Hurwitz's theorem (composition algebras)

Imagine you have a magical box with some interesting properties. This box can do math for you, but only when you put certain things called "elements" inside. These elements are like objects that have special rules for how they can be added, subtracted, multiplied, and divided.

Now, imagine that you have another magical box. This box is also able to do math, but with a different set of elements. You might think that these two boxes are completely unrelated, but that's not true! In fact, there is a special way to combine the two boxes together to create an even more powerful one.

This is where Hurwitz's theorem comes in. It tells us that there are only four ways to combine these two boxes together in a way that preserves all of their special rules. These four ways are called "composition algebras." Each composition algebra has its own set of elements, which have different rules for how they can be added, subtracted, multiplied, and divided.

But why is this important? Well, these composition algebras show up in all sorts of places in math and science. They help us understand things like rotations in three dimensions, the behavior of particles in quantum mechanics, and even the structure of DNA. So by understanding Hurwitz's theorem and the four composition algebras, we can better understand the world around us.