By combining the language of groups with that of geometry and linear algebra, Marius Sophus Lie created one of math’s most powerful tools.

In mathematics, ubiquitous objects called groups display nearly magical powers. Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries. They can tell you which polynomial equations are solvable, for instance, or how atoms are arranged in a crystal.

And yet, among all the different kinds of groups, one type stands out. Identified in the early 1870s, Lie groups (pronounced “Lee”) are crucial to some of the most fundamental theories in physics, and they’ve made lasting contributions to number theory and chemistry. The key to their success is the way they blend group theory, geometry and linear algebra.

In general, a group is a set of elements paired with an operation (like addition or multiplication) that combines two of those elements to produce a third. Often, you can think of a group as the symmetries of a shape — the transformations that leave the shape unchanged.

Consider the symmetries of the equilateral triangle. They form a group of six elements, as shown here:

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