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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