This movie shows the complete sequence of rotations from which the five in the "ZSquared Necklace" were taken. Notice that the projection into two dimensions maintains the threefold 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 "ZSquared Necklace" were taken. Here the surface is enclosed in a bounding hyperbox to make the fourdimensional 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 fourspace is a composition of two rotations, one in the xwplane and one in the yzplane. These can be seen in this movie as the "sliding box" rotation in the hypercube (due to the xw rotation) and a rotation in threespace (from the yz rotation).


The five views in "ZSquared Necklace" are projections of the complex squaring function into threespace. We can get a better idea of the shape of the projection by looking at a sequence of threedimensional images. Here we see the first projection as a threedimensional 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 "ZSquared Necklace" rotating in threespace. 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 threespace, then rotated in space. In this projection, the surface has begin to intersect itself.


This fourth projection of the squaring function shows the surface as it continues to pass through itself, rotated in threespace.


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 selfintersection.



