It was also an exciting year in geometry. The most attention-getting result of the year was the discovery of a new kind of tile that covers the plane in a pattern that never repeats. A two-tile combination that does this has been known since the 1970s, but the single tile, discovered by a hobbyist named David Smith and announced in March, was a sensation. Fans used the simple design as a cookie cutter and sewed it into quilts. We followed our news coverage with a column explaining some of the underlying math and another giving a brief history of tiling.

Speaking of needles, it was also a year of progress on the Kakeya conjecture, which asks how small a volume of space an idealized needle can occupy while spinning in all directions. A new proof of a special case of the conjecture (called the “sticky” Kakeya conjecture) gives strong evidence that the more general conjecture is true.

The conjecture turns out to have implications not only for geometry, but also for harmonic analysis and the study of partial differential equations. A follow-up explainer examines those implications. And a Quantized Academy columntakes readers through the conjecture’s underlying logic.

In other geometry news, a long-standing idea about maps between spheres of different dimensionality, called the telescope conjecture, was shown to be false. Particular types of contact structures (patterns of planes that satisfy certain mathematical properties) that had long been thought to be impossible turned out to exist.

We interviewed Emmy Murphy, a geometer who studies such contact structures. Murphy describes contact geometry (and its sibling, symplectic geometry) as existing in the middle of a spectrum of rigidity and flexibility. In rigid geometry, much depends, she said, on precise measurements, while flexible geometry tends to resemble algebra. But in between, she said, is where “visual thinking is more useful.”

In January, the mathematician Assaf Naor and the computer scientist Oded Regev proved the existence of so-called spherical cubes. These are objects whose surface area grows slowly — as does the surface area of spheres in higher dimensions — but which can completely fill space the way cubes can.

One of the most prominent geometers of the 20th century, Eugenio Calabi, died at age 100 on September 25. Jerry Kazdan, one of his longtime colleagues, said that Calabi would “ask interesting questions that no one else was thinking about.” Our obituary of Calabi explores those questions, focusing particularly on his best-known discovery, Calabi-Yau manifolds, which later became central to string theory in physics.