Getting into Shapes: From Hyperbolic Geometry to Cube Complexes

A proof announced in March resolved the last of 23 questions about 3D shapes posed in 1982 by mathematician William Thurston, marking the end of an era in the study of "three-manifolds"

By Erica Klarreich

Figure 3. When viewed through the lens of hyperbolic geometry, all the fish are the same size. The curves that run along the fishes? spines are hyperbolic straight lines, or "geodesics." Image: Illustration: Douglas Dunham, University of Minnesota Duluth

From Simons Science News (find original story here).

Thirty years ago, the mathematician William Thurston articulated a grand vision: a taxonomy of all possible finite three-dimensional shapes.

Thurston, a Fields medalist who spent much of his career at Princeton and Cornell, had an uncanny ability to imagine the unimaginable: not just the shapes that live inside our ordinary three-dimensional space, but also the far vaster menagerie of shapes that involve such complicated twists and turns that they can only fit into higher-dimensional spaces. Where other mathematicians saw inchoate masses, Thurston saw structure: symmetries, surfaces, relationships between different shapes.

?Many people have an impression, based on years of schooling, that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules,? he wrote in 2009. ?Good mathematics is quite opposite to this. Mathematics is an art of human understanding. ? Mathematics sings when we feel it in our whole brain.?

At the core of Thurston?s vision was a marriage between two seemingly disparate ways of studying three-dimensional shapes: geometry, the familiar realm of angles, lengths, areas and volumes, and topology, which studies all the properties of a shape that don?t depend on precise geometric measurements ? the properties that remain unchanged if the shape gets stretched and distorted like Silly Putty.

To a topologist, the surface of a frying pan is equivalent to that of a table, a pencil or a soccer ball; the surface of a coffee mug is equivalent to a doughnut surface, or torus. From a topologist?s point of view, the multiplicity of two-dimensional shapes ? that is, surfaces ? essentially boils down to a simple list of categories: sphere-like surfaces, toroidal surfaces, and surfaces like the torus but with more than one hole. (Most of us think of spheres and tori as three-dimensional, but because mathematicians think of them as hollow surfaces, they consider them two-dimensional objects, measured in terms of surface area, not volume.)

Thurston?s key insight was that it is in the union of geometry and topology that three-dimensional shapes, or ?three-manifolds,? can be understood. Just as the topological category of ?two-manifolds? containing the surfaces of a frying pan and a pencil also contains a perfect sphere, Thurston conjectured that many categories of three-manifolds contain one exemplar, a three-manifold whose geometry is so perfect, so uniform, so beautiful that, as Walter Neumann of Columbia University is fond of saying, it ?rings like a bell.? What?s more, Thurston conjectured, shapes that don?t have such an exemplar can be carved up into chunks that do.

In a 1982 paper, Thurston set forth this ?geometrization conjecture? as part of a group of 23 questions about three-manifolds that offered mathematicians a road map toward a thorough understanding of three-dimensional shapes. (His list had 24 questions, but one of them, still unresolved, is more of an intriguing side alley than a main thoroughfare.)

?Thurston had this enormous talent for asking the right questions,? said Vladimir Markovic, a mathematician at the California Institute of Technology. ?Anyone can ask questions, but it?s rare for a question to lead to insight and beauty, the way Thurston?s questions always seemed to do.?

These questions inspired a new generation of mathematicians, dozens of whom chose to pursue their graduate studies under Thurston?s guidance. Thurston?s mathematical ?children? manifest his style, wrote Richard Brown of Johns Hopkins University. ?They seem to see mathematics the way a child views a carnival: full of wonder and joy, fascinated with each new discovery, and simply happy to be a part of the whole scene.?

In the decades after Thurston?s seminal paper appeared, mathematicians followed his road map, motivated less by possible applications than by a realization that three-manifolds occupy a sweet spot in the study of shapes. Two-dimensional shapes are a bit humdrum, easy to visualize and categorize. Four-, five- and higher-dimensional shapes are essentially untamable: the range of possibilities is so enormous that mathematicians have limited their ambitions to understanding specialized subclasses of them. For three-dimensional shapes, by contrast, the structures are mysterious and mind-boggling, but ultimately knowable.

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