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To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful... A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere... The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside... But... a being... unable to leave the surface... could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. ...On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. ...The spaces of zero and negative curvature are infinite, that of positive curvature is finite. ...the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would... differ... by an amount too small to be appreciable... then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension....our case with reference to three-dimensional space is exactly similar. ...we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations. (en) |
