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.

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