Compactification

A dimension may be in one of two states. It is either extended, like the 4 that we observe macroscopically and can understand by general relativity, or, the dimension is compact. What is a compact dimension?  This kind of dimension is too small to be observed by our low energy experiments. It is small, compact, rolled up or curled up. 

An analogy for compactification that is often used is the garden hose. If a view a garden hose from some sufficient distance, it appears as a 1-dimensional line. However, if I can approach the hose, I would notice another dimension of the hose, namely, the circumference of the hose, on which an ant could walk across.  

The first theory to incorporate these kinds of dimensions was Kaluza-Klein theory. A 5-dimensional unification of gravity and electromagnetism posited by Theodor Kaluza in 1921. Oskar Klein, will propose a quantum mechanical explanation for as to where this 5th dimension was located. It was a small curled up circle of the Planck length.

In string theory, extra dimensions could be hidden to our low energy experiments by either compactification or brane world scenario. In superstring theory, the extra 6 dimensions take on the geometry of the Calabi-yau manifold, named for mathematicians Eugenio Calabi (1954) and Shing-Tung Yau.  (1977 and 1978)