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

[doyle_sb4.livejournal.com]

I understand that a2 + b2 = c2, which reduces to a + b = c

It doesn't, though (unless I'm missing what you mean, which is entirely possible). Squaring both sides of
a + b = c
gives you
a2+ 2ab + b2 = c2
not
a2 + b2 = c2

c is the square root of (a2 + b2) - I know bugger all about knitting so I'm imagining the stitches here as lego blocks. If you want c to be a whole number you need a, b and c to be a Pythagorean triple - so if a was 8 and b was 15, c would be 17.

[neadods.livejournal.com]

It is way, way more likely that I'm buggering the math. I'm terrible at math, and long since lost the ability to create simple equations (which is proving to be a problem.)

On the other hand, I do know A and B, both of which are 40, creating a right triangle with two equal legs.

[doyle_sb4.livejournal.com]

In that case your diagonal is the square root of 40^2 + 40^2 = square root of 3200 = 56.5685 (to 6 significant figures)

[neadods.livejournal.com]

Which rounds up to 57, as the second decimal tips it to rounding up rather than down. (ETA: except in a later comment a knitter tells me not to.)

THANK YOU!

[doyle_sb4.livejournal.com]

Actually, I don't think you need to work out the hyptonenuse at all. Say your square is 10 by 10 - you're just dropping one stitch each time. Representing this with *s:

**********
*********
********
*******
******
*****
****
***
**
*

a = 10
b = 10
c = 10

[evil_plotbunny]

40 squared + 40 squared = the 3rd leg squared.

The square root of 3200 works out to 56.57 ish. If you need an even number go with 56, an odd go with 57. Knitting is forgiving that way. ;)

[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.

I can offer no help...

[lagilman]

except to say that you've broken my brain. Owssss....

(this is only one of the reasons why teaching me to knit drove my grandmother to sighing heavily. I can balance my checkbook, figure out recipe conversions on the fly, and navigate my way around a map, but abstract numbers are just Not My Thing. I salute those of you who can manage it)

Re: I can offer no help...

[neadods.livejournal.com]

You'll notice that I'm not one of the ones figuring it out. :> I know how many stitches to cast on the usual edges and how much yarn each piece will take. And that's about it!

Re: I can offer no help...

[neadods.livejournal.com]

ETA: I can use a map, but not draw one on the fly. There's a visualization circuit just missing.

[mtgat]

Not to muddy the waters further, but there's actually a factor you can use. For right triangles where A=B (45 degree angles like you have) you can use the meme "1, 1, square root of 2." The square root of 2 is about 1.4142 (or you can just simplify it to 1.4). Multiply whatever your side length is by 1.4 and there's your hypotenuse. In this case, it comes out at 56 even, if that helps.

(Also, for right triangles that use 30 degree angles, the meme is "1, square root of 3, 2." I don't think you'd be making rectangles, though.)

[neadods.livejournal.com]

I don't think you'd be making rectangles, though.)

Not in this manner; there are rectangles in the pattern, but I'm just casting on either the known long or known short edge.

Multiply whatever your side length is by 1.4 and there's your hypotenuse

*pounces on shiny, useful knowledge* THANK YOU!