In math, there is no limit on the number of dimensions. You can think of a dimension as being a direction, but you can also think of it abstractly, as a degree of freedom, or an independent coordinate. Imagine something where you need “n” numbers to fully distinguish it from all other somethings. Your space of somethings has n dimensions. As a contrived example, my phone number has 10 digits. If I consider a dimension to be the values that one digit can take, then my phone number has 10 dimensions. (Obvious alternative: if I consider the set of integers as my dimension, then my phone number has only one dimension. Both descriptions are legit.)
String theory physicists are trying to describe reality with math. They need more than the four space-time measures that we experience to fully distinguish one physical configuration from another, so they need more dimensions for those degrees of freedom to live in. Are those dimensions “real”? (And if so, *where* *are* *they*!?) That’s why string theory is open research, not settled science.
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