(Since a full rotation brings every point on the triangle back to where it started, mathematicians stop counting rotations past 360 degrees.)
These symmetries are discrete: They form a set of distinct transformations that have to be applied in separate, unconnected steps. But you can also study continuous symmetries. It doesn’t matter, for instance, if you spin a Frisbee 1.5 degrees, or 15 degrees, or 150 degrees — you can rotate it by any real number, and it will appear the same. Unlike the triangle, it has infinitely many symmetries.
These rotations form a group called SO(2). “If you have just a reflection, OK, you have it, and that’s good,” said Anton Alekseev, a mathematician at the University of Geneva. “But that’s just one operation.” This group, on the other hand, “is many, many operations in one package” — uncountably many.
Each rotation of the Frisbee can be represented as a point in the coordinate plane. If you plot all possible rotations of the Frisbee in this way, you’ll end up with infinitely many points that together form a circle.
This extra property is what makes SO(2) a Lie group — it can be visualized as a smooth, continuous shape called a manifold. Other Lie groups might look like the surface of a doughnut, or a high-dimensional sphere, or something even stranger: The group of all rotations of a ball in space, known to mathematicians as SO(3), is a six-dimensional tangle of spheres and circles.
Whatever the specifics, the smooth geometry of Lie groups is the secret ingredient that elevates their status among groups.
Off on a Tangent
It took time for Marius Sophus Lie to make his way to mathematics. Growing up in Norway in the 1850s, he hoped to pursue a military career once he finished secondary school. Instead, forced to abandon his dream due to poor eyesight, he ended up in university, unsure of what to study. He took courses in astronomy and mechanics, and flirted briefly with physics, botany and zoology before finally being drawn to math — geometry in particular.
In the late 1860s, he continued his studies, first in Germany and then in France. He was in Paris in 1870 when the Franco-Prussian War broke out. He soon tried to leave the country, but his notes on geometry, written in German, were mistaken for encoded messages, and he was arrested, accused of being a spy. He was released from prison a month later and quickly returned to math.
In particular, he began working with groups. Forty years earlier, the mathematician Évariste Galois had used one class of groups to understand the solutions to polynomial equations. Lie now wanted to do the same thing for so-called differential equations, which are used to model how a physical system changes over time.
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