Look at a football. It is made out of 32 distinct patches or “faces”, some black, some white. The borders or edges separating the faces number 90. The vertices, or corners, or meeting points of edges, number 60.
The number of faces plus the number of vertices is pretty close to the number of edges. It’s just two more. 60+32-90=2.
It makes sense that the more faces you have, and the more corners (vertices) you have, the more edges are needed to mark them off and connect them.
If we modify the football, the calcuation still gives the answer 2. Divide each face into two by drawing a new border across it, from one corner or vertex to an opposite one, and we add 32 faces and 32 edges. Faces plus vertices minus edges still equals 2.
Divide each edge into two halves, creating a new vertex or meeting point in the middle of it, and we add 90 edges and 90 new vertices. Again, faces plus vertices minus edges still equals 2.
Or merge two faces into one by wiping out the border between then. One less face, one less edge, so faces plus vertices minus edges still equals 2.
Suppose the “football” is made of super-Play-Doh which can be stretched or squeezed as much as we like. If we just stretch and squeeze, without ripping or cutting, each face will remain a face, each edge will remain an edge, and each vertex will remain a vertex, though their sizes and shapes will change. Faces plus vertices minus edges will still be 2.
Is it always 2, however much we change the pattern?
It can be mathematically proven that we get the number 2 not just from the relatively neat pattern on a football, but from any network drawn on a sphere of faces (spaces which are triangles or combinations of triangles), separated by edges which meet at vertices. In fact, for any such network drawn on any shape that a sphere can be stretched or squeezed into, without ripping or cutting.
Take another look at the football. The black faces are pentagons, with five sides, and the white faces are hexagons, with six. Couldn’t the designers have made it more uniform? All neat pentagons? Or all neat hexagons?
Again, mathematics helps. If we want the faces to be all the same, and all neat and regular, then there are only five possible patterns, all with many fewer faces than the standard football.
The area of maths which tells us such things is called topology. It doesn’t just tell us cute facts about footballs. It is an essential tool for research into the shape of the universe, for example, and for the design of robots.
On 12 March Aditi Kar from Oxford University and Ellen Powell from Cambridge University came to CoLA to tell us a little about that sort of maths, and one of the chief figures in its development, Emmy Noether.
Emmy Noether was to 20th century algebra what Einstein was to 20th century physics, yet in her time, because of sexist prejudice, she was unable to get more than a casual teaching position at a university. (And then, in 1933, the Nazis threw her out of Germany altogether, because she was Jewish and left-wing).
These days there are many talented young women, like Aditi and Ellen, with jobs in mathematical research; but still young women are underrepresented in mathematics. Recent investigations suggest that this is entirely because girls and young women often feel less confident about maths, and confidence is more important to success in maths than in other subjects. (See here and here.)
That’s one reason why we’ve taken trouble to help CoLA students learn about the contribution of women in mathematics. On 8 January, Serafina Cuomo from Birkbeck College, and June Barrow Green from the Open University, came to CoLA to lead a session on that.
June spoke about the life and work of Sofia Kovalevskaya, the first woman in modern times to become professor of mathematics at a university, and led a session on the work of women in code-breaking in World War 1 and World War 2.
Serafina spoke about, and led a session on, the way that bits of mathematics like “Pythagoras’s theorem” have different meanings depending how you approach them.
On 3 November, Tim Gowers, professor of mathematics at Cambridge University and a former winner of the Fields Medal (the equivalent for maths of the Nobel Prize, but awarded only once every four years), was at CoLA on Monday 3 November to lecture and lead a classroom session on “What is mathematical proof?”
After the session, Tim Gowers told us that he believes that organising such beyond-the-syllabus events at state schools in worse-off areas is important for social equality. Otherwise access to real mathematics, beyond the exam boards’ “remember a formula, stick in the numbers” stuff, is restricted to students at posh schools.
Students and teachers from CoLA Hackney came to join us for Tim’s session, and Serafina and June’s; students from the City of London School for Girls also came to Tim’s. We’re pleased to have been a centre for giving their students access to real maths, beyond the syllabus.
At CoLA we make sure we cover the exam boards’ grind, training students carefully for good results in their exams. But we try to do a little more. As the old textile workers’ strike song puts it, we fight for bread – but we fight for roses too.