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Manifolds are objects that on a zoomed-in, ‘local’ scale appear indistinguishable from the plane or higher-dimensional space described by Euclidean geometry.

Every theorem in Euclidean geometry can be dualized in projective geometry, setting up a whole set of new theorems in the parallel universe of projective geometry.

�y��vlogsworth imagined that Euclidean geometry “wedded soul to soul in purest bond / Of reason, undisturbed by space or time.”

After puzzling over this for some time, Kepler hit on the idea that the number of planets might be related to the number of regular solid figures that can be constructed using Euclidean geometry.

From them, it should be possible to derive a complete system of knowledge embracing every aspect of the natural world, just as one can deduce the whole of Euclidean geometry from five axioms.