This movie shows the complete sequence of rotations from which the five in
the "*z*-Cubed Necklace" were taken. Notice that the projection into two
dimensions maintains the four-fold symmetry throughout as the image moves
from a disk to a triply covered disk containing a ramification point.

This movie shows the complete sequence of rotations from which the five in
the "*z*-Cubed Necklace" were taken. Here the surface is enclosed in a
bounding hyper-box to make the four-dimensional rotations easier to follow.

This movie shows the same sequence as the previous movie, but from a
different viewpoint in space, one where the symmetry is less
apparent. The rotation in four-space is a composition of two rotations,
one in the *xw*-plane and one in the *yz*-plane. These can be
seen in this movie as the "sliding box" rotation in the hypercube (due
to the *xw* rotation) and a rotation in three-space (from the
*yz* rotation).

The five views in "*z*-Cubed Necklace" are projections of the
complex cubing function into three-space. We can get a better idea of
the shape of the projection by looking at a sequence of
three-dimensional images. Here we see the first projection as a
three-dimensional object. From above it looks like a disk, but it is
actually a monkey saddle (the real part of the cube).

This movie shows the second projection from "*z*-Cubed Necklace"
rotating in three-space. Opposite ends of the saddle are beginning to
squeeze shut as other parts expand.

This is the third projection of the complex squaring function, projected into three-space, then rotated in space. In this projection, the surface has begin to intersect itself.

This fourth projection of the cubing function shows the surface as it continues to pass through itself, rotated in three-space.

This final movie shows the last view of the surface, which we can think of as the graph of the imaginary part of the inverse of the cube (i.e., the complex cubed root). Here, the ramification point appears as a pinch point on the surface, at the origin. The self-intersection appears as a cubic curve on the surface.

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