Can Someone Help Me With My Math?

Buck Strong's picture

I have been working on a few round flat pieces lately, baby blanket, doily, etc., and I started to think about how the increases are placed. I'm going to go through my thought process here.

1. If I cast on 6 sts and just keep knitting round after round after round, I will end up with a giant tube.
2. If I add increases, evenly spaced and every round or every few rounds, my piece will grow outwards and lie flat.
3. If I increase too little, my work will start to look like a bowl.
4. If I increase too much, my work will have ripples.

So, how do I know how much to increase and when? This is when my math teacher brain kicked in. It seems that there should be a way to figure this out with a formula. Here is my next round of thinking.

1. A lot of this type of knitting happens in sections, e.g. cast on 8 stitches and you end up working 8 sections.
2. The edge of one of these sections is the same as the radius of a circle. Think of a piece of pie; it is cut from the center of the pie to the outside of the circular pie plate. This cut is a radius.
3. Now, let's make a stitch a unit of measure. Follow my thinking here. A row is made up of several stitches side-by-side. The height of my work is the number of rows that I have made stacked on each other. Therefore, the number of rounds that I make is the number of stitches from the center of the piece to the outside edge or the radius. If I have a piece with 120 rounds then my radius is 120.
4. So, in a nice universe, 120 stitches should yield a circumference of 754 stitches ( C=2*Pi*r). However, knitted stitches do not have a 1 to 1 ratio. One stitch in height does not equal one stitch in length. Generally, you need 1.25 more stitches in height in order to match the length. This means that my circumference should be 80% of of the circumference stitches or about 603 stitches around.

Now dividing this by 8, for the 8 sections, I end up with about 75 stitches for each section at the end of my 120th rnd. I have gone from 1 to 75 stitches in 120 rnd; so, I have added in 74 stitches in 119 rounds or about .6218 stitches per rnd. Most patterns that I have looked at generally do their increases every other rnd. Following that idea, I would increase about 1.25 stitches every other round. Well, I can't increase .25 of a stitch. Of course this doesn't have to be super exact as the material will be forgiving. I could do one increase one time and two another as long as I end up in the ball park of around 75 stitches at the end.

Yeah!......Wait! In looking at different patterns, I'm finding that this is not consistent, meaning that the circumference stitches are not 80% of the radius stitches. What I have noticed is that none exceeds the 80%; however, several are much less than 80%. The majority seem to fall in the 55% to75% range with one or two as low as 43%.

I'm pretty sure that my math is correct and I can understand some deviation due to stitch types. However, this seems to be too much.

What am I missing here? It seems that there should be a straight forward mathematical calculation for figuring out the number of increases needed. Should I be looking at a range, that is, you can't do less than this percent for your increases and not more than 80%.

Thoughts?

Comments

jwhassjr's picture

Best I can surmise and

Best I can surmise and understand is this.

If you have a gauge that is 5 stitches per inch and 7 rounds per inch, then your first round mathematically is (1/7)*2*pi*5 (seven being your rounds per inch; let's call this "r" for rounds, and five being your stitches per inch; let call it "s" for stitches). Replace constants with our variables, and the equation looks like this "(1/r)*2*pi*s". Plug this into a spreadsheet with a column for rows starting with 1 and ending with say 32 and then set up a formula in the adjacent column to calculate the number of stitches in each round, you essentially will get the number of stitches with which you should cast on for the beginning round, and the number of stitches needed for sequential rounds that follow but notice that each time you double the rounds, i.e., round 2, 4, 8, 16, etc., you get a result that correlates to EZ's instructions for the pi shawl.

My opinion, EZ once again has got it figured out. Follow her advice and you can't go wrong.

rmbm612's picture

Elizabeth Zimmerman in the

Elizabeth Zimmerman in the Knitter's Almanac I think discussed this relationship in the Pi Shawl printed material. You lost me with your description almost immediately but math isn't my strong subject. I was impressed with your logic and ability to put that thinking into logical text.
If you have a copy of the Knitters' Almanac, check out the Pi Shawl story and see if the Matriarch of Knitting has a twist on your theory. Is always fun to read your posts.

steve kadel's picture

i have a problem for

i have a problem for you:

one train leaves the station at 4:15 heading to bakersfield averaging 45 mph. another leaves at noon heading to san francisco averaging 41 mph. which one serves cocktails?

answer: it doesn't matter. no one in their right mind takes a train to bakersfield

we won't just get a cat, nubby nu nu, will manifest a kitten from our love...and lint from our hemp socks

michaelpthompson's picture

That's the best answer yet,

That's the best answer yet, Steve.

