With the help of a friend, this morning I played some elaborate mathematical estimation and supposition games. From this emerged a possible solution to yesterday’s hat problem.
To restate the problem, a hat has to have a top or crown of 24″ perimeter, and the opening of the hat has to have a similar sizing. Together, we worked backwards from the assumption that C = 24″. First off, 2-pi is 6.28 (or really 6.25, because measuring anything in pi with scissors and fabric, you’re not going to get much closer than a 1/8″ to your line). If we start with that assumption, then everything inside the square-root bracket has to be less than 4″, because 6×4 = 24, and that’s our maximum size. That means that a^2 + b^2 / 2 has to be less than 16, and that a^2 + b^2 has to be less than 32. If a perfect circle worked, then a and b would both be equal to 4 (and I might have to try that next, for reasons shown below).
The distance between the brow and the top of the spine, where the skull meets the spinal cord, is a longer distance than the distance between the ears. SO I started with an assumption that a = 3.5″, and b = 4.375″. And the result was that I got a perfectly mathematical hat, entirely appropriate for a Jupiter Day. How jovial!
But, of course, the critical element of all of this is seam-allowance. In sewing, you always have to remember that you’re going to lose a 1/4″ minimum, maybe 5/8″, to the business of joining edges of fabric together, especially along the line of the crown. BUT, you’re losing a 1/4″ all the way around, so really it’s a half-inch all the way around, and a half-inch on the sides… So, I lost at least an inch of diameter, and maybe I wasn’t sewing carefully enough…
Ideally, you should allow an additional 1/4″ to 1/2″ all around to make sure that the seam is correct. And I kind of forgot to do that. So the size of the hat drops from around a 24″ to about a 21.5″, and the resulting hat is too small.
So, the resulting hat is too small for my head, and too tall in any case for anyone’s head. Maybe a child would like it, or a person with a small head.
But the mathematics worked. And now I have a procedure to try on future iterations — a larger crown, and more care to slope the brim of the hat a little wider.