Humiliating knitting geometry question

[neadods]

How sad is it that I can't work out the number of stitches along the diagonal of a square when I know the number of the stitches along the two sides? (ETA: 40 stitches per side. In theory. It can be argued that maybe the reality is 40 stitches on a and 39 for b, depending on how you count...)

I understand that what I'm looking for is the hypotenuse of a right triangle. I understand that a2 + b2 = c2, which reduces to a + b = c. I understand that as I work on the problem from the other angle (casting on in the corner and adding a stitch every row) the preliminary numbers of stitches along side (a and b) vs. number of stitches on needles (c) is exactly a + b = c.

I also understand that when I worked a mitered version of the same square, I cast on a + b and then reduced 2 stitches every other row, so that by the time I was knitting along c, it was well under the sum of a and b.

And that's when the cognitive dissonance hits.

I would like to be able to provisionally cast on along the diagonal, reduce every row to one corner, pick up the live stitches, and reduce every row to the other corner. It would be easier, and thus more zen to do. I'm not doing too well at picking up the "knit in the bar" increase, although that's the only one that gives me the look I want (I've tried the classic knit-front-back and I don't like the look for what I'm doing).

However, I cannot cast along the diagonal until I know what the diagonal properly is...

Comments

[msp-hacker.livejournal.com]

a2 + b2 = c2 doesn't reduce to a + b = c.

As you can have a 3 x 4 x 5 right triangle:

3^2 + 4^2 = 5^2.
9 + 16 = 25.

And you can't have a triangle that a = b where c is a whole number either. But you can come close: 3 4 5, 20 21 29, 119 120 169, 696 697 985.

At least, geometrically. I don't know how to knit...

[neadods.livejournal.com]

And you can't have a triangle that a = b where c is a whole number either

Which is breaking my brain just a bit too, on the account that a = 40, b = 40, and there aren't any partial stitches along c. So the reality must fudge the pure math... but up or down? So far I've got 10 stitches on each side... and exactly 20 on the needle.

(Unless... I've only been counting stitches up one side. I wonder if the reality is a = 10, b = 9, c = 20.)

*sigh* My engineer father and brother would be having fits if they knew about this.

[msp-hacker.livejournal.com]

My father is a engineer. I feel your pain.