In string theory, a model used in theoretical physics, a compact dimension is curled up in itself and very small (Planck length). Anything moving along this dimension's direction would return to its starting point almost instantaneously, and the fact that the dimension is smaller than the smallest particle means that it cannot be observed by conventional means.
An example of a compactdimension would be rolling the x axis into a closed circle of radius R, which then has a finite volume of 2pR.
Extra noncompact dimensions would change the force law of gravity away from being the inverse square law that has been and still is measured experimentally.
As in the Newtonian limit, the Newton's constant measured in four spacetime dimensions is again derived from the full gravitational coupling constant in the five-dimensional theory, divided by the volume (in this case a circumference) of the compactdimension.
In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of very small radius, so that a particle moving a short distance along that axis would return to where it began.
This extra dimension is a compact set, and the phenomenon of having a space-time with compactdimensions is referred to as compactification.
The phenomenon of having a higher-dimensional manifold where some of the dimensions are compact is referred to as compactification.
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