Like the *z*-Squared and *z*-Cubed tetraviews, the images "*z*-Squared Necklace" and "*z*-Cubed 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 *u*-axis, so that what we see is just a disc in the
*xy*-plane) and ending with the graph of the real part of the
inverse relation (viewed from the negative *x*-axis, so we see a
doubly or triply covered disk in the *uv*-plane). The intermediate
images show views after rotating the surface in both the the *xv*-
and *yu*-planes by an angle of ,
for several values of between 0 and 90
degrees. As a projection into 2-space, each view shows a three- or
four-fold symmetry. The on-line 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 *xy*-plane. This can be parameterized as
(*x*, *y*) = (cos *t*, sin *t*). Since we have shown
(in the discussion
of the tetraviews) that the complex squaring function has the graph
(*x*, *y*, *x*^{2}-*y*^{2},
2*xy*), the image of this circle is then

(cos *t*, sin *t*,
cos^{2} *t* - sin^{2} *t*,
2 cos *t* sin *t*)
=
(cos*t*, sin*t*, cos 2*t*, sin 2*t*).

Rotating this through an angle of in
the *xv* and *yu*-planes gives

(*x*, *y*) =
(cos cos *t* +
sin sin 2*t*,
cos sin *t* +
sin cos 2*t*),

which equals

(*x*, *y*) =
cos (cos *t*, sin *t*) +
sin (sin 2*t*, cos 2*t*).

Plotting this curve reveals that it does indeed have the required 3-fold 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 three-fold 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 from the center of the small circle.