It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better — H.Poincaré, Science and Method (1908, Part II. Ch. 2, p. 129)
We all know what a circle and a sphere look like. The first we can draw with a compass, the second occurs as the surface of the earth, or of a ping-pong ball. A circle is one-dimensional, since a point is determined by giving an angle (see our discussion of dimension). For the same reasons, a sphere is two-dimensional: a point is determined by two numbers, latitude and longitude. One might ask: is there a three-dimensional analogue of the sphere, what we might call a three-sphere, or a hypersphere?
Description using words. The short answer is "yes." A somewhat longer answer is (1) the circle is the set of points in the plane at fixed distance (the radius) from a given point (the center); (2) the sphere is the set of points in three-space at fixed distance from a given point; (3) thus by analogy the hypersphere is the set of points in four-space at a fixed distance from a given point. This way of thinking about the circle, sphere, and hypersphere makes evident the family resemblance of these shapes.
Description using equations. A somewhat different answer can be given using equations. The translation of (1) is that the circle is the set of points where
x2 + y2 = r2.
This is the equation for a circle of radius r centered at 0, 0. The translation of (2) is similar:
x2 + y2 + z2= r2.
This is the equation for a sphere of radius r centered at 0, 0, 0. The translation of (3) follows the same pattern:
x2 + y2 + z2 + w2= r2.
This is the equation for a hypersphere of radius r centered at 0, 0, 0, 0. Again, the family resemblance of these shapes is made apparent by the family resemblance of the equations. We could call the shapes defined the one-sphere, the two-sphere, and the three-sphere, respectively.
Riemann's description. Here is yet another way of thinking of the hypersphere, or three-sphere. (1) Take two wires. Join the ends of one to the ends of the other, as in the FIGURE. The result is a misshapen circle. It can be reshaped, without cutting or tearing the wire, into a true circle.
(2) Consider a two-sphere. Cut it along the equator. Two pieces result, the northern hemispherical cap, and the southern one. Now glue the two pieces together along the cut, which is a circle. The result is a two-sphere. What are the two pieces? Well, look at the northern hemispherical cap from directly above. It looks like a disk: a circle, together with the material inside it. Glue two disks together along the "boundary circles" to obtain a somewhat mishapen two-sphere. If the odd shape bothers you, connect a tube to the inside and pump it full of air. You will then have a round two-sphere.
(3) With this little thought experiment complete, let us try something more ambitious. Take two solid balls in three-space. Each one consists of a two-sphere and the material inside it, just as a disk consists of a circle and the material inside it. Now imagine that the two-sphere for one of the balls is glued (magically) to the two-sphere of the other ball in such a way that latitudes and longitudes of one sphere correspond to latitudes and longitudes of the other sphere. The result is the hypersphere, or three-sphere. This description goes back to Riemann. See XXXX:Osserman.
Dante. A model of the three-sphere similar to the one just described appears in Dante. The earth is a ball. We live on its surface, a two-sphere. Below the surface of the earth are concentric but smaller spheres in lower rings of the underworld. These spheres contract to a single point, the center. Perhaps the abode of Hades, or the Devil.
As you pass upwards from the surface of the earth into the sky, then the heavens, you enter another realm, also composed of concentric spheres. When you go high enough, the spheres start to become smaller and smaller. Eventually the spheres shrink to a single point. This is the abode of the Deity.
The universe, in Dante's view, is made of two three-dimensional balls. A point in each is determined by latitude, longitude and depth (or height). While we can visualize each ball as a separate entity, we cannot visualize the hypersphere in its entirety. Nonetheless, we can understand it well enough to say what happens as we travel from part to part.