The Map is Not the Territory

When I found out about a new exhibit of African textiles and baskets I knew I had to go.

I was in the middle of the first flurry of exploring mathematical paper weaving inspired by Patrick Honner's guest post over at Moebius Noodles.  At the time, I was having fun experimenting with warps made from various combinations of multiples as well as the different patterns created by a weaving algorithm and its inverse, and using Fibonacci numbers to create interesting designs.  So, a chance to see traditional, often geometric, weaving designs up close and personal was definitely something I did not want to miss.

The Woven & Constructed exhibit was fabulous.  It was something else entirely to see these incredible, LARGE pieces hung with care, side by side, ceiling to floor. Photos are one thing, but these pieces had a presence that is hard to describe, but felt by everyone in the gallery.

Over the next few weeks I will be working on woven paper designs inspired by the motifs in some of my favorite pieces from this exhibit.  Here is my first design which is, I realized after the fact, inspired by a design that is pieced rather than woven.  This may have been why it was so hard to weave!

Here is a close up view of the picture above, a pieced design from the Democratic Republic of the Congo:
















Here is a sketch in my book, as I tried to figure it out (and a sneak peek at some other ideas):

























And here I worked to graph it out and figure out the basic pattern.  The weave pattern was 4 and an inverse of half (basically), and the warp was built on multiples of 4.






































And here is what the woven piece looks like: 

























Oh my gosh, it was SO hard to keep track of the six individual lines of weaving; I had to start over a couple times.  Working with slightly different paper also slowed me down a little. I had decided to use 1/4" strips instead of 1/2" so I could get more iterations of the design in the piece.  I liked the thinner strips but will probably like them better cut from colored copy paper rather than this high-end construction paper used here.

I am really enjoying this dual artistic/mathematical inquiry through paper weaving.  It's easy and satisfying to have an idea and finish a piece in a short amount of time.  And, every time I finish one piece I have questions that lead me right to the next, but different, piece. It's an exhilarating feeling, honestly.  This is exactly the kind of work students and I do in Math in Your Feet and I feel certain that this kind of double inquiry can be similarly structured for paper weaving, even as early as first grade.

What I find fascinating about this little project is that I used a different set of skills today than in previous weavings.  Compared to my previous efforts, there was a lot more analysis and a lot less "I wonder what would happen if...?" kind of thinking.  If I had to choose, I'd probably pick the "I wonder.." approach, but I can appreciate how reproducing other people's designs can inform both my design process and my mathematical understanding.  It was certainly a good challenge to move a design from the inspiration piece to the paper version. 

More to come!

p.s. Join us at the Math in Your Feet Facebook page where I share fun stuff like the math my daughter and I find all around us as well as interesting links to things I find around the web relating to elementary math learning...

Below is some experimentation with keeping the weft (horizontal) snug, in yellow, and a little looser, in green.  I love the look, but for a child's mathematical inquiry, I think it's better to keep both the warp and the weft snug, so the final design is as mathematically accurate as possible.

Since I now had some flexibility with the number of vertical strips I started wondering if the number would affect the ultimate design.  Curious about working with threes I started with a warp of nine strips. At this point I was just playing around to find a design I liked.  There's a basic reflection from top to bottom and left to right in each design.

























Here is another multiple of three, a six-strip warp -- a frieze pattern, I think!  The design is made possible because of the two-color warp.

It's also where I started of thinking about inverse operations.  Weaving technique requires the use of some combination of overs and unders.  Row 1, from right to left: [3-over, 1-under, 1-over, 1-under].  Row 2 & 3: both the inverse of Row 1, but with different colors.  Row 4: repeating Row 1, but in a different color.  Then repeat!  Even now it seems like magic.  I can't believe I figured this one out.


























Why do inverse operations matter?  From what I've read, and the number work I've done with my daughter, I know you can't really fully understand addition until you also understand subtraction.  Same for multiplication and division.  Add/subtract and multiply/divide are each two sides of same process.  It seems simple to our adult brains but it can be a very hard concept for a child to grasp fully.  I'm thinking that a focus on creating a weaving algorithm and it's inverse might really be a supportive numeracy effort. 

Here's another multiple of 3 warp:

























After this I tried a warp of four strips.  Each woven 'unit' is made up of three horizontal yellow strips.  I love how the red and the yellow are rotations of each other from left to right, and reflections of each other from top to bottom. 

Also, notice that the second unit is the inverse of the first.  Instead of having the first yellow strip go 1 under, 3 over, the second unit starts 3 over, 1 under.  Interestingly, I usually weave from right to left, but when faced with an inverse, I found myself weaving from left to right.  I did it every time.  It was quite fascinating to watch myself in this process, which is why I suspect this would be really great for kids.  There's so much to learn and understand as you work to create a visually pleasing design. 

























Here is a multiple of four, an 8-strip warp.  In this case, it's a basic over/under weave, but the two colored warp gives it some real variety.  I love this one.

























