Archive for the ‘Math’ Category.

Making Fractals with MetaPost

Sierpinski's Sieve rendered with MetaPost I’ve played a lot with MetaPost today — I’ve been making fractals. It’s a lot of fun to make graphics with MetaPost, and making the fractals was no exception.

I’ve made three kinds of fractals: Koch Curves, Hilbert Curves and finally the one shown at the right, Sierpinski’s Sieve. The last one is also know as Sierpinski’s Triangle, for obvious reasons.

You can download the sourcecode and a document here: /downloads/metafractals/. If you’re trying to remake the document, then please note that the images are called filenameN.mps instead of just filename.N. This is because PDFLaTeX works best with those filenames. So you’ll have to rename the files generated by MetaPost before running LaTeX/PDFLaTeX. If you’re running Zsh, then you could use this very little shell-script:

for basename in $*; do
    mpost $basename
    for file in ${}.<0->; do
        mv $file ${file/./}.mps

A Long and Exciting Day

Yesterday was a rather long day — or to put it more precisely, it became a long day, when Manitou (Jérémy) and I decided to attend a lecture about Models and Logic (the course is called dModLog at DAIMI). We had started our day at 8 O’clock and was finished five hours later. But then we decided to go to the lecture — it lasted for another three hours.

The lecture was really interesting, we heard about Finite Automata, which I believe is also called Finite State Machines. We heard about languages and how to define a Finite Automata in terms of five parameters: Q, Σ, δ, q0, and F:

  • Q is a finite set of states. The states are the memory of the machine, and in any given moment, the machine will be in exactly one of the states.

  • Σ is a finite input alphabet. It could be binary digits, the ASCII characters or another finite set of symbols.

  • δ is a transition function δ: Q×Σ → Q. δ is the function that makes the machine act on it’s input.

  • q0 ∈ Q is the start state of the machine, and

  • F ⊆ Q is the set of final, accepting states. If the machine is in a state in F, then it will answer “Yes”, if not, then it will answer “No”.

You can take a look at the slides that were used if you want to know more. I hope to be able to follow the lectures loosly, as they’re going to prove Gödel’s incompleteness theorem later. Ever since I first heard about it, I’ve wanted to understand what it really says — I’ve only heard the informal explaination of it, namely that it says, that there are things in every closed system that cannot be proved, although they’re known to be true. That sounds like a very fundamental theorem, and I’m really looking forward to learning about it — if not this year, then definitely next year, where dModLog will be part of our mandatory courses.

My Exam is coming up…

As you might have guessed, then I’m studying for my upcoming exam in linear algebra, also know as Mat10. The exam is the 15th so there’s still plenty of time. I’ve now worked by way through all the subjects we will be tested in, and have made a (Danish) document in LaTeX with all the theorems and proofs. I’ve now written something for every subject — there’s even a proof of the nasty Cayley-Hamilton Theorem! At first I thought it was very confusing, but then I found a proof on the net (sorry — I didn’t save the link) that took the time to explain things properly, and that helped.

The exam is without preparation, so I’ll probably have to rehearse some more… I’ve never tried this kind of exam so I don’t know what to expect. It sounds a bit scary, that I have to be able to talk about a random subject — just like that. My last exam in math was with 20 minutes of time for preparations — plenty of time to look things up in the books and so on… But on the other hand, we wont get a grade for this exam — we either pass or fail. That should make things much easier.

Notes for Mat10

I’ve placed my notes for the forthcoming exam in Mat10 (linear algebra) on the web. You’ll find them here. (The notes are in Danish)

Mersenne Primes

M_p = 2^p-1 Last Monday I finished by big Danish paper on Mersenne primes, or Mersenne primtal as it’s called in Danish.

I made it to my school only ten minutes before the deadline, because it took over an hour to print the darn thing! I had underestimated the printing, because when I printed it during the week, I only printed the important pages, and skipped the title page, the table of contents, the appendix etc. But with all those pages the report is 36 pages long, and since I have to print at “Best” quality on my Deskjet 970Cxi it took a loong time to print… But I made it — with time to spare :-)

I’ve put the paper up in the Danish section of my site, so that you can take a look. You’ll find several different formats to choose from, so everybody should have a chance. But if your browser can’t display the image above, in PNG format, then don’t bother trying the HTML versions. Due to patent problems, all the images are in PNG format.