# The field of constructible lengths, part 2

Sorry for the delay. It turns out graduate school is time consuming. I think I will probably knock down my daily posting to 5 days a week. I suppose I’ll play it by ear.

This field of constructible lengths, $\mathbb K$, that we talked about last time is somewhat mysterious, and we want to illuminate it a bit more. What does it look like? There are two important ideas here. The first is that instead of thinking about constructible distances, I want to think about all the real numbers that can be coordinates of a point we can construct. A little thought tells us that these are the same fields. We can translate points to the origin, and then draw a circle to rotate one onto the $x$-axis.

The second idea is going to be looking $\mathbb K$ by starting with $\mathbb Q$ and building it up piece-by-piece.