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

This movie shows the complete sequence of rotations from which the
five in the "*z*-Squared 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*-Squared Necklace" are projections of the complex
squaring 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 saddle
surface (the real part of the square).

This movie shows the second projection from
"*z*-Squared Necklace" rotating in three-space. One part of the
saddle is beginning to squeeze shut as the other expands.

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

This fourth projection of the squaring 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 square (i.e., the complex square root). Here, the ramification point appears as a pinch point on the surface, at the end of the curve of self-intersection.

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