The Remarkable Theorem

Jack Dikian

September 2011

Ever since I read Flatland: A Romance of Many Dimensions by the English schoolmaster Edwin Abbott my mind turns to the idea of higher dimensions, and whether we humans have the capacity to visualize the fourth dimension. I don’t mean using time as a fourth dimension viz a viz Special Relativity – rather trying to imagine the existence of a 4-dimensional being looking back at us and our world.

Abbott‘s in his 1884 satirical novella wrote pseudonymously as "A Square", in the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture. A 3-dimensional being, of course, is could see everything in their world, and all at once.

In the same way, a 4-dmensional being looking back at us could look inside our stomach, and remove, if they want to the lunch we just had without cutting through our skin, just like we can remove a dot inside a circle (flatland) by moving it up into the third dimension perpendicular to the circle, without breaking the circle.

Then years later, I learned about Carl Friedrich Gauss and his Theorema Egregium (the remarkable theorem in Latin). How for example, can an Ant (say a 2-dimensional being) stuck on the surface of our curved world, and can’t stand back to see the curvature of our planet ever realize that the surface is curved.

The theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. An absolutely amazing insight! This however only applies to curved surfaces which are 2-dimensional.

It would take a brilliant student of Gauss, Bernhard Riemann at the age of just 26 to develop and extend Gauss's theory to higher dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. That is generalizing Gauss’ work to describe the curvature of space in any dimension. Again, how do we, non-mathematicians, visualize a curved 3-dimensional space. What encapsulates it? The genius of Riemann was to show that we don’t need to step into the fourth dimension to tell if space is curved. We can do it form the inside.

Albert Einstein, as we know, came along and used the theory of Riemannian manifolds to develop his General Theory of Relativity. In particular, his equations for gravitation are restrictions on the curvature of space. He took the mathematics of Gauss and Riemannian and used it to develop a revolutionary picture of our physical world showing that we live in the curved worlds of Gauss and Riemannian.

So we get to finally that gravity is not a pull downwards but rather an object falls following the simplest path through bend space. Of course, Einstein didn’t stop there and showed that the presence of mass that bends space.