For most interpretations, superstring theory requires a large number of extra space dimensions to be mathematically consistent: M-theory requires ten space dimensions. With the introduction of branes as multidimensional objects in string theory, it becomes possible to construct and imagine wildly creative geometries for space that correspond to different possible particles and forces. It’s unclear, at present, whether those extra dimensions exist or are just mathematical artifacts.

The reason string theory requires extra dimensions is that trying to eliminate them results in much more complicated mathematical equations. It’s not impossible, but most physicists haven’t pursued these concepts in a great deal of depth, leaving science (perhaps by default) with a theory that requires many extra dimensions.

From the time of Descartes, mathematicians have been able to translate between geometric and physical representations. Mathematicians can tackle their equations in virtually any number of dimensions that they choose, even if they can’t visually picture what they’re talking about.

One of the tools mathematicians use in exploring higher dimensions is analogy. If you start with a zero-dimensional point and extend it through space, you get a 1-dimensional line. If you take that line and extend it into a second dimension, you end up with a square.

If you extend a square through a third dimension, you end up with a cube. If you then were to take a cube and extend into a fourth dimension, you’d get a shape called a hypercube.

A line has two “corners” but extending it to a square gives four corners, while a cube has eight corners. By continuing to extend this algebraic relationship, a hypercube would be a 4-dimensional object with 16 corners, and a similar relationship can be used to create analogous objects in additional dimensions. Such objects are obviously well outside of what our minds can picture.

Humans aren’t psychologically wired to be able to picture more than three space dimensions. A handful of mathematicians (and possibly some physicists) have devoted their lives to the study of extra dimensions so fully that they may be able to actually picture a 4-dimensional object, such as a hypercube. Most mathematicians can’t (so don’t feel bad if you can’t).

Whole fields of mathematics — linear algebra, abstract algebra, topology, knot theory, complex analysis, and others — exist with the sole purpose of trying to take abstract concepts, frequently with large numbers of possible variables, degrees of freedom, or dimensions, and make sense of them.

These sorts of mathematical tools are at the heart of string theory. Regardless of the ultimate success or failure of string theory as a physical model of reality, it has motivated mathematics to grow and explore new questions in new ways, and for that alone, it has proved useful.