--
"All knitting is just one stitch at a time."
Tallguy

jwhassjr's picture

LOL. I love your quick wit!

LOL. I love your quick wit!

steve kadel's picture

you lost me at math problem

you lost me at math problem :)

we won't just get a cat, nubby nu nu, will manifest a kitten from our love...and lint from our hemp socks

MMario's picture

The rule of thumb for MOST

The rule of thumb for MOST people (individual gauge or wierd yarn/needle combos can change this) is 4 stitches per round will give you a flat piece.

This is usually done as 8 stitches every other round, but can be 12 every third, or 16 every fourth....

But because knitting is inherently flexible you can usually get away with some differences

Placement of the increases means your piece may be circular, ar a specific polygon.

michaelpthompson's picture

I could have used this

I could have used this information when I was making a jacket for my niece for Christmas. The pattern only went to size 10, I had information she was 10/12, so I wanted to make it bigger. My math was not as good as either of yours, and I was lost trying to figure out how many stitches I would need to end with at the armhole, as the sleeves were supposed to be done from the cuff up. Most of the armhole was rows rather than stitches, in garter stitch, so I didn't know whether to end with one stitch per row, or every other row or what. Finally wound up picking up at the shoulder and going the other way instead. Less brain damage.

--
"All knitting is just one stitch at a time."
Tallguy

Buck Strong's picture

So, I am close in my

So, I am close in my calculations and that their are variations. I'm thinking that 4 stitches per round is a bit arbitrary. In looking at a lot of patterns, the increases tend to be in multiples of the original cast on. So, I cast on 6; all of my increases are multiples of 6. This is also true when I look at other patterns; their increases are multiples of the original cast on. However, just to note there are lots of instances when there are no increases for several rounds. This may bring the average to about 4 per round (I'll look into this one).
Looking at my example, of 120 rows, and starting with 8 cast on, (using 4 inc per rnd) I end up with 488 stitches. If 603 is 80% of the radius, 488 is about 67%. Yup, MMario you rock; that's just a bit more than the 10% threshold that you were talking about. This fits in with what I have been observing. The majority of increases are between 55% and 75%. Again, dropping down to 55% probably takes into consideration materials and blocking before construction begins.

Don't you just love math? :)

To sit with a dog on a hillside on a glorious afternoon is to be back in Eden, where doing nothing was not boring-it was peace.
~Milan Kundera
www.canzonestandardpoodles.com

MMario's picture

looking over your original

looking over your original premises - I would challenge the1.25 to 1 ratio of rows to stitches. I think most knitting falls more between 1.6 to 1 and 2 to 1.

I'm not sure how that changes your figures.

the 10% I mentioned was 10 % from the theoretical of 4 per row.

Buck Strong's picture

I got the 1.25 to 1 as an

I got the 1.25 to 1 as an average of gauges from yarns from lace through sport. Those numbers I got from a general discussion about yarn weights. Maybe I'm getting to much variation due to too much averaging. I'll try it with your numbers and see what I get.

To sit with a dog on a hillside on a glorious afternoon is to be back in Eden, where doing nothing was not boring-it was peace.
~Milan Kundera
www.canzonestandardpoodles.com

MMario's picture

yeah - I got mine from what

yeah - I got mine from what tends to work when picking up stitches along an edge.

MMario's picture

I try not to go more then 10

I try not to go more then 10 % UNDER but you can go almost 100 % OVER the theoretical number per round depending on whether you are continueing to work the piece or not.

TomH's picture

One easy solution is the Pi

One easy solution is the Pi formula pattern for circular knitting where you consistently double the number of stitches at the appropriate round. But may not be a solution that interests you since it doesn't include continuous increasing. NOTE: This is where you step in, Mario.