The introduction of a constant wasn't all that new or exciting at the time. After all, physics equations throughout history have involved random constants that express the strengths of various relationships. Isaac Newton's formula for universal gravitation had a constant, called G, that represents the fundamental strength of the gravitational interaction. The speed of light, c, tells us about the relationship between electric and magnetic fields. The spring constant, k, tells us how stiff a particular spring is. And so on.

But there was something different in Sommerfeld's little constant: It didn't have units. There are no dimensions or unit system that the value of the number depends on. The other constants in physics aren't like this. The actual value of the speed of light, for example, doesn't really matter, because that number depends on other numbers. Your choice of units (meters per second, miles per hour or leagues per fortnight?) and the definitions of those units (exactly how long is a "meter" going to be?) matter; if you change any of those, the value of the constant changes along with it.

But that's not true for the fine-structure constant. You can have whatever unit system you want and whatever method of organizing the universe as you wish, and that number will be precisely the same.

If you were to meet an alien from a distant star system, you'd have a pretty hard time communicating the value of the speed of light. Once you nailed down how we express our numbers, you would then have to define things like meters and seconds.

But the fine structure constant? You could just spit it out, and they would understand it (as long as they count numbers the same way as we do).