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In the field of basic physics, a dimension is described as one of the three basic directions an individual can travel from a specific point, namely: height, width, and length, with the fourth dimension being time (TNG: "Where Silence Has Lease"). These four dimensions make up the realm of space. Or, "space-time", as scientists have come to refer this dimensional existence. ( TNG: "Tapestry", ENT: "Storm Front")
It is believed that beyond the universe itself, few things, like other dimensions and higher levels of being exist. Unfortunately, during the 23rd century, the existence of these concepts could not be proven logically. (Star Trek: The Motion Picture)
The Androids of planet Mudd had an advanced understanding of dimensions, having formulated computations that would inevitably lead to a total description of the parabolic intersection of dimension with dimension. (TOS: "I, Mudd")
Other universe, such as the mirror universe, coexist with our universe, but on different dimensional planes. (TOS: "Mirror, Mirror") It was later determined that the dimensional structure of each alternate universe was totally dissimilar, and that any use of power would disturb its structure. (TOS: "The Tholian Web")
The galactic creation point extends through space and time into another dimension where the logic of things was totally different, where magic, the ability to control reality with thoughts, was part of the laws of nature. (TAS: "The Magicks of Megas-Tu")
According to Spock, the antimatter universe-as well as the negative universe-occupy the same space, as our universe, but in different dimensions. ( TOS: "The Alternative Factor", TAS: "The Counter-Clock Incident")