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
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
For an angle of 45 degrees we see a quasiperiodic image
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.