If you’ve ever been stuck in traffic on a rainy afternoon, you’ve probably watched raindrops racing each other down the car window. When pairs of droplets collide, they merge into a new droplet, losing their separate identities.

That merging is possible because the water droplets are just about spherical. When shapes are flexible — as raindrops are — attaching a sphere doesn’t change anything. In certain areas of mathematics, a sphere attached to a sphere is still a sphere, though perhaps a bigger or lumpier one. And if a sphere gets glued onto a doughnut, you still have a doughnut — with a blister. But if two doughnuts merge together, they form a two-holed shape. To mathematicians, that’s something else completely.

That quality makes spheres a crucial test case for geometers. Mathematicians can often transfer lessons learned on spheres to more complex shapes by looking at what happens when you sew the two together. In fact, they can apply this technique to any manifold — a class of mathematical objects that includes simple shapes like spheres and doughnuts, as well as infinite structures like a two-dimensional plane or three-dimensional space.

Spheres are especially important in a subdiscipline of geometry known as contact geometry. In contact geometry, every point on a three-dimensional manifold — such as the 3D space we live in — corresponds to a plane. The planes can tilt and twist from point to point. If they do so in a way that satisfies certain mathematical criteria, the entire set of planes is called a contact structure. A manifold (like 3D space) together with a contact structure (all the planes) is called a contact manifold.

Though contact structures might seem to amount to little more than decoration, they bring fundamental insights into the manifolds they live on, as well as links to physics. Modern mathematicians can use contact manifolds to reformulate theories about how light behaves and about the way water flows through space.

Results about three-dimensional contact manifolds frequently come back to spheres. If you glue a contact sphere onto another contact manifold, such as a 3D doughnut, the 3D version of the sphere can donate parts of its contact structure to the union. If you want to prove that a doughnut can have a contact structure whose planes twist a thousand times as they circle the doughnut hole, you can first build that structure on the sphere and then add it to the doughnut by cutting a small hole in both shapes and patching them together along the edges. Mathematicians exploring which contact structures can exist on a given manifold frequently rely on this framework, said John Etnyre, a mathematician at the Georgia Institute of Technology. “They do a lot of work to reduce the problem to understanding what happens on the sphere,” he said.

As Jonathan Bowden, a mathematician at the University of Regensburg, puts it: “If you can’t understand a sphere, how can I possibly understand anything else?”

We tend to think of spheres as simple shapes: They’re merely all the points that are a fixed distance from a center point. Examples include a circle, which is one-dimensional, as well as the two-dimensional surface of an ordinary ball like a basketball. But when you add in contact structures, spheres can get more complicated than you might expect. And as mathematicians attempt to sort through a disorganized ocean of contact manifolds, new types of spheres can give them clues about what they might fish out of the depths.

In a recent paper that was substantively updated last week, four mathematicians — Bowden, Fabio Gironella, Agustin Moreno and Zhengyi Zhou — have uncovered a new type of contact sphere and, with it, an infinite number of new contact manifolds.

Full Contact Sport

As a field, contact geometry emerged gradually over the course of centuries. Though modern mathematicians looking back see hints of contact geometry in the study of optics in the 17th century and thermodynamics in the 19th, only in the 1950s was the phrase “contact manifold” first used in a paper, according to the mathematician Hansjörg Geiges’ history of the subject.

By that time, mathematicians were already aware of some examples of contact manifolds. For technical reasons, contact manifolds only come in odd dimensions. Standard three-dimensional space has a contact structure consisting of rows of planes that gradually tilt forward. This structure naturally extends to what mathematicians call the three-dimensional sphere. (This is the surface of a four-dimensional ball, much as the two-dimensional mathematical sphere is the surface of an ordinary three-dimensional ball.)

Starting in the late 1960s, mathematicians began to present new examples of contact manifolds. In 1968 Mikhael Gromov made progress on finding new contact structures on certain manifolds, such as three-dimensional space, and Jean Martinet followed in 1971 with examples on so-called compact shapes (which are finite with a clear boundary) like the 3D sphere. In 1977, Robert Lutz figured out how to create a new contact structure on any three-dimensional manifold. Lutz’s construction involved slicing open the contact manifold, twisting it up, and sewing it back together in a way that kept the underlying shape the same, but forced the contact structure into a new configuration. It resulted in a new contact structure for infinite 3D space, the 3D sphere, and any number of even stranger objects, such as a cube where, if you stick your hand through the bottom, you’ll see it dangle down from the top.

Still, those results left late-20th-century mathematicians with many unanswered questions about contact manifolds. What kinds of contact structures were out there? How should they be categorized? “When mathematicians come to some subject, they always want to classify or understand objects,” said Yakov Eliashberg, a mathematician at Stanford University who was instrumental in the early development of contact geometry.

In dimensions five and higher — remember, contact manifolds can only have an odd number of dimensions — these questions still aren’t answered. In the three-dimensional case, much of the progress was made almost single-handedly by Eliashberg, who arrived in Berkeley, California, in the 1980s as an immigrant from the Soviet Union.

Twist and Shout

Prompted by a question from a new Berkeley acquaintance named Jesús Gonzalo Pérez, who had been studying Lutz’s technique for creating new contact manifolds, Eliashberg noticed that all the three-dimensional contact manifolds you could get using Lutz’s strategy had certain commonalities. In 1989, he published a seminal paper describing these manifolds in detail. He called the new class of contact manifolds “overtwisted” because of the way the planes of the contact structure rotated multiple times, beyond the twisting required to qualify as a contact structure. Eliashberg’s 1989 paper answered practically any questions mathematicians might have about overtwisted manifolds in three dimensions, but any other contact manifold — which Eliashberg called “tight” because of how little its contact structure twisted — was much harder to get at.

“Whereas overtwisted structures exist in abundance, tight contact structures are more rare or, at least, way more poorly understood,” said Moreno, a mathematician at Heidelberg University.