Banchoff and his colleagues have devised striking images that illustrate properties of four-dimensional objects. The images below, for example, depict the rotation of a four-dimensional hypercube in four-dimensional space. To appreciate the images consider the shadow cast by an ordinary Cube on a plane: the shadow can resemble a square inside a square. If the appropriate faces of the cube are shaded, the shadow is a square with a square hole in it.
Rotation of a four-dimensional hypercube through dimensions 2 and 4, projected into ordinary three-dimensional space.
Projections of the cube and the sphere.
Similarly, when a hypercube is illuminated from a point "above" ordinary
space in the fourth dimension, the three-dimensional "shadow" cast by the
hypercube can resemble a cube inside a cube. The inner cube is surrounded
by six six-sided polyhedrons that can be regarded as distorted cubes. The
four distorted cubes adjacent to the sides of the inner cube fit together
to form the solid figure whose surface is the boxlike torus shown in
Banchoff's images. The other two distorted cubes, the inner cube and the
outer cube also form a solid torus, which is not shown. As the hypercube
rotates, the square hole in the visible torus seems to move toward the
viewer. Those who write the program
HYPERCUBE will see similar
changes, albeit not so realistic or continuous.
The images are from a forthcoming film by Banchoff and his colleagues Huseyin Kocak, David Laidlaw and David Margolis: The Hypersphere: Foliation and Projections. The hypersphere is a far more complex object than the hypercube, and I shall not describe it in detail. Nevertheless, one can begin to appreciate the images by considering an ordinary sphere. If the sphere is initially at rest on a plane tangent to its south pole and a light is fixed at the initial position of its north pole, the shadow cast on the plane by the lines of latitude is a series of concentric circles [see bottom illustration on page 20]. If the sphere is rotated while the light is kept fixed, the images of the circles may become nonconcentric, and the image of any circle that passes through the source of light is a straight line.
Sequence of nested toruses, analogous to the latitude lines on a sphere, projected from the hypersphere into three dimensions.
Similarly, the three-dimensional "shadow" cast by a hypersphere can be viewed as a series of concentric toruses. The toruses are made more readily visible in Banchoff's images by cutting away parts of one torus along strips that wind around it. When the hypersphere is rotated, the toruses appear to swell up and sweep past one another. Any torus that passes through the source of light becomes infinitely large.
Projected motion of toruses as the hypersphere is rotated, analogous to the projected motion of latitude lines during the rotation of a sphere.