As in, "The Hitchhiker's Guide". Sandra says, "Rule of thumb: number of shafts, squared, less 2 = number of possible combinations of shafts that can be lifted." You don't want to know how many times I tried to figure this out.
EDIT: I made a flippant remark here and in the comment and now I have to dig up some old notebooks in the margins of which I may have made comments/observations, since I used to think about this a lot. Although there is also the possibility I got rid of those notebooks; we'll see.
Meanwhile, I tended to see shaft lifting combinations as combination with no repetitions. Today, Thursday, while rechecking the formula, (one of very few I remember from junior high,) I found a website that included information on Pascal's Triangle Cally mentioned. So I add the link for your reading pleasure.
I'm relieved Sandra has given us the formula, but I would like to try and recall if I was so off track or pretty much on course. Not that that's gonna make much difference to how or what I want to weave.
Yikes.
9 comments:
With 4 shafts, it's easy:
1, 2, 3, 4
12, 23, 34, 41
123, 234, 341, 412
13, 24
If you can think of any other [different] combinations, please do let me know. As far as I'm aware, the "rule of thumb" holds true for any number of shafts.
Happy counting!
I think I was thinking of 8. I think I worked out the 4 bits.... Uh huh.
Hold on, I just checked the notebook that hadn't been touched for a few months. I don't know what motivated me, but I've been figuring out different ways to fill in the squares consisting of 4 x 4 grid, but I can't remember the rules I set for myself at the start. How odd, I may have been making up components to use to fill in the 16 by 16 tie up plan. Or some other big idea... I can't remember now.
I think you want Pascal's triangle but without the '1's at either end (since lifting all shafts or no shafts is not an option). In the case of 4 shafts, as Sandra has shown above, you have 4 ways to lift one shaft, 6 ways to lift two shafts, 4 ways to lift three shafts - see row 5 of the triangle. Row 4 (1-3-3-1 but without the ones) applies to three shafts, row 6 to five shafts and so on.
My brain hurts...
Did not know about Pascal's Triangle until this morning, though his Wager has troubled me since I was in the Convent School. I have to go away to my time out space and think about what I thought on the subject. LOL.
But you have to remember, this isn't about weave structure - there's no "twill" rule, just the mathematical analysis of how many combinations of 4 digits can be made. I never said all those combinations made "good cloth," just that one could conceive of those combinations. Only a fool would try to actually use all those combinations in a single cloth. If you think about the 8-shaft combinations and using them all together, the cloth would end up ultra-firm (unbendable, if you will) in some areas and ultra-sleazy in others. In my presentation, the only goal was to educate nonweavers about how the complexity of the loom can sometimes have an impact on the complexity of the cloth that can be created in a loom-controlled way. In the real world, only the weaver's sensible choices, using a small subset of the apparently almost infinite choices (as in jacquard) results in "good cloth" that is pleasing to the observer. Just because you have a complex loom doesn't make the cloth superior to cloth made on a simple loom; only the weaver makes superior cloth, not the loom.
I think you misunderstand me. I've always been interested to see if I could figure out the math. (Then again you may not know the distance or hours I would go to see if I can figure something out. I was a Philosophy major. Until I decided I wanted to get out of blizzard Minnesota one bad Friday morning, and discovered I had enough credit if I switched to English major.) It's bothered me for years I couldn't figure out the math for 8.
I don't think I could be bothered creating a draft using all possibilities for the heck of it, much less weaving it. But then again, you never know; you might have given me a seed of an idea.
Actually, I did use all 14 in this series, http://www.megweaves.co.nz/2012/03/prayers.html, at least in the early stages.
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