Gaming Graph Theory

The Seven Bridges of Königsberg is an important historical problem in mathematics, explored by Leonhard Euler in 1735, that laid the foundations of graph theory and topology.

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges.

 Mathematically the problem can be stated like this:

Given the graph above, is it possible to construct a path (or a cycle – a path starting and ending on the same vertex) which visits each edge exactly once?

Euler proved that it was not possible. He proved that for the existence of what he coined “Eulerian trails”, it is necessary that no more than two vertices have an odd degree; this means the Königsberg graph is not Eulerian. Mathematically, if there are no vertices of odd degree, all Eulerian trails are circuits. If there are exactly two vertices of odd degree, all Eulerian trails start at one of them and end at the other. A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian.

This type of mathematics can be quite abstract for students (and teachers) and most find it quite difficult even when presented in context of network optimization, railway systems, traffic flow, shortest path problems etc. 

One Touch Drawing by Ecapyc Inc. is a nice puzzle game on iOS that deals exclusively with Graph Theory. It could be a nice way to get students thinking about Graph Theory before the topic is introduced and then used as a context for discussion and problem solving to give the topic more meaning to students. I have only played the first few levels, the the difficulty ramps quite quickly. Recommended.

Pyramide du Louvre

The Pyramide du Louvre is in the Cour Napoleon of the Palais du Louvre. It aligns with the Arc de Triomphe and the obelisk in the Place de la Concorde giving a truly spectacular view that sweeps from the Louvre through the Luileries and continues up the Champs-Elysees to the Place de i’Etoile.

In Dan Brown’s Da Vinci Code, Robert Langdon states at one stage that the Pyramid contains 666 panes of glass embedded within the numerous rhomboids and triangles making up it’s outer surface. According to various sources this is one of the myths that Brown took a creative licence with. 

From Wikipedia – 

“counting the panes is remarkably easy: each of the four sides of the pyramid has 18 triangular panes and 17 rows of rhombic ones arranged in a triangle, thus giving 153 rhombic panes. The side with the entrance, however, has 11 panes less (9 rhombic, 2 triangular), so the whole pyramid consists of 4cdot153-9=603 rhombi and 4cdot18-2=70 triangles, 673 (outside) panes total. If the entrance was designed to be six panes wide at the base (just as it is in reality) and four (instead of two) panes high, the number of panes would be indeed 666 (598 rhombi and 68 triangles).”

(I still love the Da Vinci Code…)

The Louvre itself is spectacular and I could of easy spent a week wandering the galleries and corridors. Our tour guide, Charlotte told us that,

“if you spent only 3 seconds at each piece at the Louvre, it would take 4 months, both day and night, to see the entire museum collection.”

Seeing the works of Da Vinci was one of many highlights.

Incorporating the works of Da Vinci into the mathematics classroom would be a great way to instill in students a sense of history and culture. Using the Last Supper, Mona Lisa and Vitruvian Man, students could explore and investigate concepts like perspective, depth, the Golden Ratio, Golden Rectangle, proportion and both 2D and 3D geometry.