Gauß Year¶
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2005 is Gauß year because Gauß died 150 years ago. Gauß was quite a famous mathematician in his day, and the things he did are still used today.
Famous facts he figured out include the formula for adding up the first n integers,
which he allegedly made up in primary school because his teacher told him to add the numbers from 1 to 100 as a punishment… just to be surprised that Gauß had finished the task after just a minute or so. Another extremely popular invention of his is the ‘Gauß curve’, also known as the normal distribution in statistics.
But his works goes much further than that. Gauß gave us the first proof of the fundamental theorem of algebra, gazillions of facts in number theory (if I recall correctly, including an algorithm to compute the day of week for any date that was only slightly improved by Knuth much later). He also had practical interests, which included triangulating the landscape in Lower-Saxony and making sure the university got an observatory.
Last but not least, Gauß was interested in geometry. Every student of differential geometry will learn about the Gauß map and the famous theorema egregium which are in a framework that opens the doors to thinking about non-Euclidian geometry. But here I want to focus on a more down-to-earth fact of geometry that Gauß proved: regular polygons.
The nice thing about this topic is that it’s easy to explain. A regular n-gon is just a geometrical figure consisting of n vertices connected by n edges. A equilateral triangle or simply a square may be the best known examples. Now, in school geometry you construct things just by using a pencil, a ruler and a pair of compasses. That’s how constructions are done since the ancient greek times.
And an equilateral triangle and a square can both be drawn using just these tools. In fact that’s not hard to do and everybody probably did it at school. So we’ve covered the numbers 3 and 4 this way but what about the 5? Well, we can also construct a regular pentagon with these tools. Indeed, this was already known in Euclidian times. And what about a regular hexagon? That’s quite easy as well as we already have a triangle and it’s not a problem to double the number of vertices (can you do that or are you simply taking my word for it?).
So far everything was known by Euclid already. But what about the regular heptagon? Euclid couldn’t construct that. And many others had tried since and didn’t manage to construct it. And Gauß proved a theorem that shows that Euclid and all the others weren’t daft or anything but that it’s impossible to construct a regular heptagon with those tools. In fact the theorem ist much more general. It says that any polygon with a prime number of edges can be constructed the classical way if and only if the prime number is of the form 22m +1 for a positive integer m.
So is this good or bad? Well, let’s look at the numbers that turn up for different values of m
| m | pm |
| 0 | 3 |
| 1 | 5 |
| 2 | 17 |
| 3 | 257 |
| 4 | 65537 |
| 5 | 4294967297 |
| 6 | 18446744073709551617 |
Which means we’re left with very few regular polygons that can be constructed. And the 17-gon (heptdecagon as some people say) is the one that Gauß discovered first. It’s on his tombstone. Constructing it is quite a hassle but a nice homework for undergrads. A long time (i.e. a century or so) ago there was a PhD student here who executed the computations for the construction of one of the larger ones which is a thick volume that’s hidden somewhere in our library.
To present this problem to the public, one of our professors prepared a few posters. On these he compares the abilities of architects, Euclid and Gauß. Starting with 3 the question is whether the corresponding polygon can be found in architecture, whether it could be constructed by Euclid and whether it could be constructed by Gauß. The posters are full of photos he has taken of things like church windows. There you frequently find polygon subdivisions and patterns. Those are frequently triangles, squares, hexagons, octagons or dodecagons. But with a bit of luck you’ll also see pentagons and decagons and with a lot of time at your hands you’ll also be able to find some sort of heptagon.
That, of course can’t have been constructed strictly mathematically, but of course that’s largely irrelevant for an architect. The only thing that apparently hasn’t been built in any of the buildings we saw is a 17-gon. I somehow suspect that there may have been some maths-architecture geek who has actually built this, but we haven’t seen it.
