Another rotation from four- to eight-fold rotational symmetry

There are different possibilities to orient a four-dimensional periodic pattern around the two-dimensional drawing plane such that we see a periodic pattern of four-fold rotational symmetry. In contrast to the last post we will now use a symmetric choice. The drawing plane (x,y) then has the coordinates u=x, v=y, v=x and z=y for the four-dimensional object. There the pattern is as before

f=sin(u)+sin(v)+sin(w)+sin(z) ;

and we thus see in the drawing plane two identical square waves

f = f1 + f2 with f1 = f2 = sin(x) + sin(y).

We then rotate the four-dimensional space around the drawing plane to turn the square wave f1 to the left and f2 to the right. Depending on the rotation angle α we have

u = x cos(α) + y sin(α),

v = – x sin(α) + y cos(α),

w = x cos(α) – y sin(α) and

z = x sin(α) + y cos(α).

We then see a pattern f(x,y) = f1 + f2 with

f1 = sin[x cos(α) + y sin(α)] + sin[- x sin(α) + y cos(α)] and

f2 = sin[x cos(α) – y sin(α)] + sin[x sin(α) + y cos(α)].

For an angle α of 22.5 degrees we get a quasiperiodic pattern of eight-fold rotational symmetry. The animation below shows this transition. As before, the sum of the square waves f = f1 + f2 controls brightness. But here we use colour too. The hue is given by the absolute value of the difference |f1 – f2|.

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