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To make the math work. Ever since Einstein determined that time is actually another dimension, Physicists have used that notion to expand the conception of the Universe to include added (by not sensible) dimensions to get their math and theories to work. Of particular note is Witten's unification of string theories which "only" required the addition of yet ...


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It is relatively easy to imagine 4th dimension. That would be time. But time as if we had a time machine with which we we could arbitrarily move through it. Higher dimensions would be more difficult but possible as if "destinies". For example imagine that in destiny1 you see a car going from A to B in a given hour but in alternate destiny2 you see the same ...


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In differential geometry, a space of a given number of dimensions can be curved rather than Euclidean, so for example the surface of a sphere is understood to be a 2-dimensional space in spite of the fact that we can't help but visualize the sphere sitting in a higher-dimensional 3D Euclidean space. This 3D space that we imagine the 2D surface sitting in is ...


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The definition of dimension used here is that of a dimension of a manifold - essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime). Manifolds may carry a notion of length, and one of volume. They may also be compact or non-compact, roughly1 corresponding to finite and infinite. E.g. a sphere of radius ...


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The torus is special because it's so simple, and because it provides the most tractable example of Mirror Symmetry https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory), a generalization of T-duality (which relates Type IIB with Type IIA with one another). Toric compactifications are rather special, they're a special case of an incredibly large ...



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