# The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)

### Postlude. ENTERING THE SANCTUM

Let us end where we began, looking to the past in the hopes of gleaning hints about the road ahead. The year was 387 B.C. or thereabouts, when, in an olive grove in the northern suburbs of Athens, Plato established his Academy, which is sometimes referred to as the world’s first major university. (The Academy he founded remained in use for more than 900 years, until the Roman emperor Justinian shut it down in A.D. 526—a record that makes the 370-plus-year lifetime of my school, Harvard, seem paltry by comparison.) As legend has it, Plato placed an inscription above the entryway to the school that read: *Let no one ignorant of geometry enter here*.

The exact wording of this is in doubt, as I’ve seen it rendered several ways. Some experts deny whether it said that at all. Piers Bursill-Hall, a scholar of Greek mathematics at the University of Cambridge, suggests the inscription might just as easily have been “No parking in front of these gates.”__ ^{1}__ For many, however, there’s little reason to doubt the claim. “It was asserted by one of the ancient authorities to be true, and there’s nothing to suggest that it’s apocryphal,” argues Donald Zeyl, a Plato expert at the University of Rhode Island. “It makes sense to me, given that Plato considered geometry an essential prerequisite to the study of philosophy.”

^{2}I, of course, am neither a historian nor a classicist and am thus in no position to arbitrate such a dispute. However, given the little I know about Plato, and the good deal more that I know about geometry, I’m inclined to side with Zeyl on this matter, if for no other reason than that despite the 2,400 or so years that separate Plato from me, we do see eye to eye on the importance of geometry. Plato regarded the truths of geometry as eternal and unchanging, whereas he regarded the knowledge derived from empirical science as more ephemeral in nature, unavoidably subject to revision. On these points, I heartily agree: Geometry can carry us far toward explaining the universe on scales both big and small (though perhaps not all the way down to the Planck scale), and when we prove something through rigorous mathematics, we can be sure it will stand the test of time. Geometrical proofs, like the diamonds advertised on TV, are forever.

Even though the particulars of Plato’s “theory of everything,” as outlined in *Timaeus*, strike the modern sensibility as absurd (if not borderline psychotic), there are many parallels between his picture of the universe and that embodied in string theory. Geometrization—the idea that the physics we observe springs directly from geometry—stands at the core of both approaches. Plato used the solids named after him to pursue this end (unsuccessfully, I might add) in much the same way that string theory relies on Calabi-Yau manifolds, though we’re hoping the results will be better this time around.

The Platonic solids are literally built on symmetry, as are contemporary theories of physics. In the end, the search for a single, all-encompassing theory of nature amounts, in essence, to the search for the symmetry of the universe. The individual components of this overarching theory have their own symmetries as well, such as in the inherently symmetrical gauge fields, which we’ve considered at length, that provide our best current descriptions of the electromagnetic, strong, and weak interactions. What’s more, the symmetry groups these fields inhabit are actually related to the symmetries of the Platonic solids, although not in the ways the ancient Greeks imagined it.

Today’s physics is awash in dualities—the idea that the same physical reality can be described in two mathematically distinct ways. These dualities associate four-dimensional quantum field theories with ten-dimensional string theories, link ten-dimensional string theory with eleven-dimensional M-theory, and even reveal the physical equivalence between two Calabi-Yau manifolds that, on the face of it, have little in common. Likewise, the Platonic solids have their own dualities. The cube and octahedron, for instance, form a dual pair: Each can be rotated twenty-four different ways and still remain unchanged. The icosahedron and dodecahedron belong to a larger symmetry group, remaining invariant after sixty twists and turns. The tetrahedron, meanwhile, is its own dual. Curiously, when my mathematics colleague Peter Kronheimer (whose office is just a few doors down the hall from mine) tried to classify a group of four-dimensional Calabi-Yau manifolds by symmetry, he discovered that they followed the same classification scheme as the Platonic solids.

I’m by no means trying to suggest that Plato, promulgating his ideas in the early days of mathematics, got it all right. On the contrary, his notions about the origins of the elements are flat-out wrong. Similarly, the astronomer Johannes Kepler’s attempts to explain the orbits of our solar system’s planets by means of nested Platonic solids, lying within concentric spheres, were also doomed to failure. The details in these scenarios don’t add up, and they don’t even come close. But in terms of the big picture, Plato was on the right track in many ways, identifying some of the key pieces of the puzzle—such as symmetry, duality, and the general principle of geometrization—that we now believe any workable attempt to explain it all ought to include.

In view of that, it seems plausible to me that Plato would have made some reference to geometry at the entryway to his famed Academy. Much as I share his respect for the discipline that I chose so many years later, were I to mount a sign above the doorway to my decidedly uncelebrated Harvard office, I would amend the wording to this: *Let no one ignorant of geometry leave here*. The same words, I trust, would apply to readers now “leaving” the pages of this slim volume and, I hope, viewing the world somewhat differently.

**A FLASH IN THE MIDDLE OF A LONG NIGHT**

*Once I stood on the edge of a bridge,* *strolled along the lakeshore,* *striving to catch from afar* *a glimpse of your matchless beauty.*

*Once I climbed atop a pavilion,* *seeking a road at land’s end,* *yearning for a vision* *of your fair, ineffable presence.*

*Ten thousand yards of silk* *your lightness cannot contain.* *The moon hovers miles away,* *its shadow broad, expansive.*

*The boundless sky, the neverending stream,* *waves breaking without surcease.* *All excite the unfathomable depth* *of which we call eternity.*

*The road is long, the view obscured,* *with thousands of threads entangled.* *Pursuing the truth, you toil endlessly.* *In dreams you’ve surely been there.*

*Yet inspiration has struck, time and again,* *lifting us onto the shoulders of giants.* *From Euclid to Descartes, Newton to Gauss,* *and Riemann to Poincaré.*

*O! The phantom of all things,* *So hard to divine in daylight’s glare.* *Then suddenly, when the radiance dims,* *she reveals a glimpse of her unseen form.*

*A chance encounter, unrivalled splendor,* *from genius through the ages.* *Let us celebrate the poetry of the universe* *and the geometry through which it sings.*

SHING-TUNG YAU

CAMBRIDGE, 2006