On 18th May I was lucky enough to get involved with my first RBS Museum Lates at the National Museum of Scotland. These events happen about 3 times a year and are a chance for the (over 18) public to come back into the museum after hours and to get cosy with the exhibits with a cocktail and live band. It’s also a chance for science (and arts!) communicators like me to run an activity and get some surreptitious education into the evening.
The theme for this month’s Museum Late was “A Night in Wonderland”, so there were lots of top hats, white rabbits and red queens! (See lots of photos of the event on the Museum’s Flickr page.) Knowing that Lewis Carroll (real name Charles Dodgson) was a mathematician and logician as well as nonsense-poem writer, it seemed wrong for there not to be a mathematical component to the evening so I got together with Madeleine Shepherd (from ICMS) to brainstorm some ideas…
Our first idea was to get the public to make some Fortunatus’ purses. A Fortunatus’ purse appears in the novel Sylvie and Bruno by Lewis Carroll and is based on the old tale of Fortunatus, who has a purse which replenishes itself with money as often as coins are drawn from it. If you read the book you’ll find instructions for making such a purse by sewing together the edges of 3 handkerchiefs in an unexpected way.
The mathematical object created is one which has no inside or outside – it is called non-orientable, and is (of course) not possible to make in 3 dimensions without part of the purse intersecting itself. Some of you may be thinking that this is a Klein Bottle, but it is actually a different creature called a Projective Plane.
However, whilst doing the practice run for the purse-making, we found that it took quite a long time, was fairly fiddly and would involve giving drunk people sharp needles. Probably not the best idea. (But we might do this in a future maths/craft event!)
So instead we came up with the “Snark Constellation Challenge”, inspired in equal parts by the Lewis Carroll poem The Hunting of the Snark and by a mathematical object in graph theory called a snark. Visitors were invited to play a game which involved colouring the lines between stars in a constellation, and were challenged to colour the lines using only 3 colours.
There were two games the visitors could play: working collaboratively to find a colouring of all the lines, or working competitively to be the last person to draw a valid line. Have a go at the puzzle and see if you can colour the lines before reading on!
Hopefully you had a go at colouring in the picture and didn’t get too frustrated when you didn’t succeed. The reason you failed is because the picture can’t be coloured with only 3 colours: it is a special type of graph called a snark, whose defining feature is exactly that it can’t be coloured in less than 4 colours.
Snarks were invented back in 1880 by the Scottish mathematician P.G.Tait when he was trying to prove the more famous problem, the Four Colour Theorem. This states that any map can be coloured with at most 4 colours so that no two adjacent regions share the same colour. It was a problem that remained unsolved until the 1970s, and even now remains controversial as we have never found a proof that does not involve using a computer to check cases. Tait’s idea was to change the question about maps into a question about graphs, and showed that the Four Colour Problem was equivalent to showing that no snark was planar; i.e. that no snark could be drawn on paper without at least two lines crossing over each other.
You’d probably be surprised to hear that the first snark was not discovered until 1898 – nearly 20 years after the definition was made! The name of ‘snark’ (made by Martin Gardner many years later) was given because in Lewis Carroll’s poem, the snark is an elusive creature which disappears as soon as anyone sees it. It seemed a fitting description of these strange mathematical pictures which were extremely hard to discover! Even today there aren’t so many known snarks. You can see most of them on this page. And there are still open questions about snarks waiting to be solved, such as the Cycle Double Cover Conjecture.
Going back to our museum event, we wanted our visitors to have some artistic fun as well as learning deep and exciting maths, so we set up a table with a “Snark Hunters’ note book” and invited guests to draw their impressions of what a snark might look like.
It was a really great evening – we had a lot of fun and met lots of interesting characters! I shall leave you with just one more photo of our intrepid snark hunters. Contact us if you ever need help with your own snark hunting!