Another four multiple, this time putting together the previous two ideas: the weaving algorithm of the first and the two colored warp of the second.  I wasn't trimming the edges on my designs at this point, which I think affects how your eye sees the patterns.


























Now I was curious to see what it would look like all one color.  I think using green in both warp and weft brings out the structure in the weaving in a whole new way.  I like the green one untrimmed.

























And finally, onto multiples of five.  This one is nice...

























But this one is by far my favorite!
























The first row is double strips that go [1 under, 2 over, 1 under, 1 over] and the second row is its inverse.  So simple yet very effective.  A ten strip warp is interesting because it's comprised of two, five-strip units.  A nice juxtaposition of odd and even.

It was such a fun day!  I think I understand the symmetries and the inverse weaving at this point.  I'm not sure how or if building a warp out of multiples affects things.  If you have any insights or see anything in my post that is mathematically shaky please feel free to correct me.  And, if I can clarify anything, please do let me know if you have questions.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!

Patrick Honner's Moebius Noodles guest post about mathematical weaving has been in the back of my head for over a month and a half.  Mathematical weaving employs one of my favorite making materials - colored paper! - and I thought it would be fun to try with my seven year old.  I know the rudiments of weaving, but I wasn't sure how to get started, so yesterday I played around to try and figure out a few things.  It was actually sort of challenging, but I landed on some solutions and new grid games, so I thought I'd share.

I'm not done with my exploration, but what I have discovered so far is a perfect little unit for young children.  I am imagining that the weaving and the games would be completed in an enjoyable collaboration between adult and child over the course of a day or two.

I started by experimenting with loose 1" strips of paper as the warp (vertical strips) but soon found that much too unwieldy for even my adult hands.  The pieces were not connected to each other so they slipped all over the place and I had to use a lot of tape to keep the weft (horizontal strips) connected to the warp, which wasn't ideal.  So, I searched for some advice on how others have made paper weavings.  A quick Google search and I found this video (which is cool, but still too persnickety for the young ones) and this video which, although the cutting is somewhat haphazard, led me to a solution for how to weave paper without tape...

I first decided that a 3/4" width for vertical and horizontal strips made a more pleasing final product to my eyes than 1".  To make the vertical strips I folded a piece of paper in half and used my paper cutter to cut 3/4" strips from folded edge to about 3/4" away from the open edges closest to me.  Essentially, I was creating a paper warp that was still basically one piece of paper.



















As you can see, below, the horizontal strips weave in very nicely and don't need any glue or tape to keep them in place if you focus on pushing them gently, but snugly, downward.  For the young ones, at least, a basic over/under/over/under weave is challenging enough.  Using two (or more?) horizontal colors creates visual interest and perhaps even a conversation about the patterns you see: alternating colors both vertically, horizontally and diagonally.  You can also make a connection to odd and even numbers.  Yellow squares in the design show up 2nd, 4th, 6th... places.  Green squares are 1st, 3rd, 5th...

























The minute I finished the piece I thought - A GRID!  It's a grid!  The Moebius Noodles blog is very inspirational and a great source of grid games (my favorite so far is Mr. Potato Head is Good at Math) and I always have grids at the back of my mind these days because of them!  Here are some of the ideas I came up with using a newly woven paper math and one of my favorite math manipulatives -- pennies!

Adult: Oh look!  There are three different colors of squares in our woven grid.  I've got some pennies -- I wonder if we could make a square by putting pennies down on only one of the colors?

























Adult: That does look like a square. Let's count and see if there are the same number of little squares (yellow, blue, yellow, blue...) that make up each side?  There are!  How many little squares are there on each side?

Adult: But, wait! Look what happens when I push a corner penny in toward the center!  Yep, it lands on a green square!  Let's do it with the rest of the corners and see what we get.  Oh, lovely.  A rhombus.

























Adult:  The corners on the rhombus are on the yellow squares.  I wonder what would happen if we pushed them one square toward the middle?  Ooooh, look!  We have another square.  Is it bigger or smaller than our first square?  Each side on our first square was six little squares long.  This square has sides that are...three little squares long.  Cool.

























Another exploration:

Adult: Here's a little story about a tiny X who wanted to get bigger.  Can you help him figure out how to help the X get bigger?


























Or, how about the tale of some square numbers who also wanted to get bigger?  What little kid doesn't want to grow up?

























And, here's my favorite.  It's a 'let's make a rule' kind of game.  The first penny goes in the bottom left hand corner, and you start counting from there.  The first rule here (pennies) was two over, one up.  Each time you repeat the rule, you start counting from the last token on the grid.




















You're probably wondering about the buttons?  Well, that's a different rule: one over, one up.  Isn't it cool how they overlap, but not always?  Kids can make up their own rules after a little modeling or you can challenge them to guess a rule you made up and keep it going. 

And then, of course, the final thing would be to leave the pennies and the paper grid mat out to explore at leisure.