Reflections on maths, learning and maths learning support, by David K Butler

Tag: art

  • Quarter the Cross: Colouring

    Quarter the Cross is one of my favourite activities of all time, whether in maths or just life. I learned about it way back in 2015 and have been mildly or very obsessed with it ever since. This blog post is about one particular version of the Quarter the Cross problem you might like: the colouring version!

    You can read the rest of this blog post, and four other related posts, in PDF form here 

        The titles of the five posts in the series are:

        • Quarter the Cross (2016)
        • A Day of Maths: Quarter the Cross (2016)
        • David Butler and the Prisoner of Alhazen (2016)
        • Quarter the Cross: Colouring (2020)
        • Quarter the Cross: Connect the Dots (2020)

        Some resources linked from this post:

      • Too much time on his hands

        On the train a while ago I overhead some people talking about Heston (the celebrity chef). Apparently he had been doing a series on giant food. It involves him trying to figure out the physics and logistics of trying to produce food on a giant scale – for example, a three-metre tall soft-serve ice-cream cone.

        After describing all the care and effort Heston took to produce this giant ice-cream, the first person declared, “He’s very clever.” Her friend’s response was, “He has too much time on his hands.”

        Clearly this person could easily do a better job than Heston if she wanted to, but she chooses not to because she has so many more important things to do with her time. Apparently Heston’s not clever, he’s just idle.

        I had a strong desire to lean over and ask her if JK Rowling had too much time on her hands, or if Stephen Spielberg had too much time on his hands, or Michaelangelo had too much time on his hands. If you think about it, what they did was more or less for their own enjoyment too and wasn’t “important” either.

        Of course, it wasn’t Heston I was really indignant about. The statement brought up several unpleasant memories when people have said this to my face when they have seen me making models of fractals, crocheting hyperbolic coral, drawing digits of pi on the pavement or solving puzzles, or even just doing maths in my own time. They seemed to feel that they needed to make those things seem trivial.

        Perhaps they felt cheated that they don’t spend more time doing things they actually enjoy. Perhaps they felt like I was making them look stupid and they needed to make me feel bad for it. Or perhaps they are just grumps who are unable to share in others’ fun.

        Now that my indignation has faded a little, I feel sorry for them. I remember what it was like to be in a situation where I felt it was somehow wrong to choose to do things I enjoyed, and it wasn’t a pleasant place to be. It can colour your view of the world and frankly it does make it difficult to enjoy other people’s fun.

        Still, it’s no real excuse for making people feel bad about things they have spent a lot of time acheiving. Sure, they may have a lot of time on their hands, but at least they are using it well!

      • The Road to Royalty

        Last week, I met His Royal Highness Edward the Duke of Kent. I’d like to tell you the story of how this came about.

        His Royal Higness was in Adelaide because he is the patron of the Royal Institution of Australia and was presenting an award to a scientist there. But it just so happened that the Hyperbolic Crochet Coral Reef exhibition was still on display at the Royal Institution during his visit, so he dropped in for half an hour to see it.

        “How nice for him,” you might say, “but how did you get to be there while he dropped in?”

        Good question! Well, less than a week before, the curator of the exhibition Julie had emailed me to ask if I would like to attend afternoon tea with His Royal Highness in the Reef exhibition. She said that I was one of the very few men who was involved in crocheting the reef, and moreover she would greatly appreciate the knowledge I would bring about the mathematical aspects of the exhibition. After making sure the students in the Drop-In Centre would be looked after, I said yes – and that’s how I got to be there.

        “But David,” you might continue, “how did you get involved in crocheting the reef in the first place?”

        Another good question! Well, let’s see… another RiAUS staff member Cobi who I happened to know dropped in to my office in February asking me if I knew anything about hyperbolic geometry, and also if I happened to have any 3D models that might be used as part of an art exhibition. And of course I knew about hyperbolic geometry, and I had a couple of ready-made models of hyperbolic quadrics rigt there.

        That was the start of a year-long involvement in the project. During the course of the year I wrote a poster for the exhibiton to explain the geometry, learned to crochet, crocheted about forty corals, and ran three crochet coral workshops. I was probably the most involved of the very few men who were involved.

        “Wow! You really committed yourself to this, David,” you are probably saying. “But how did you happen to know Cobi?”

        Ah! Yet another good question! Well Cobi had once worked as a tutor for a course called Research Methods in Media at the University. The very first tute was supposed to be refreshing the students’ memories of statistics, and she had come to me as the coordinator of the Maths Learning Service for some help running that tute. I jumped at the chance and we had a great time getting the students to draw graphical representations of data on the windows. It’s one of my first memories of working at the Maths Learning Service.

