23 / The Mobius band is the simplest nontrivial example of a fiber bundle. The fibers are shown in red. The twist given to the fibers makes the Mobius band topologically different from an ordinary cylindrical band. /


24 / A cross-section of a quintic hypersurface. Reprinted with permission from Paul Nylander, /

The cross-fertilization of ideas between the mathematical sciences and theoretical physics continues to this day. In the late 20th and early 21st centuries, string theory was formulated as an approach to unifying gravity and quantum physics into a theory of everything. Like all other theories in physics, it is highly mathematical—but the necessary mathematics has not yet been invented. There is still no rigorous context for the calculations that string theorists do, nor do mathematical scientists know the extent to which these techniques are valid.

However, the study of string theory has led to some important applications of the mathematical sciences. For example, since the 1800s, mathematicians have studied the solution sets of polynomial equations, such as a fifth degree polynomial in four variables. An example of such a polynomial is x5 + y5 + z5 + s5 + t5 = 0, which is known as a “quintic hypersurface” (see Figure 24.) These surfaces contain figures that Galileo would have recognized.

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