5. The Golden Rectangle

‘Quick, Alan. We’ve got to go NOW or we’ll lose them. They’re only a tenner each.’

I didn’t question Sheila further. We jumped into the car and sped off in the direction of our local second-hand furniture shop. What she had spotted was a collection of ammunition boxes – leftovers from World War II that probably originated from the US army. But of course, what Sheila was after was wooden boards for her paintings. 

Normally Sheila paints on commercial canvases. She was delighted to have the opportunity to paint on wood but also to be free to make this wooden ‘canvas’ to any size and shape she fancied. Now, as I am a big fan of a particular rectangular shape – the so-called ‘golden’ rectangle’ – and also because, I was the one wielding the saw, I suggested that we should cut up the box lid to create a board of this ‘golden’ shape.

A week later, this little ‘Apple Blossom’ beauty was hanging on our kitchen wall.

Apple Blossom

This isn’t the only art object that Sheila has produced that was inspired by this particular 'golden' proportion. Here is her Golden Window, painted from a wrought-iron window frame in a crumbling fromagerie in Normandy.

Golden Window

So what exactly is a golden rectangle and why is it considered to be special? Have a look at this 3-minute video to get a quick overview.

Golden Rectangle

Maori drawings

A few years ago, we spent a wonderful two months travelling around New Zealand. I had arranged a deal with a number of NZ mathematics advisors that I would provide a series of ‘inspiring’ talks/workshops in return for a night’s board and lodging in each location. That way we had a great time meeting and chatting to many teachers and their families in their own homes. But there was also plenty of time for sight-seeing. We particularly enjoyed having the opportunity to explore Maori culture and art. 

On our return to England, we decided to mark the trip by commissioning, from my friend Roger the glass artist, a pair of small stained glass windows that celebrated the fusion of mathematics (upper) and Maori art (lower). 

The Golden Spiral

A Symbol from Maori Art

The mathematical one shows a logarithmic spiral (sometimes referred to as the ‘golden’ spiral) inscribed inside a golden rectangle. Here's a detail to clarify the mathematical part of the picture.

A Detail of the Golden Spiral

Starting out from the bottom left corner, the spiral takes a path between opposite corners of each square in turn. In order to construct it, Roger used compasses to form an arc inside each square which he drew as a quarter circle, centred at one of the other corners of its square. 

Golden rectangles have been known about and celebrated since the time of Euclid. He didn't use the term 'golden' but referred to this special ratio as the "extreme and mean ratio". It's sometimes said that shapes formed from 'golden' proportions have a particular aesthetic appeal. I'm not totally convinced by this claim but it would be lovely if it were true!. 

Anyway, this 'special' ratio is all very well, but it's not getting these lightbulbs sorted out ... .

Postscript

This post looked at the idea of a golden rectangle, which has proportions of approximately 1.618 to 1. The number, 1.618… is often referred to with the Greek letter Φ (phi). If you're interested in finding out where this number comes from and indeed other matters 'golden', have a look in my website, MathinPictures.co.uk

4. The Simpsons and Mathematics

I recently celebrated my n+1th birthday (yes, a year older…) and received from Sheila, ‘The Simpsons and their Mathematical Secrets’ by Simon Singh. Having already read several of Simon Singh’s books – Fermat’s Last Theorem, The Code Book: The Secret History of Codes and the fabulous Trick or Treatment?: Alternative Medicine on Trial – I was confident that this one would prove to be an entertaining page-turner. And I’m pleased to say that it is indeed a little cracker so I thought I’d share an extract from it in this post.

The Simpsons

Unless you’ve been hiding out in another planet for the past 20 years, you’ll know already that The Simpsons has been one of the longest running and most successful TV series in history. Singh's book explains how it came about in the first place.

Back in 1985, the British entertainer Tracey Ullman was having a successful run of shows in the US. Her sketches were performed before a live audience and many of them required extraordinary makeup and prosthetics. Scriptwriter James L. Brooks thought that it would give Ullman more time between sketches if they could include some cartoon animations. He met with cartoonist Matt Groening and minutes prior to this meeting Groening thought up the characters of the Simpson family. Later, some of the short animations featuring the Simpsons were pieced together into a longer show. To everyone’s surprise, they went down very well. Their first full-length show was ‘Simpsons Roasting on an Open Fire’ which was aired as a Christmas special on 17 December 1989. 

'Simpsons Roasting on an Open Fire'

Where’s the mathematics?

If you are a regular follower of The Simpsons, you’ll be aware of the many amusing mathematical references it has included over the years. This is down to the strongly mathematical backgrounds of many of the writers, some of whom gave up teaching and research posts in mathematics to join the writing team. Singh includes many examples of mathematical references but don’t be put off if your own mathematical background is a bit thin – the emphasis is just as much on the humour as on the mathematics.

Various mathematical topics are touched upon in the book but I’ll mention just one – the number known as pi. An episode, first aired in 2001 and called “Bye, Bye, Nerdie” is about the bullying of nerds, a question that Lisa decides to investigate. She discovers and isolates a pheromone emitted by ”every geek, nerd and four-eyes” that could be responsible for attracting bullies. She decides to deliver a paper on this theme to the ‘12th Annual Big Science Thing’. The conference is hosted by John Nerdlebaum Frink Jr, Springfield’s favourite absent-minded professor. Frink attempts to introduce Lisa but the audience is so excitable that he is unable to bring them to order. In desperation, he eventually calls out:

“Scientists… Scientists, please!  

I’m looking for some order.  

Some order, please, with the eyes forward…  

And the hands neatly folded…  

And the paying of attention…  

Pi is exactly 3!”

Suddenly the noise stops. Professor Frink has correctly judged his audience. Stating that there was an exact value for pi has stunned this audience of geeks into silence!

If you have only a hand-waving grasp of this number, pi, about where it comes from, or indeed, why this story is funny, you might like to look at the 3-minute video below where I have provided a cheap and cheerful explanation of pi and how it is linked to a fundamental question about what makes a circle a circle.

The Circle and Pi

Of course, this story just scratches the surface of the many gems contained in Singh’s book – definitely 5 stars from me.

Now on to weightier matters. How are those lightbulb operatives doing, I wonder.

Postscript

If you are interested in reading more stories about pi and this delightful book about the Simpsons, have a look here in my website, MathinPictures.co.uk.