A South African teacher, known on Twitter as @Pyfagoras, sent me this photo of his board. It shows how to turn a cube into an Eulerian graph by adding in the diagonals across each face of the cube. In an Eulerian graph, every vertex has even degree (that is, an even number of edges coming out of it) and it is so called because Euler proved that for such graphs it is possible to start anywhere and traverse every edge in the graph exactly once, returning to the start. Furthermore, such a circuit is impossible if the graph contains a vertex of odd degree. For the cube on the board, there are 6 edges coming out of each vertex, meaning it is not only Eulerian, but also a regular graph. You can read more about it in its incarnation as the 16-cell.
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Network flow analysis
This is Julian Hall‘s whiteboard. Julian works in the area of optimization, and he is trying to figure out better ways to implement the simplex method to solve very large linear programming problems. Network flow problems are special cases of linear programming problems; for example, how a company should organise deliveries from various warehouses to various shops, or how a company should organise its workforce to complete a set of activities most efficiently. On the board here, these network problems are being analysed to understand why the simplex method sometimes runs much faster when the objective costs of a linear programming problem are perturbed randomly.
If you have access to a black or white board of interesting mathematics, please take a photo and send it to me (along with a short description) at Julia.Collins@ed.ac.uk!
Set theory Marley
Jane Walker from ICMS sent me this photo that she took at the Theory of Infinity Conference (also known as the Third European Set Theory Conference). The concept of infinity raises many difficult questions in Set Theory, many of which were written about by Georg Cantor and Bertrand Russell, and they are still being discussed today. The little dog in the picture is called Marley, and his photo is being posted around the world in order to raise money for the Dog’s Trust. He has a blog and can be followed on Twitter as #flatmarley.
A 15-year legacy
Simon Willerton drew these knots on a board at the University of Edinburgh when he was a graduate student in 1995, and today they are still there! You can see in the bottom left corner there is a message to new students that these knots must never be rubbed off, so there is every hope that they will persist for many years to come. Simon’s thesis was about Vassiliev invariants, which are a way of constructing knot invariants using the extended space of singular knots (i.e. not just the usual knots, like those drawn on the board, but knots which may intersect themselves or have crossings where more than two strands are present). It is still an open question in mathematics whether Vassiliev invariants can be used to distinguish any pair of knots.
Cambridge
Our photo today is from James Grime, who is the Enigma Project Officer at Cambridge University. He came across this board whilst wandering around the maths department there: unlike at most universities, there are boards in the hallways (and even, so I am told, in the lifts and the toilets!) to encourage collaboration. This board seems to have a good mix of maths, physics and humour (see the calculation at the bottom!).
Cayley graph
Our photo this week is by Christian Perfect, a PhD student at the University of Newcastle. He was drawing a Cayley graph, which is a picture encoding the structure of a group. This particular picture is a drawing of the structure of F2, the free group on two generators. This group is generated by the elements {x,y,x-1, y-1} with x and x-1 being right and left arrows, and y and y-1 being up and down arrows respectively. Each element of the group is then a vertex of the graph. For example, to find the vertex corresponding to xyx2y-1 you start in the middle and go right, up, right, right, down.
This picture can also be interpreted (if you are a topologist!) as the universal cover of the ’8′, or infinity, symbol.
Christian actually took a series of photos as he was drawing the graph. I’ve put these together into an animated gif for your viewing pleasure (but beware – it’s 30Mb!).
Instanton geometry
This week’s whiteboard is a joint effort between José Figueroa-O’Farrill and PhD student Paul Reynolds, who study objects in mathematical physics called instantons. These are special solutions to the Yang-Mills field equations which play important roles both in Physics and Mathematics. For example, in quantum physics instantons describe the process of tunnelling, where particles can get through barriers that classically they would be unable to surmount. In mathematics, they were instrumental in Donaldson’s work on 4-dimensional manifolds, for which he was awarded the Fields Medal.
Instantons appear in all sorts of dimensions, and José and Paul are specifically interested in 8 dimensions. Four-dimensional instantons are intimately related to the quaternions, whereas the 8-dimensional instantons are intimately related to the octonions.
The first bit of the blackboard (to the left) explains that the 8-dimensional instanton equations (which consist of 7 independent conditions) can be viewed as the components of a momentum mapping. This generalises a similar fact about 4-dimensional instantons. In the 4-dimensional case the geometry is that of a hyperkähler momentum mapping, but the geometry of the 8-dimensional equations remains unknown.
Hyperkähler manifolds can be constructed by taking the quaternions and quotienting out by certain group actions. Paul is trying to do something similar with the octonians: trying to find the right quotients that will give interesting manifolds which fit the geometry of the 8-dimensional instantons. The part of the board to the right describes his attempts at finding the geometry of a quotient in a special case.
Therefore 7 is prime
This photo of a blackboard was posted by an anonymous user (“yesmanapple”) on Reddit, who I hope doesn’t mind it being posted here. I love how the difficult Galois theory and topology surrounds the statement “therefore 7 is prime”. Brilliant. If you are yesmanapple, please reply and tell us more!
Colouring dodecahedra
This week our photos are from the blackboard of Toby Bailey, where he and I were trying to figure out how to colour the edges of a dodecahedron.
First I asked whether it was possible to colour the edges with 3 colours so that at every vertex 3 different colours met. Initially we thought it would be impossible and set out to find a proof of this. However, our investigations showed that it was actually possible! The colours around a vertex can be ordered in two ways: blue-red-green or blue-green-red. It turns out that out of the 20 vertices, 16 of them will have the same colour ordering and 4 of them with be the other one. And these 4 vertices (circled in the top picture) form the vertices of a regular tetrahedron! Magic.
After this challenge, the next obvious question was whether we could colour the edges with 5 colours so that around every face of the dodecahedron there are 5 different colours. This was a much easier task but the results were, in many ways, much more beautiful.
Since there are 30 edges to be coloured in 5 colours, there must be 6 edges of each colour. Toby noticed that there are 5 different cubes that can be embedded in a dodecahedron, and wondered if there would be a relation between the cubes and the colouring. The photo below shows where the 5 differently coloured cubes sit inside the dodecahedron. A colouring of the faces is therefore obtained by looking at the pentagram formed inside a pentagonal face, and colouring the opposite edges in the colours given.
Here is a picture of our final 5-colouring:
See if you can spot all the symmetries in this picture! How many different 5-colourings are there in total?
Category O and hexagons
These beautiful blackboard diagrams were drawn by Rollo Jenkins of Edinburgh University during the first year of his PhD, when he was trying to understand a mathematical object called Category O. This object contains a collection of nicely behaved modules related to 3×3 matrices with trace zero.
For each module in Category O there is an associated pair of numbers called a weight, and each weight can be thought of as a single point in a diagram like the one in the top-right of the photo. The geometry of the diagram completely determines the behaviour of the modules.
One thing you notice looking at the green/yellow picture in the top-right is the hexagonal pattern of vertices. It turns out that if you look at 6 of the green/yellow vertices which are in a hexagon, then these 6 modules alone tell you everything you need to understand about all of the modules. But why is it hexagons that are important and not some other shape? Rollo has noticed that if you look at the following matrix with zeros on the diagonal then the non-zero numbers appear to form a hexagon if you tilt your head the right way. Coincidence?
- The 6 stars (non-zero entries) form a hexagon. Coincidence?
(Rollo did all this work in the computer room instead of working in his own office. Two years later nobody has had the heart to remove the diagrams!)
