
Like the ZSquared and Zcubed tetraviews, the images "ZSquared Necklace" and "ZCubed Necklace" also show views of the complex squaring and cubing functions. The sequence begins with the graph of the real part of the function (viewed from above, i.e., from the uaxis, so that what we see is just a disc in the xyplane) and ending with the graph of the real part of the inverse relation (viewed from the negative xaxis, so we see a doubly or triply covered disk in the uvplane). The intermediate images show views after rotating the surface in both the the xv and yuplanes by an angle of q, for several values of q between 0 and 90 degrees. As a projection into 2space, each view shows a three or fourfold symmetry. The online gallery provides movies that give the complete sequence of which the five in each necklace are a part.
One way to see the symmetry is to look at the boundary of the unit disc in the xyplane. This can be parameterized as (x,y) =
(cost, sint) . Since we have shown in (in the discussion of the tetraviews) that the complex squaring function has the graph(x, y, x^{2}y^{2}, 2xy) , the image of this circle is then
(cost, sint, cos^{2}t  sin^{2}t, 2 cost sint) =(cost, sint, cos 2t, sin 2t) .Rotating this through an angle of q in the xv and yuplanes gives
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x y u v ö
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cos q 0 0 sin q 0 cos q sin q 0 0 sin q cos q 0 sin q 0 0 cos q ö
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cost sint cos 2t sin 2t ö
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Multiplying the matrices and then taking the orthogonal projection into the xyplane gives the curve
(x,y) = (cos q cost + sin q sin 2t, cos q sint + sin q cos 2t), which equals
(x,y) = cos q (cost, sint) + sin q (sin 2t, cos 2t). Plotting this curve reveals that it does indeed have the required 3fold symmetry. It is left as an exercise for the reader to verify that this curve is a hypocycloid formed by a small circle rolling along the inside of a larger circle with radius three times that of the small circle (thus the threefold symmetry). The point that traces the cycloid may be anywhere along the radius of the small circle (indeed even outside it). In fact, if the radius of the inner circle is normalized to be of unit length, then the point is at a distance of
2 tan q from the center of the small circle.

