Three-Dimensional Non-Euclidean Geometry

Bolyai, Lobachevski, and Gauss had created two-dimensional non-Euclidean geometries. For any point, the surrounding space looked like a piece of the plane. To check on the possible curvature of the space it might suffice to make some very careful measurements. In fact if the curvature of the space is too gradual or if we draw too small a triangle, we might not realize that the intrinsic geometry is non-Euclidean. To see this, we only have to go back to Gauss and his activity as a surveyor. If he confines himself to small enough regions, then the angle sum will indeed he 180 degrees, within the tolerance of his measuring instruments. But for a large enough triangle the difference is appreciable.

The idea that there could be different kinds of geometries on surfaces was strange enough, but even more threatening to convention was the suggestion that there could be different kinds of three-dimensional geometry. Surely there was only one way to think of space? At least that is the firm opinion stated by the followers of Immanuel Kant, who felt that any alternative was unthinkable. But it wasn't unthinkable for Gauss. He not only considered the possibility of non-Euclidean three-space, he also speculated about whether or not this non-Euclidean model might be the true description of the space we live in.

Is it possible that our space is curved rather than flat? One of Gauss' most important insights was that we can tell the shape of the space we are in by measuring angle sums for triangles, not just in two dimensions, but also in three-dimensional space. To show that space is non-Euclidean, all we have to do is find a triangle with an angle sum observably different from 180 degrees. Gauss set out to measure the angle sum for the largest triangle he could find. He did not want to lay it out along the surface of the earth, where he knew that angle sums of spherical triangles could be greater than 180 degrees. Instead he used what he thought of as the straightest lines in space, represented by light rays. In order to make a large triangle of light rays, he positioned beacons on the tops of three high mountains, where the curvature of the earth would not block the light rays from the sight of observers positioned on each mountain. The observers measured the angles and totaled up the answers, but the experiment was inconclusive. The sum was 180 degrees, up to the accuracy of their surveying instruments. It is quite difficult to prove that the sum of the angles of a triangle of light rays is precisely equal to 180 degrees. Even a modern computer cannot establish that two numbers are exactly equal, although it can check easily enough that two numbers are equal up to a desired tolerance.

We still do not know whether or not our three-dimensional space satisfies the axioms of Euclid's solid geometry, but we do know that we can't use light rays as our straight lines in this geometry. One of the crucial discoveries of physics is that light rays are deflected as they pass near a very massive object. Thus a ray might bend as it passed a star, and the bend would alter the angle sum of a triangle of light rays. That does not mean that our geometry is a non-Euclidean three-dimensional geometry, but it does mean that we have to be careful in trying to apply such a geometry to the study of light rays traveling interstellar distances.

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