# An Introduction to String Theory

Slide 29 of 37
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Even though we first noticed the symmetry between momentum level and winding number in the spectrum of allowed energies, it turns out that it matches for the entire theory: interactions, vibrations, complicated diagrams with lots of loops... everything. Theoretical physicists are always excited to find symmetries in their equations: they give important hints about the underlying structure of the system. And, of course, they make the math easier.

The pictures here illustrate T-duality between two "universes" whose radii are close to the string length. The string of winding two (and momentum zero) on the larger cylinder corresponds to the string of momentum two (and winding zero) on the smaller cylinder: the math describing those two objects is exactly the same, even if we interpret it differently. This is fascinating, even if it is rather abstract.

But this T-duality has deep implications even at a level that beginners can understand: it says that there is honestly no difference at all between a "circular" universe that is very large and one that is very small. That could even be relevant to the three dimensions that we actually see! If it turned out that our universe wasn't really infinite but just too large for us to see it start repeating, then T-duality would tell us that we could just as accurately think of it as extremely small instead. What we think of as momentum could equally well be described as strings "wrapped around the (tiny) universe". This is an extremely weird thought!

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