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the end. Witten was now collaborating with Candelas, Horowitz, and Strominger, trying to figure out the shape, or geometry, of the six “extra” dimensions of string theory. The physicists proposed that these six dimensions were curled up into a minuscule space, which they called Calabi-Yau space—part of the same family of spaces, which Calabi originally proposed and I later proved to exist.14 (See Figure 7.) String theory, again, assumes that spacetime has ten dimensions overall. The three large spatial dimensions that we’re familiar with, plus time, make up the four-dimensional spacetime of Einstein’s theory. But there are also six additional dimensions hidden away in Calabi-Yau space, and this invisible space exists at every point in “real space”, according to string theory, even though we can’t see it (Figure 8). The existence of this extradimensional space is fantastic on its own, but string theory goes much farther. It says that the exact shape, or geometry,
farther. It says that the exact shape, or geometry, of Calabi-Yau space dictates the properties of our universe and the kind of physics we see. The shape of Calabi-Yau space—or the “shape of inner space”, as we put it in our book—determines the kinds of particles that exist, their masses, the ways in which
Figure 9. If string theory is correct, at any point in four-dimensional spacetime there’s a hidden, six-dimensional Calabi-Yau space. [Xianfeng (David) Gu and Xiaotian (Tim) Yin in The Shape of Inner Space. (Calabi-Yau images courtesy of Andrew
Figure 10. A two-dimensional cross-section of a six-dimensional Calabi-Yau space. [Andrew J.
While Einstein had said the phenomenon of gravity is really a manifestation of geometry, string theorists boldly proclaimed that the physics of our universe is a consequence of the geometry of Calabi-Yau space. That’s why string theorists were so anxious to figure out the precise shape of this six-dimensional space—a problem we’re still working on today. (See Figure 10.) Wi
Before we get too carried away, we should bear in mind that string theory, as the name suggests, is just a theory. It has not been confirmed by physical experiments, nor have any experiments yet been designed that could put that theory to a
definitive test. So the jury is still out on the question of whether string theory actually describes nature, which was, of course, the original intent. (See Figure 14.) On the positive side of the ledger, some extremely intriguing, as well as powerful, mathematics has been inspired by string theory. Mathematical formulae developed through this connection have proved to be correct, and will always remain so, regardless of the scientific validity of string theory. Although it is empirically unproven, string theory now stands as the only consistent theory that unifies the different forces. And it is beautiful. Moreover, the effort to unify the different forces of nature has unexpectedly led to the unification of different areas of mathematics that at one time seemed unrelated. We still don’t know what the final word will be. In the past two thousand years, the concept of geometry has evolved over several important stages to the current state of modern geometry. Each time geometry has been transformed in a major way, the new version has incorporated our improved understanding of nature arrived at through advances in theoretical physics. It seems likely that we shall witness another major development in the twentyfirst century, the advent of quantum geometry—a geometry that can incorporate quantum physics in the small and general relativity in the large. The fact that abstract mathematics can reveal
the end. Witten was now collaborating with Candelas, Horowitz, and Strominger, trying to figure out the shape, or geometry, of the six “extra” dimensions of string theory. The physicists proposed that these six dimensions were curled up into a minuscule space, which they called Calabi-Yau space—part of the same family of spaces, which Calabi originally proposed and I later proved to exist.14 (See Figure 7.) String theory, again, assumes that spacetime has ten dimensions overall. The three large spatial dimensions that we’re familiar with, plus time, make up the four-dimensional spacetime of Einstein’s theory. But there are also six additional dimensions hidden away in Calabi-Yau space, and this invisible space exists at every point in “real space”, according to string theory, even though we can’t see it (Figure 8). The existence of this extradimensional space is fantastic on its own, but string theory goes much farther. It says that the exact shape, or geometry,
farther. It says that the exact shape, or geometry, of Calabi-Yau space dictates the properties of our universe and the kind of physics we see. The shape of Calabi-Yau space—or the “shape of inner space”, as we put it in our book—determines the kinds of particles that exist, their masses, the ways in which
Figure 9. If string theory is correct, at any point in four-dimensional spacetime there’s a hidden, six-dimensional Calabi-Yau space. [Xianfeng (David) Gu and Xiaotian (Tim) Yin in The Shape of Inner Space. (Calabi-Yau images courtesy of Andrew
Figure 10. A two-dimensional cross-section of a six-dimensional Calabi-Yau space. [Andrew J.
While Einstein had said the phenomenon of gravity is really a manifestation of geometry, string theorists boldly proclaimed that the physics of our universe is a consequence of the geometry of Calabi-Yau space. That’s why string theorists were so anxious to figure out the precise shape of this six-dimensional space—a problem we’re still working on today. (See Figure 10.) Wi
Before we get too carried away, we should bear in mind that string theory, as the name suggests, is just a theory. It has not been confirmed by physical experiments, nor have any experiments yet been designed that could put that theory to a
definitive test. So the jury is still out on the question of whether string theory actually describes nature, which was, of course, the original intent. (See Figure 14.) On the positive side of the ledger, some extremely intriguing, as well as powerful, mathematics has been inspired by string theory. Mathematical formulae developed through this connection have proved to be correct, and will always remain so, regardless of the scientific validity of string theory. Although it is empirically unproven, string theory now stands as the only consistent theory that unifies the different forces. And it is beautiful. Moreover, the effort to unify the different forces of nature has unexpectedly led to the unification of different areas of mathematics that at one time seemed unrelated. We still don’t know what the final word will be. In the past two thousand years, the concept of geometry has evolved over several important stages to the current state of modern geometry. Each time geometry has been transformed in a major way, the new version has incorporated our improved understanding of nature arrived at through advances in theoretical physics. It seems likely that we shall witness another major development in the twentyfirst century, the advent of quantum geometry—a geometry that can incorporate quantum physics in the small and general relativity in the large. The fact that abstract mathematics can reveal
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