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In summary, Kaluza-Klein theory proposes the existence of an extra compact spatial dimension in addition to the familiar three dimensions. Matter is described as a 3-dimensional surface moving in these 4 spatial dimensions in a cyclical manner. The shape of this surface determines the physics. As observers, we only perceive the three dimensions of x, y, and z. At a given instant, matter is distributed in 4-space, appearing as a 2-dimensional surface in 3-space. The theory was originally intended to unify electromagnetism and gravity, but has fallen out of favor due to its treatment of fermions and the prediction of a hypothetical particle, the dilaton, that has not been observed. However, it has influenced many

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https://en.m.wikipedia.org/wiki/Kaluza–Klein_theoryIt was originally formulated as a way to geometrically unify electromagnetism with gravity. Solving the Einstein field equations in 4+1 dimensions (where one of the spatial dimensions is periodic) gives the 10 original EFEs for 3+1 dimensions along with Maxwell’s equations. There’s also an extra scalar equation that predicts the existence of a hypothetical particle called the dilaton which has never been observed. KK theory has fallen somewhat out of favor because it only treats fermions in an ad hoc manner, but it is an important jumping off point for many modern beyond-Standard-Model theories that incorporate dimensional compactification.

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