495 words on Uni
David Weinberger writes about the four colour theorem. It's nice to see that there are reports about mathematical subjects popping up more frequently in general interest media. It's quite astonishing as well, since the interest of the public (with the state as its voice) or companies in funding research of (pure) mathematics is decreasing at the same time.
Communicating the importance of mathematics more effectively seems to be the goal of the Gauß lecture that took place in Göttingen on Friday. It was intended for the 'interested public' and they made an effort: It took place in the university's historical Aula and the university's and the DMV's vice presidents were there to open it, along with a few musicians playing a piano quartet by Dvořák (I thought the pianist sounded a bit numb, though). Then there was a historical lecture on Gauss and his Disquisitiones arithmeticae, before the main lecture on a topic from numerical simulation of transport processes began. The speaker was quite vivid, making sure nobody was bored or fell asleep despite it being late Friday afternoon. The concept he presented in his talk shared aspects with image processing: Trying to get rid of noise while not blurring sharp information (edges). He showed some images of a colleague of his that made anybody who tried improving images in Photoshop want those algorithms (although I suspect, that they simply tried many times with different parameters as well). The whole event ended with a little gathering featuring drinks and a few bites.
Back to my original topic: David asks what happens in dimension three. An interesting question that has already been answered in the comments with reference to a nice discussion and example showing that you'd indeed need infinitely many colours in three dimensions.
Why is that? Apart from the example given, we can note that the situation in two dimensions is quite special. There are only few things we can do with segments of the plane. We can't really twist them around one another very much to cause serious trouble in terms of the number of colours needed. In three dimensions, however, things are quite different. We have an extra dimension to bend things into, enabling us to make any solid meet with virtually every other one. This is another case of the phenomenon I tried to describe earlier this week: Having an extra dimension may make things easier.
In fact using three-space and three dimensional building blocks is only one way of trying to generalise the four colour theorem. And as it so easy to solve, nobody really cares. There are other generalisations questions as well: For example to maps on other surfaces than the plane. If I'm not mistaken, the seven-colour-theorem for the torus (think of the tube in a bike's tyre) was actually proven first and people found out about the number of colours needed for other surfaces as well.
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