PI Theory

Can anyone give me the formula for PI? I’d like to try my hand at designing one of these shawls, but I haven’t been able to find a source that explains the mathimatics of figuring PI. :??

well i thought PI was a mathematical constant, and that it is still trying to be figured to it’s correct decimal value, but now instead of mathematicians and philosophers doing it, we have computers (it’s gone from having 35 decimal places to over 100,000,000).

It is the expressed ratio between the circumference of a circle and it’s diameter, from what I remember from maths… and you would simply just use PI (as we currently know it 3.1416). So technically, i guess i’m trying to say, you can’t figure PI, or if you do, you’ll be spending an awful long time on it (someone spent their whole lives!), but you can use PI to figure things.

I probably haven’t helped you out much at all!

EDIT: to be clearer for you. PI will always be the same no matter what size circle you are using… you divide the circumference of a circle by it’s diameter, and you will get PI, but as this is always the same, this is why i said to use 3.1416 - here are some more complex formulas for pi:

Vieta’s Formula

2/PI = 2/2 * ( 2 + 2 )/2 * (2 + ( ( 2 + 2) ) )/2 * …c

Leibnitz’s Formula

PI/4 = 1/1 - 1/3 + 1/5 - 1/7 + …

Wallis Product

PI/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * …
2/PI = (1 - 1/22)(1 - 1/42)(1 - 1/62)…

Lord Brouncker’s Formula

4/PI = 1 + 1
2 + 32
2 + 52
2 + 72 …

(PI2)/8 = 1/12 + 1/32 + 1/52 + …

(PI2)/24 = 1/22 + 1/42 + 1/62 + …

Euler’s Formula

(PI2)/6 = (n = 1…) 1/n2 = 1/12 + 1/22 + 1/32 + …
(or more generally…)

(n = 1…) 1/n(2k) = (-1)(k-1) PI(2k) 2(2k) B(2k) / ( 2(2k)!)
B(k) = the k th Bernoulli number. eg. B0=1 B1=-1/2 B2=1/6 B4=-1/30 B6=1/42 B8=-1/30 B10=5/66. Further Bernoulli numbers are defined as (n 0)B0 + (n 1)B1 + (n 2)B2 + … + (n (n-1))B(N-1) = 0 assuming all odd Bernoulli #'s > 1 are = 0. (n k) = binomial coefficient = n!/(k!(n-k)!)

I think this may be a little too detailed for you tho :teehee:

You may have to actually get an EZ book for her PI calculations, I couldn’t find much online about using her technique (prolly cause the content is copyrighted)

Pi shawl are a form of binar math.

this is totally ‘standard math’ --but not usually applied (or workable to create circles!)in standard math.

but you have to remember that knitting has a 2:1 ratio (garter) and 3:4 ratio (stocking knit) and yet another ratio for the *K1, YO rows! and most formulas for creating a circle (on a cartesian plane) do not have these pecuilar ratios (but rather have a 1:1 ratio.
so things work out differently in knitting than they would on standard graph paper-(which is why knitters use knitters graph paper, with its rectangles, not squares!)

a basic Pi shawl "formula"
every time the number of rows doubles, double the number of stitches.
(start with 4 (you can start with other number, and a similar pattern appears, it is most evident with 4 (or 8)
cast on 4
row 1 4stitches
row 2 (you’ve double the number of rows, now double the number of stitches-8 stitches.
row 3 -8 stitches
row 4 (double again, so double again–16 stitches)
row 5, 6, 7, --16 stitches
[color=red]row 8 --double stitches (32 stitches)[/color][color=red]row 16-- 64 stitches[/color]

but if you just increase 8 stitches everyother round, you get to the same numbers (of stitches) at the same rows.

row1-- 4
row 2-- 8stitches
row 3-- 8
row 4-- 16stitches
row 5-- 16
row 6-- 24

row7-- 24
[color=red]row 8-- 32 stitches[/color]
row 9-- 32
row 10-- 40 stitches
row 11-- 40
row 12-- 48 stitches
row 13-- 48
row 14-- 56 stitches
row 15-- 56
[color=red]row 16-- 64 stitches [/color]

it is binaray/hexidecimal math at its most basic! You don’t even have to know or understand binary or hexidecimal --just look at row 8 or row 16 in each “pattern” in both cases, you end up with the same number of stitches.

both of these patterns work, because, in the end, the increases are the same!
NOTE: both patterns require YO’s for the increase, not another type of increase.

many, many circular shawls work with 8 increase every other round… and Pi (the knitting pi shawl) is a that ratio simplified.

(i am an amuteur math geek when i am not knitting)

[color=indigo]Gee, aren’t you glad you asked? :doh: [/color]

Thanks all. That helps a lot, more so than any of the sites I searched. Those that did give some form of explanation seemed to be talking out of both side of their mouth. It’d be great if there were a site that had downloadable patterns (that used Pi). Then the mork would be done for me. :teehee:

I got a book from the library a few weeks ago, and there was a PI shawl from Elizabeth Zimmermann. Math is not my thing at all, but this I could understand. Her explanation: It is just the circumference of a circle doubling itself as the radius doubles. For example: cast on 9 stitches and knit one round. Then, increase stitches to 18 and knit two rounds. Next, increase stitches to 36 and knit 4 rounds. (The number of stitches was increased by k1, yo) I really didn’t feel like making an entire shawl, only to have something go wrong, so I just practiced by making a doily. Pretty neat technique.