The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)
PRELUDE. THE SHAPES OF THINGS TO COME
God ever geometrizes.
In the year 360 B.C. or thereabouts, Plato published Timaeus—a creation story told in the form of a dialogue between his mentor, Socrates, and three others: Timaeus, Hermocrates, and Critias. Timaeus, likely a fictitious character who is said to have come to Athens from the southern Italian city of Locri, is an “expert in astronomy [who] has made it his main business to know the nature of the universe.”1 Through Timaeus, Plato presents his own theory of everything, with geometry playing a central role in those ideas.
Plato was particularly fascinated with a group of convex shapes, a special class of polyhedra that have since come to be known as the Platonic solids. The faces of each solid consist of identical polygons. The tetrahedron, for example, has four faces, each a triangle. The hexahedron, or cube, is made up of six squares. The octahedron consists of eight triangles, the dodecahedron of twelve pentagons, and the icosahedron of twenty triangles.
Plato did not invent the solids that bear his name, and no one knows who did. It is generally believed, however, that one of his contemporaries, Theaetetus, was the first to prove that five, and only five, such solids—or convex regular polyhedra, as they’re called—exist. Euclid gave a complete mathematical description of these geometric forms in The Elements.
0.1—The five Platonic solids, named for the Greek philosopher Plato: the tetrahedron, hexahedron (or cube), octahedron, dodecahedron, and icosahedron. The prefixes derive from the number of faces: four, six, eight, twelve, and twenty, respectively. One feature of these solids that no other convex polyhedra satisfy is that all their faces, edges, and angles (between two edges) are congruent.
The Platonic solids have several intriguing properties, some of which turn out to be equivalent ways of describing them. For each type of solid, the same number of faces meet at each of the corner points, or vertices. One can draw a sphere around the solid that touches every one of those vertices—something that’s not possible for polyhedra in general. Moreover, the angle of each vertex, where two edges meet, is always the same. The number of vertices plus faces equals the number of edges plus two.
Plato attached a metaphysical significance to the solids, which is why his name is forever linked with them. In fact, the convex regular polyhedra, as detailed in Timaeus, formed the very essence of his cosmology. In Plato’s grand scheme of things, there are four basic elements: earth, air, fire, and water. If we could examine these elements in fine detail, we’d notice that they are composed of minuscule versions of the Platonic solids: Earth would thus consist of tiny cubes, the air of octahedrons, fire of tetrahedrons, and water of icosahedrons. “One other construction, a fifth, still remained,” Plato wrote in Timaeus, referring to the dodecahedron. “And this one god used for the whole universe, embroidering figures on it.”2
As seen today, with the benefit of 2,000-plus years of science, Plato’s conjecture looks rather dubious. While there is, at present, no ironclad agreement as to the basic building blocks of the universe—be they leptons and quarks, or hypothetical subquarks called preons, or equally hypothetical and even tinier strings—we do know that it’s not just earth, air, fire, and water embroidered upon one giant dodecahedron. Nor do we believe that the properties of the elements are governed strictly by the shapes of Platonic solids.
On the other hand, Plato never claimed to have arrived at the definitive theory of nature. He considered Timaeus a “likely account,” the best he could come up with at the time, while conceding that others who came after him might very well improve on the picture, perhaps in a dramatic way. As Timaeus states midway into his discourse: “If anyone puts this claim to the test and discovers that it isn’t so, his be the prize, with our congratulations.”3
There’s no question that Plato got many things wrong, but viewing his thesis in the broadest sense, it’s clear that he got some things right as well. The eminent philosopher showed perhaps the greatest wisdom in acknowledging that what he put forth might not be true, but that another theory, perhaps building on some of his ideas, could be true. The solids, for instance, are objects of extraordinary symmetry: The icosahedron and dodecahedron, for instance, can be rotated sixty ways (which, not coincidentally, turns out to be twice the number of edges in each shape) and still look the same. In basing his cosmology on these shapes, Plato correctly surmised that symmetry ought to lie at the heart of any credible description of nature. For if we are ever to produce a real theory of everything—in which all the forces are unified and all the constituents obey a handful (or two) of rules—we’ll need to uncover the underlying symmetry, the simplifying principle from which everything else springs.
It hardly bears mentioning that the symmetry of the solids is a direct consequence of their precise shape or geometry. And this is where Plato made his second big contribution: In addition to realizing that mathematics was the key to fathoming our universe, he introduced an approach we now call the geometrization of physics—the same leap that Einstein made. In an act of great prescience, Plato suggested that the elements of nature, their qualities, and the forces that act upon them may all be the result of some hidden geometrical structure that conducts its business behind the scenes. The world we see, in other words, is a mere reflection of the underlying geometry that we might not see. This is a notion dear to my heart, and it relates closely to the mathematical proof for which I am best known—to the extent that I am known at all. Though it may strike some as far-fetched, yet another case of geometric grandstanding, there just might be something to this idea, as we’ll see in the pages ahead.