinates (u,v,w,z) of the object we define the wave pattern by

f(u,v,w,z)=sin(u)+sin(v)+sin(w)+sin(z) .

If the object is rotated by an angle α then the coordinates x and y of the drawing plane correspond to

u=x cos α , v=(x+y) sin α /√2 , w=y cos α and z=(y-x) sin α /√2 .

Note that if the angle α=0 we get a simple periodic image as

f(x,y)=sin(x)+sin(y).

For an angle of 45 degrees we see a quasiperiodic image

f(x,y)=sin(x/√2)+sin(y/√2)+sin[(x+y)/2]+sin[(y-x)/2].

In the animation below I am showing the transition between these two cases. For intermediate values of α we have in general quasiperiodic designs of four-fold rotational symmetry.

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