Saturday, September 8, 2012

What the doctor ordered


Today is one of those days where staying indoors and making friends with one-self is exactly what the doctor ordered.

I promised myself that I’d read the first book my eye catches amongst books that I haven’t yet shelved. So it was Flatland: A Romance of Many Dimensions by Edwin Abbott.



Ever since I first read Flatland years ago I’ve always been fascinated with the idea of higher dimensions, and whether we mere humans have the capacity to visualize the fourth dimension. I don’t mean Space-time viz-a-viz Relativity theory – rather trying to imagine the existence of a 4-dimensional being looking back at us and our world in a similar way we look at an ant like insect navigating a 2-dimensional 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, 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 wanted 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.

Flatland is a relatively easy book to read and given its only 82 pages it isn’t at all burdening. What is interesting however is the whether we are able to conceive curvature form the inside. Can an ant for example calculate the curvature of the earth whilst being stuck on a 2-dimensional surface?


Carl Friedrich Gauss and his Theorema Egregium (the remarkable theorem in Latin) truly is a remarkable piece of insight.

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. 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.

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