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Movies of the Z-Squared Necklace

[Plain Rotation]

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.

[Boxed Rotation]

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.

[Rotated View]

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).

[First View]

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).

[Second View]

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.

[Third View]

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.

[Fourth View]

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

[Fifth View]

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.

See the description of movie formats and plug-ins if you are having difficulty getting these movies to play.

Surfaces Beyond the Third Dimension
Last modified: 08 Oct 2000 09:32:09
Comments to: Thomas F. Banchoff
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