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Shadows from Higher Dimensions

Ivars Peterson


From the graceful, flowing forms of minimal surfaces to the starburst of edges in a hypercube, many striking computer-generated mathematical images are coming out of exploratory forays into the realm of geometric shapes in three- and four-dimensional space (three-space and four-space, for short). A particularly rich source of images is the bizarre world of three-manifolds in four-space.

When topologists started to explore the world of geometric forms in higher dimensions, they found that neither the intuition nor the vocabulary of ordinary geometry was sufficient to describe and classify the new forms they were discovering. They introduced manifold as a general term for describing a certain common type of higher-dimensional, geometric object.

Manifolds are surfaces and shapes, sometimes very complex, that appear to be euclidean when a small region is examined. On a large scale, however, these forms fail to follow the rules for a euclidean geometry. Roughly speaking, the earth's surface is an example of a manifold. To a gardener marking off a square plot in a field, the earth's curvature is barely noticeable, if at all. The gardener doesn't have to worry about opposite borders that aren't quite parallel or corners that aren't quite right angles.

A circle, although it curves through two dimensions, is an example of a one- dimensional manifold, or one-manifold. A close-up view reveals that any small segment of the circle is practically indistinguishable from a straight line. Similarly, a sphere's two-dimensional surface, even though it curves through three dimensions, is an example of a two- manifold. Seen locally, the surface appears to be flat.

Picturing a three-manifold is somewhat more difficult. Standing in one spot and looking around within a three-manifold wouldn't tell an observer much about the manifold's shape. It would look like ordinary three-dimensional space. Indeed, just as it's hard to imagine the two-dimensional surface of a sphere without putting it into three- space, a three-manifold is most easily viewed from four-space.

The theory of manifolds arose in the nineteenth century out of a need to understand solutions to sets of equations expressing relationships between two or more variables. For instance, all the possible solutions to a two-variable equation can be plotted as a set of points in the plane. Each point represents a pair of values that make the equation true. Those points typically fall on a curve. Similarly, the set of solutions to an equation that has three variables can be plotted as a two-dimensional surface in three-dimensional space. The points could, for example, fall on the surface of a sphere or some other two-manifold. For an equation with more than three variables, the set of solutions may look like a multidimensional manifold embedded in a space one step higher in dimension than the manifold itself.

It takes considerable mental agility to picture even a simple three-manifold, such as a three-dimensional sphere. That contorted object can be thought of as the result of gluing together the skins of two balls so that one sphere is turned inside out over the other. Astonishingly, the mathematics of three-manifolds turns out to be exceedingly intricate and more difficult than the mathematics of manifolds in any other dimension.

The place to start on a trip into this strange world is back in everyday three-space -the three-dimensional, euclidean space familiar to all of us. The two-sphere is a convenient starting point. Assigned a radius of 1 unit, it has a remarkably straightforward algebraic formula: x12 + x22 + x32 = 1,where the coordinates x1, x2, and x3 pinpoint the location of each point on the sphere with respect to three axes at right angles to one another.

To be seen, the surface of a sphere ought to be drawn or represented in three- dimensional space, just as a globe represents the earth's features. But it's also possible to represent the surface of a sphere in two dimensions: on a flat surface or on the screen of a video monitor. Mapmakers do this all the time, having developed a variety of ways to map continents and oceans onto pages in an atlas.

The standard stereographic projection is one means of achieving the transformation from a globe's surface onto a flat map. Every point on the sphere ends up at a particular spot on the planar map. If the north pole happens to be at the point (0,0,1), that is, one unit up the x3 axis, then the projection maps every point on the sphere onto a plane, defined by the x1 and x2 axes. That plane slices through the globe along its equator. All the sphere's points, except the north pole, end up somewhere in the plane (see Figure 4.7).

FIGURE 4.7 The stereographic projection of a sphere onto a plane preserves the shapes of some geometric forms and alters others.

