The best thing to do when designing a hat according to something you’ve researched and looked at dozens of pictures of, is to make one for yourself, along roughly the lines that you intend to make. So… it turned out more or less right. It’s got the tall sides, of course, so that’s all right.

But there’s something about it that’s… just… not quite right. I can’t put my finger on quite what it is, exactly…

Oh, yes. I remember.

It has something of an airplane tail fin on it. And this turns out to be the challenge with using a circle as the base of your pattern, rather than an oval — some extra fabric winds up needing to go somewhere.

So, back to the drawing board, quite literally, to construct an ellipse. And then to figure out how to construct an ellipse of a given circumference.

Which turns out to require calculus:

I’m not expert a calculus. I don’t actually know anything about calculus at all. I get that this formula wants me to multiply pi times 2, and then multiply that times the height and width of the ellipse. But none of that tells me what width to set my compass to, if I want an ellipse of a given perimeter.

But, I’ve read that it was Newton’s mastery of Renaissance and Enlightenment-era geometry that made it possible for him to invent/discover calculus. So, there’s probably a way to do it geometrically. I wonder what it is? Most of the methods of drawing ellipses require knowing a given height and given length of the ellipse, and working from those; but if you want a finalized Perimeter, something rather different has to be going on.

In the meantime, there’s a rather different solution. Make an ellipse, measure its perimeter, and then adjust upward or downward until I hit the perimeter of the right length. By trial and error, I may discover the methodology or relationship between an initial proportion for the compass and the the final proportion of the ellipse.