Pascal's Triangle of Doom

[velvetpage]

I'm sure there are more than a few people on my friends list who think Pascal's Triangle spells doom without adding anything to the mix, but bear with me.

My student teacher and I took a grade four activity about probabilities in coin tosses (what is the likelihood of throwing two heads on two coins? How about three heads on three coins? Two heads and one tail on three coins?) The probabilities with coin tosses just happen to fit in beautifully with Pascal's Triangle, one of the more famous (and complex) number patterns out there. The National Library of Virtual Manipulatives" has an excellent starting point a ways down that page.

Well, it occurred to me that this might make an interesting quilt pattern. Could I choose colours that would reflect certain multiples and go from there? I realized pretty quickly tht making it in the shape of a triangle was at best problematic, but I could make it more like fraction strips (which I'm sure you can find if you poke about on that website a bit) so it would be a rectangle cut into fractions to represent the numbers in the pattern.

Well, so far, so good. The problem came when I realized I didn't just want to divide the strip in half, then in quarters, then in eighths, with the numbers of the sequence in each piece. What I wanted to was to represent the numbers themselves as part of the fractions. So, the zero line is one whole strip; the first line has two ones, which can be represented as halves of the strip; but the second line is 1, 3, 3, 1. I don't want to represent that as quarters; I want to represent it as eighths, where the two threes in the middle are three times larger than the ones at the side and set apart from each other using colour or texture. The next line after that is 1, 4, 6, 4, 1; that's a total of sixteen. The ones can remain the same colour as the other ones to give the impression of a triangle as I descend the quilt, and I kind of want to bring multiples into it, so the six would have to be a similar colour scheme to the two threes in the line above, and the two fours would have to be totally different.

It then occurred to me that the possibility of pulling colour theory into this exists in primary, secondary, and tertiary colours (though it gets a bit hazy after that and may need to be accomplished with texture and multiples of those colours.) So, if one is black, I could start with red for multiples of three and blue for multiples of two. So the zero strip would be black; the first would also be black but halved using texture somehow; the second would be black, redredred, a different redredred, black; the third would be black, royal blue in a four square, purple (for 2x3) in six parts, blue again in a four square, and black.

The next row has another prime number, five, which then needs to be - yellow, I guess, to use up the primary colours? - then there are tens in there so yellow + blue = green. The next row has six again, so purple, followed by fifteen, which is 5x3 to yellow + red = orange, then twenty which brings in gray for the first time, multiplying 2x3x5, or all three primary colours. Or I could skip the primary colours for five and go straight to a tertiary colour to make it more interesting.

My head's starting to hurt but I just CAN'T STOP.

Can anyone else follow this at all?

Comments

[mrs-dm.livejournal.com]

I'm still stuck on the Girl-O's description of "Cookie Math". The kids are each given 4 (paper) cookies and told to divide them up so that 6 people can partake of the cookies (not necessarily equally).

The exciting part was, "Liam ate the cookie, even though it was paper!"

Too cute.

[velvetpage.livejournal.com]

That's pretty advanced fractions for grade one. I approve!

[kisekileia.livejournal.com]

I get it. It sounds like it would get very complex very quickly, possibly too much so to be doable, but it would be very interesting.

[dornbeast.livejournal.com]

I had to open up a spreadsheet and build a triangle to make sense of it, but it does make sense. I don't think you could get in too deep, since it's going to get into very small pieces somewhere around line seven or eight*, which should limit the complexity.

* - Well, I think it'll get into very small pieces at that point. I have no idea how large a quilt generally is, so I could be wrong about this.

[velvetpage.livejournal.com]

I was hoping to go to about line nine, but I hadn't actually worked it out to see if I could. I may try it on graph paper with pencil crayons today. It occurred to me that, rather than working with primary and secondary colours, I could work with colour shades for multiples; so three would be red and six would be, say, burgundy. It would keep the colours a little more manageable, at least.