A simple mathematical formula gives every starting point on the sphere its particular destination on the plane. A point on the globe's equator stays on the equator, now seen as a circle on the planar map. The point (3/5, 0, 4/5) in the northern hemisphere is sent to the spot (3, 0). In fact, every point in that hemisphere ends up somewhere on the plane outside of the circle marked by the equator. A point, such as (3/5, 0, -4/5), in the southern hemisphere goes to a planar point (1/3, 0), that falls inside the equatorial circle. In general, each point (x1 , x2, x3) on the sphere arrives at [x1/(1-x3),x2/(1-x3)] in the equatorial plane. The south pole sits at the center (0, 0), while the north pole is banished to infinity.

Although the resulting map is a decidedly distorted view of the globe, with every southern point crowded inside a circle and all the northern points spread out over the rest of the plane, no point on the sphere has been lost and no points overlap. Structures and patterns visible on a globe appear in warped but recognizable forms on the resulting map. By carefully studying what happens to the shape of particular global structures when they are transferred to a flat surface, mathematicians can learn to deduce or construct global properties merely by examining features on a flat map. For four-dimensional objects, which can't be seen directly, mathematicians have to rely entirely on three-dimensional "maps" to get a sense of their four-dimensional counterparts.

One way to see what's happening under a stereographic projection is to imagine a transparent globe on which circles of latitude have been drawn. If a light is fixed at the usual position of the north pole and the globe's south pole rests on a flat surface, then the shadow cast by the circles of latitude is a set of concentric circles. If the globe is tilted while the light stays fixed in position, the images of the circles shift and become distorted. Any circles that momentarily happen to pass through the light's position would cast a fleeting straight-line shadow.

Thus, any circle on the two-sphere reappears as a circle in the plane. The only exceptions are circles that happen to pass through the north pole. They project to a straight line. Thus, a set of stripes, all of equal width and parallel to circles of latitude, would project into a bull's-eye of concentric rings on the plane. Each ring gets wider as its distance from the center of the plane, or south pole, increases (see Figure 4.8).

FIGURE 4.8 Projections of bands encircling a sphere onto a plane

The procedure used to create planar images of the two-sphere can be extended to generate images of a hypersphere, or three-sphere, as it would appear in three-dimensional space. The three-sphere in four-dimensional euclidean space is defined by the equation x12 + x22 + x32 + x42 = 1. Compared with the two-sphere, all that's new is the addition of the coordinate x4.

What does the three-sphere look like? The experience of "A Square" in Flatland suggests one way to visualize the object. When Flatlanders view a sphere descending from above into their planar world, they see only the part of the sphere that intersects their plane. Their first view is of a point, followed by circles of increasing diameter up to the largest circle, when the sphere's equator passes through Flatland. Then diminishing circles appear until they dwindle to a point and vanish. By analogy, a three-dimensional human visited by a hypersphere would first see a single spot, a tiny sphere, growing steadily through a sequence of ever larger spheres. Then the spheres would begin to shrink, at last disappearing altogether.

Projections, such as the four-space equivalent of the stereographic projection, provide some mathematically more useful glimpses of a hypersphere. One example of such a projection is the Hopf map, named for German mathematician Heinz Hopf. This map takes points on the three-sphere and systematically finds places for them on the two-sphere. Mathematically, each point (X1, X2, X3, X4) on the three-sphere becomes the point (x1, x2, x3) on the two-sphere, where

x1 = 2 (X1X2 + X3X4),
x2 = 2 (X1X4 - X2X3),
x3 = (X22 + X32) - (X22 + X42)

Under the Hopf map, every point on the two-sphere represents a circle (called a Hopf circle) on the three-sphere. Reversing the projection by going from the two-sphere to the three-sphere, that is, finding the map's inverse, reveals that a circle of latitude on the two-sphere represents a doughnut-shaped surface, or torus, in the three-sphere. This is enough to build up a picture of the hypersphere. Just as two points (representing the north and south poles) and a series of adjacent bands on a planar map can be used to build up a complete picture of a two-sphere, two linked circles and a family of tori make up a full three-sphere. A sliced three-sphere can be imagined as a sequence of doughnuts within doughnuts, with each successive doughnut swelling in size as its distance from the center increases (see Figure 4.9).

