Sound and Fury

Today’s Guardian had a cool little special section about CERN. It is available here. I now have a false colour bubble chamber image as my desktop background. So the LHC hasn’t blown us all up yet, which is nice. If it ever does, I might feel the tiniest bit responsible, since I’m signed up to the LHC BOINC project…

In other news, the nine-point circle is my favourite fact about triangles. I will try and mention it and its relation to incircles and excircles in my dissertation. Hopefully with gratuitous use of diagrams drawn in Kig. If I can be bothered to learn how to get pictures to work in LaTeX… Another thing I’d like to do is learn how to write chapters as separate files and include them in some master file. For 15,000 words it’s hardly worth it, but it’ll be useful for next year. I’d also like to change the default font to Gentium, for no other reason than to make a move away from default LaTeX formatting…

So I looked into it and Topology and sigma-algebra are distinct things. Apparently a sigma-algebra need only be closed under countable intersection, whereas a topology requires arbitrary intersections to be included. Also topologies won’t necessarily be closed under complement, whereas sigma-algebras will be. But the smallest sigma-algebra containing some topolgy is called the “Borel sigma-algebra.” So that’s that all cleared up…

I took out a book on Euclidean geometry today written by none other than Charles L. Dodgson; better known as Lewis Carroll. Cool huh? And what is more, it’s written as a dialogue! I also recently managed to find Alfred Renyi’s Dialogues on Mathematics. So I’m pleased with both of them.

What with various talks and my being quite lazy I haven’t really managed to get a whole lot done today. And I’m to a conference tomorrow. So this week has been a bit of a write-off really. The real proper work on my dissertation kicks off on Monday. Hoorah!

Yesterday’s Guardian had some maths puzzles from Key Stage 2, 3 and GCSE level. I was pleased to note I could manage them all except number 12. At least, that’s what I thought. For example number 8. The puzzle goes like this:

Look at these diagrams:

Diagram A {1,2,3,4} 10

Diagram B {2,3,4,5} 16

Diagram C {3,4,5,6} 18

Which one is the odd one out? Explain why.

The first four numbers there are arranged clockwise around the last number. Surely the answer is that in A and C, the outside numbers add up to the middle number. So B is the odd one out. The answer given says that “There isn’t one answer for this but you must be able to justify your answer” Fair enough. But I think my answer is one of the most obvious, so why don’t they list it in the “for example…” The ones they list are the following: “A has no numbers greater than 10, the other two do; if you add the numbers on the left hand side [that'll be the first and fourth of my lists] then double them, you should get the middle number but in B you don’t; it is only on C that two outside numbers, 3 and 6, multiply to make the middle number.” These all seem way more convoluted than my fairly elegant solution.

But it gets sillier. One of the puzzles (Number 11) doesn’t even have a solution. It is undetermined by the information given. So it’s one of those things where the numbers in the row below add up to the number in the row above. It is a triangle, three circles on the bottom, two above them and one above that. Only two entries are filled in. A little reflection will show that no two entries can determine the whole answer. Here is what the puzzle looked like, I have added letters to the empty squares to make it easy to talk about.

So the two numbers simply do not determine anything about the solution. D will have to be greater or equal to 8.2 + 5.1, A and B have to add up to 8.2 (therefore have to be less than 8.2) and C has to be at least 5.1. We are still miles from a solution. I ended up putting 8.2 in A, 0 in B. C and D were then fixed (5.1 and 13.3 respectively) But the rather arbitrary solution they give is 4.2, 4, 9.1 and 17.3 for A-D respectively. That is crazy! Where did those numbers come from?

These two questions would have bugged me loads in the exam. Maths (at school at least) is about getting THE solution. If any number of solutions are possible, isn’t this just going to trip kids up?

Incidentally, I’ve been thinking about the relationship between a sigma-algebra and a topology. Both are kinds of structure you can impose on a set X and are subsets of X’s power set. Maybe tomorrow when I’m in a LaTeX mood, I’ll write my tentative conclusions here…

From now on, if I can’t think of a title for my posts, they will all be Futurama references. Until I run out of them (in, like, 400,000 posts time) when I shall move on to Kung Pow, then Commando and so on, in that fashion.

So I went to Rock am Ring at the weekend. I got rained on and sunburned. I also saw lots of bands and so on.

My new favourite thing is the Burnt Pancake Problem because bacteria are better than computers at solving it. Also because both Bill Gates and David X. Cohen have written about it. Incidentally, if computing the nth digit of pi P or NP hard? Apparently there is an algorithm which allows you to compute digit n without having computed all lower digits. That is pretty cool.