[dornbeast.livejournal.com]

I made my comment based on a theoretical 256 cm quilt. ("Assume a perfectly spherical cow...") I'm not sure when working with fabric starts getting unmanageably small, so I arbitrarily decided that it was somewhere around the 1 cm range.

Hmm. Now that I think about it, you might be able to go to line nine.

[hendrikboom.livejournal.com]

I can't understand how you're representing the triangle if it goes exponential.

[dornbeast.livejournal.com]

Could you expand on your question?

[hendrikboom.livejournal.com]

I reread the original post, and discovered I misread it. The sums of the values in the rows indeed do go exponential -- powers of two, in fact. I had missed that in each row you have sizes of quilt pieces proportional to the value of the number.... And the ones at the ends are going to be the smallest pieces, in successive rows they are exponentially smaller.

In other words, I missed that you weren't going to use triangle geometry for the triangle.

[velvetpage.livejournal.com]

I've pretty much decided that it would be far easier to knit this, maybe even knit and felt it, than it would be to sew it. That way the pieces can get as small as one stitch. The fractions are much easier to do that way.

[velvetpage.livejournal.com]

It doesn't end up looking much like a triangle. Mind you, the graph paper representation only got to the fifth row before running out of squares widthwise.

I've decided upon mature reflection to knit it in pure wool and felt it instead. It still won't look very triangular but it will look pretty damn cool.

[hendrikboom.livejournal.com]

Do the Pascal triangle modulo 2:

1
1 1
1 0 1
1 1 1 1
1 0 0 0 1
1 1 0 0 1 1
1 0 1 0 1 0 1
1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 1

You get a definite pattern.

Modulo 3

1
1 1
1 2 1
1 0 0 1
1 1 0 1 1
1 2 1 1 2 1
1 0 0 2 0 0 1
1 0 0 2 2 0 0 1
1 1 0 2 1 2 0 0 1
1 2 1 2 0 0 2 0 0 1

Interesting.

Doing it modulo 2 and 3 and combining colours by colour-mixing is equivalent to doing it modulo 6.

Now you've got me going.

-- hendrik

[hendrikboom.livejournal.com]

When I edited it, the preview showed me a properly centered view. Oh, well.

[velvetpage.livejournal.com]

This - this is cool. But I've never come across either of these before. Can you link me to an explanation?

I think I could make the Modulo 3 work, since it would have just three colours.

[hendrikboom.livejournal.com]

And modulo n would take n colours. How many colours do you have?

Explanation? It's just Pascal's triangle, with every number replaced by its remainder on dividing by three.

Now I wonder what four would look like. Or five. Or six? And how would four be related to two? And I think choosing colours related to how the numbers factor might show some structure. Modulo 6, I suspect you could choose white, pink and red for 0, 1, 2, (repeat for 3, 4, 5) and mix blue into the 3, 4, 5. Playing with the intensities of the promaries might make structure show up differently. Sounds like something to try on the CRT or mixing paints before you sew.

-- hendrik

[hendrikboom.livejournal.com]

And I got part of it wrong. Here it is again, with extra equals-signs stuff stuck in to make it align right. I won't be able to see whether I got it right until after I've posted it.

I hope the numbers are right this time.

-----------1
----------1 1
---------1 2 1
--------1 0 0 1
-------1 1 0 1 1
------1 2 1 1 2 1
-----1 0 0 2 0 0 1
----1 1 0 2 2 0 1 1
---1 2 1 2 1 2 1 2 1
--1 0 0 0 0 0 0 0 0 1 <- Hey! Another row of zeros!
-1 1 0 0 0 0 0 0 0 1 1
1 2 1 0 0 0 0 0 0 1 2 1

And so we have some new subtriangles which will grow like Pascal's triangle until they collide. Will they annihilate again? Will we have another row of zeros?

-- hendrik


Actually, you can imagine all space to the right and left of the triangle to be filled in with zeros, which might give a clue how to fill it out if you want a square quilt/