        “Cool!” I’m sure you are saying. “But why did she think you would know about geometry and have 3D models?”

        You are full of the great questions today! Well if you know me at all you’d know that I can’t help being excited about maths in general and geometry in particular, and also that I also have a collection of models of geometries that I pull out at every opportunity. So it shouldn’t really be a surprise that I did talk excitedly to Cobi about geometry, and show her the models when we met, even though my task was to help teach statistics.

        “But,” you must certainly say, “that doesn’t explain why you happened to have the models of quadrics there.”

        That is an excellent point. Well, I’ve always been a model-maker (I remember making things even in primary school). So it was a natural thing for me to try to make models of the quadrics I was studying when I came back to Uni to do my PhD. Towards the end of my first year of PhD I spent a few weeks making paper-mache models of quadrics, constructing the underlying structure with carboard and string. I have fond memories of sitting in the School of Maths tea room with my hands covered in maper-mache glue and paper and cardboard all over the table.

        Later, towards the end of my PhD, I ended up on a team designing interactive activities for open-day and the string quadrics seemed like a reasonable thing to get passers-by to engage in. When I went to the Maths Learning Service, I took the string models with me.

        Now you probably have more questions at this stage, but our conversation has been going on for some time now, and I think I’d better make some sort of point, don’t you?

        If you trace the story back, you’ll find that there are two reasons I ended up meeting His Royal Highness. The first is that I never shyed away from doing things that would be considered play – things like paper-mache and crochet. Other people were too busy or too embarassed to do this sort of thing, but that never stopped me. The second is that I was always willing to share my love of maths – with Cobi, with Cobi’s students, with the passers-by on Open Day, with the visitors to the reef, and finally with His Royal Highness himself.

        If you want to learn anything from this, then learn what I did: never shy from playing, and never give up sharing the things you love with others. You really never know what good may come of it if you do.

      • The Pied Mathematician of Hamelin

        Have you ever been in a situation and felt like you were reliving a scene from a book or movie? Well it happened to me the other day when I went to visit my daughter’s school. I felt exactly like I was the piper in the Pied Piper of Hamelin, because an ever-growing crowd of children followed me across the oval as I walked in.

        And why were they following me? Well I had brought a Stage 4 model of the Sierpinski Sponge with me for my daughter’s Show and Tell. She had come with me to university the day before to help me make Stage 6, and we thought the other kids would be interested – and my goodness they were!

        At this point I should probably tell you more about the Sierpinksi Sponge Project…

        The Sierpinski Sponge is a fractal constructed in the shape of a triangular pyramid. It has a giant hole in the centre, and each of the corners around this hole is a copy of the whole thing. This means that each corner is a pyramid with a big hole in the middle, and each corner of those pyramids is a pyramid with a hole in the middle, and each … and you go on like this forever until you have an object with a great number of holes (infinitely many in fact) – hence the name “sponge”.

        Of course, you can’t make a real Sierpinksi Sponge because it goes inwards forever; you can make a decent model though. What you do is you get four small pyramids and you join them together at the corners to get a bigger one with a hole in the middle. Then you take four of these bigger pyramids and you join them together to get a bigger one with a hole in the middle. And so on. The individual small pyramids are called Stage 0, then when you join four together you get to Stage 1, and then Stage 2, and so on. We made a Stage 6, which contained 4096 individual pyramids.

        (To see video of the Stage 6 Sponge and us making it, check out the YouTube videos: http://youtu.be/A7YbmITSck8  , http://youtu.be/W0uLbhRR-Hw  .)

        So back to the story… on the day after we made Stage 6, I walked through the school yard with a Stage 4 Seirpinski Sponge, and all the kids crowded around to see this remarkable thing and ask questions. And then in the classroom, the kids just couldn’t keep quiet with the questions and fought over who would be first to hold the smaller models. Their little eyes lit up as they imagined standing inside Stage 6, and imagined the awesomeness of Stage 7, Stage 8 and Stage 100.

        But it did make me think: We made our Stage 6 in a public place in the Uni where hundreds of people walk past a day. Several people looked at it as they walked past, and a few came close to touch and ask questions, and a handful of those actually stopped to help make it bigger. Before construction day, I had spent every train journey sticking pyramids together, usually reaching Stage 4 by the time I got to my destination. On one particular day I finished a Stage 5 on the train and carried the metre-wide pyramid it through the crowds. Not one person stopped to look or ask about it. Yet the schoolchildren couldn’t keep themselves away.

        When did all the adults lose their sense of wonder? Because even the the Stage 4 Sponge is truly wonderful to me (and to hordes of children too).

        What would it take for adults to crowd around like the children in the Pied Piper of Hamelin?