Sunday, 12 July 2015

#tiling again

Well, John Golden has posted about his pattern block explorations, so I will too. Along with Dani Ruiz Aguilera he should accept a little of the "blame" for me spending so much time shuffling small shapes around. Dani has been doing some industrial-scale pattern block workshops himself:
I don't know whether I should put this post here or over on seekecho.blogspot.fr. I put my own things over there usually, including some of the things I've done with pattern blocks. But, although I've kind of "got the bug" with this, hopefully there are spin-offs for the classroom, like Teflon in space travel and the kitchen.

I've posted about some of the work I did in the class with a particular set of pattern blocks, the square, the hexagon and the thin white rhombus, already.

We've used Math Toybox's great Pattern Block tool - and a great thing about this is that you can save your creations onto the gallery, and even edit them again later on. John used it to respond to what we'd done in class:
I liked it that there were some interesting rules for these line patterns:
We looked at that and one or two other things, and then I asked them to work in bigger groups to extend one of them:
Meanwhile, something was intriguing me: a non-periodic, almost free-form, tiling using these shapes:
It has a really interesting balance between being forced to put the tiles in certain places, and having some freedom. To start with, if you begin with a hexagon, there are a lot of different ways of surrounding it. Here are some of them:
Similarly, there are a lot of ways of surrounding a square (finding them all might be an interesting task). You can see I've restricted myself. I never put two rhombuses against each other. Or two squares. I've got a lot of questions about this, and here are some of them:
  1. It feels like it would carry on forever in all sorts of ways. Can we prove it does?
  2. Once there's a "line" of squares and rhombuses, the line won't go away. What are the rules for it's behaviour?
  3. I've got an intuitive feel for an algorithm for making it carry on. Can we write down an algorithm for making sure it continues to grow without getting into any "impossible" situations?
  4. It feels like this could make a good game. What would the rules be?
  5. You can get areas of different regular tessellation - how many kinds of these are possible? 
  6. Can we - it feels like we can - create writing or pictures with this?
  7. Can I have a large public space, St Mark's Square in Venice for instance, to tile in this way? Or failing that, a playground?
Meanwhile, I was exploring other patterns:
Another thing that caught my attention was dodecagons.
Those holes can hold a dodecagon:
These can "point" in one of twelve directions. The idea of embedding these pointers and maybe other dodecagons, in the tessellatin grabbed me, and for a while I thought about how numbers can be patterned in this kind of space.
Here are some rotating dodecagons embedded in a simpler matrix:
John meanwhile had created his own beauty:
(At the same time, I've been reading Roger Penrose on, among other things, his work on non-periodic tiling. I guess I'm thinking about those 17 wallpaper groups, and trying to push other symmetries embedded within a simple translation.)

On a more practical note, I saw that you can get pattern block stickers!
These would really help the students recording a creation directly into their books. And something they could do this for - why hadn't I thought of it before? - popped into my head: pattern block equations:
 As Mary Pardoe tweeted:
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