Today I went on a long shopping trip with my youngest daughter. She is off to college in the fall and has one course left to choose, the other three are freshman requirements at the University of Richmond. She either will take general chemistry or calculus. The mention of calculus lead her to say how the unit circle drove her nuts in her high school calculus course. "Why did we have to memorize all of those fractions of pi and the square root of two?" she said. It turns out her teacher did not explain why the unit circle is so useful. It's not that a circle with a radius of one ever occurs in real life, the point is that every other circle can be converted into the unit circle then all the calculations relating to it are divisible by one. And the sines and cosines relating to the position of any point on the circle read directly--they don't need to be factored. The unit circle above is the way she learned it: static, with key points to memorize.
But the unit circle is better understood live. When it moves, it makes sense immediately, as you can see here.
OK, enough geek stuff. The point of this post is just that talking about abstract ideas makes me happy, so these two weeks in America really are a rest from the concrete reality of carrying a weapon, walking on rocks and riding in sand. It's raining now in Lancaster. I am going outside to enjoy it.