# With Snowflakes and Unicorns, Marina Ratner and Maryam Mirzakhani Explored a Universe in Motion

Dr. Mirzakhani was a young superstar from Iran who worked nearby at Stanford University. Just 40 when she died of cancer in July, she was the first woman to receive the prestigious Fields Medal.

I first heard about Dr. Mirzakhani when, as a graduate student, she proved a new formula describing the curves on certain abstract surfaces, an insight that turned out to have profound consequences — offering, for example, a new proof of a famous conjecture in physics about quantum gravity.

I was inspired by both women and their patient assaults on deeply difficult problems. Their work was closely related and is connected to some of the oldest questions in mathematics.

The ancient Greeks were fascinated by the Platonic solid — a three-dimensional shape that can be constructed by gluing together identical flat pieces in a uniform fashion. The pieces must be regular polygons, with all sides the same length and all angles equal. For example, a cube is a Platonic solid made of six squares.

Early philosophers wondered how many Platonic solids there were. The definition appears to allow for infinite possibilities, yet, remarkably, there are only five such solids, a fact whose proof is credited to the early Greek mathematician Theaetetus. The paring of the seemingly limitless to a finite number is a case of what mathematicians call rigidity.

Something that is rigid cannot be deformed or bent without destroying its essential nature. Like Platonic solids, rigid objects are typically rare, and sometimes theoretical objects can be so rigid they don’t exist — mathematical unicorns.

In common usage, rigidity connotes inflexibility, usually negatively. Diamonds, however, owe their strength to the rigidity of their molecular structure. Controlled rigidity — that is, flexing only along certain directions — allows suspension bridges to survive high winds.

Dr. Ratner and Dr. Mirzakhani were experts in this more subtle form of rigidity. They worked to characterize shapes preserved by motions of space.

One example is a mathematical model called the Koch snowflake, which displays a repeating pattern of triangles along its edges. The edge of this snowflake will look the same at whatever scale it is viewed.