FIGURE 4.9 A view of a hypersphere with its image sliced open to expose its inner structure.

By manipulating the computer-generated image of a three-sphere, mathematicians can turn this abstract mathematical object into the star of an animated film and use it to explore properties of three-manifolds. By tinkering with the equations that define these objects, they can highlight particular surface features. They can slice the objects open to get a better view. They can watch the objects move, linking strings of images to reveal coherent patterns, such as the smoothly twisted tori that swell up and sweep past one another when a hypersphere rotates (see Color Plate 4).
Pictures alone, however, don't tell the whole story. The Hopf map itself was the first example of a special type of projection of points from one sphere to another sphere of lower dimension. Its existence inspired further mathematical research in various geometric topics, including the sorting out of several types of three-manifolds in four-space.

Three-manifolds, such as the hypersphere, show up in applied mathematics and physics. Different sets of mathematical equations, which may be useful for describing physical processes as diverse as fluid flow and crystal growth, generate different curves and shapes in space. The changing behavior of complex systems over time turns out to correspond neatly to motion on the curving shapes of certain surfaces. In many instances, understanding the solutions of particular equations can be turned into a question of geometry. Although some equation solutions show up as collections of points on a sphere or a torus, others may reside on a more exotic object, such as a one-sided, one-edged Möbius strip or a three-dimensional tube called a Klein bottle, whose inside surface loops back on itself to merge with the outside.

The hypersphere, for one, makes an appearance in the analysis of motion in phase space, in which each dimension represents one of the variables in the equation used to model a particular system. For a mechanical system, the variables may be just the positions and velocities of each particle in the system. Beginning at a point representing the initial values of all the variables, the equation generates a trajectory that winds through phase space. The location of a point on the trajectory at any time contains all the information needed to describe the state of the system - that is, the values of all the variables - at that particular time. Although the motion itself isn't directly visible, such phase-space portraits provide a useful collection of information about the motion.

Plate 4 Sliced dounuts

For example, in the motion of a pair of oscillators, such as two windshield wipers, it takes one variable, the angle, to define each wiper's position and another variable to define each wiper's velocity. Together, the positions and velocities of both wipers trace out a path on the surface of a hypersphere in four-dimensional space. A similar analysis applies to a pendulum bob fixed to a stiff rod and mounted so that the bob is physically free to trace out a path on the surface of a sphere. It takes two coordinates to define the bob's position and two to specify how quickly it is moving. Tracking the pendulum's path through phase space - that is, plotting its position and velocity at different times-gives a graphical history of the pendulum's behavior. Four-space, however, has so much room that this curving, four- dimensional path could wander almost anywhere. Luckily, physical constraints - the laws of conservation of energy and momentum - keep the system within strictly defined bounds. The four-dimensional phase-space curves are confined to a three-manifold in four-space.

In normal situations, these curves, as seen in four-dimensional phase space. fall on the surface of a doughnut, or torus. Each torus represents a certain combination of energy and momentum, equivalent to the initial push given to a pendulum to get it started. In a hypersphere picture. the Hopf tori represent the constant "energy- momentum" surfaces, and the Hopf circles on them are curves representing the changing positions and velocities of each pendulum (see Figure 4.10). These curves are essentially solutions to differential equations, called Hamilton's equations, which define how the motion changes with time. We take a closer look at the behavior of differential equations in Chapter 6.
Computer graphics provides a way for researchers to study what the solution curves look like for various physical systems and sets of initial conditions. These pictures greatly enrich the study of dynamical systems-the way things change in time or space.

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