Architecture on the Radio

I missed the broadcast of the Radio 3 programme called The Secret Mathematician so I caught up with it on iPlayer.  http://www.bbc.co.uk/programmes/b06yrr34  

The first episode featured Marcus du Sautoy talking about the mathematics deployed by various architects, and in particular by Zaha Hadid, creator of the Aquatics Centre, built for the London Olympics.

Initially I was frustrated and annoyed by the programme:

Architecture and its links to mathematics – on the radio?  That would be architecture, a visual/tactile art, being talked about on an aural medium?  A mathematician talking very fast and enthusiastically about geometrical shapes and other parts of mathematics and linking them to particular buildings?  Buildings that I could not see?  Grr!

It seemed to me that this would work far better as a 45-minute TV show (which could show footage of the building and graphics of the mathematics and could then link them together), rather than being a 14-minute radio programme. 

In my frustration I started browsing for images of the different buildings that were mentioned.  And then I searched for more information about the mathematics.  I paused the programme a couple of times, found an image and then returned to the programme.  I searched for more information about Hadid and her other designs. 

Then I did the same for episode 2 (Art).  I had been dimly aware that Jackson Pollock’s paintings were fractal in nature, but didn’t know the reasons for this.  I also saw paintings that I had never heard of and found out more about the mathematics included in certain works of art.

I suddenly realised that this had turned out to be a brilliant piece of programming.  I had engaged so much more deeply with the subject-matter because it had appeared on the radio.  Had this been a TV programme I wouldn’t have had to have searched for so much information, wouldn’t have seen so many different pieces of art (whether architecture or paintings) and wouldn’t have found out as much about the mathematics involved.  Sitting at my laptop listening to it on iPlayer turned out to be a far richer experience than I had expected.  (It wouldn’t have worked had I been driving, which is why it was imperative that it was available online!)

Year 10 Homework

I explained the story above to my Year 10 class, and gave them a half-term task to listen to one (or both) of the first two episodes and then to respond to it by searching for information (if necessary) and writing about something they were interested in.

Here are some of the things they did:

1. Drew a diagram to show how squares with sides that are Fibonacci numbers fit together to make a rectangle (he did this freehand and then wrote that it seemed obvious – which suggests he didn’t Google it).
2. Printed a picture of the Aquatics centre and explained why the angles in a hyperbolic triangle don’t add up to 180 degrees.
3. Explained why Pollock’s work resembled that created by a double pendulum.
4. Explored other fractals, such as Sierpinski’s Triangle
5. Printed out Corpus Hypercubus and discussed the net of a 4-dimensional shape.
6. Extended the idea of a ‘net’, starting with a line of four segments being the ‘net’ of a square.
7. Searched for several different formulas for ϕ (the golden ratio).  These included one that involved sqrt 5 and a continued fraction (which wasn’t familiar to me).
8. Explained a recursive formula that generates the Fibonacci Sequence.
9. Explored Platonic and Archimedean solids.

I gathered together scans of the different ideas and we spent half a lesson looking at these. 

The Sierpinski work mentioned recursion and some interesting sequences.  We returned to these later in the term when we studied recursive sequences.  The recursive statement of Fibonacci’s sequence was also useful at that point.

We were able to use our knowledge of quadratic equations to find the exact value of ϕ, which none of them had done by themselves (they had looked it up) but which was accessible to them and rather satisfying.

The continued fraction for ϕ was this:

It turned out to be particularly interesting, because the boxes shown below all contain exactly the same thing. 

This means that it is self-similar and provided a neat (and entirely unexpected) link to one of the definitions of fractals that had been used.

Conclusion

This initially annoying set of radio programmes has turned out to be real gift.  I got lots of out of ‘listening to it with a search engine’ and so did my students particularly when we later studied recursion.