Contour Maps of Death Revisited

This is a follow-up to yesterday’s post. As I was trying to fall asleep yesterday I was thinking; those plots look pretty good, but they’d look much better if we could make them more colourful. So today I’ve played around with my code and this is the result:

all I had to do to get such a really neat result was changing a few lines of the code I posted yesterday. For simplicities sake I will post the entire new Matlab code; so here it is

a = 3;
k = 0.015;

for x = -a:k:a
for y = -a:k:a
z = x+y*i;
f = z*(z-(1+i))*(z-2)*(z-2.5*i);
g = abs(f);
frac = g – fix(g);
h = fix(10*frac);
h2 = mod(h,8);
if h2 == 1
plot(x,y,’.k’,'MarkerSize’,4.5)
hold on;
elseif h2 == 3
plot(x,y,’.r’,'MarkerSize’,4.5)
hold on;
elseif h2 == 5
plot(x,y,’.g’,'MarkerSize’,4.5)
hold on;
elseif h2 == 7
plot(x,y,’.m’,'MarkerSize’,4.5)
hold on;
end
end
end

And here are a few more really cool plots. Just click on the thumbnails to see them in their full size.

How cool is Jon Stewart?

Insanely cool and we all know it. I’ve previously told you how much I like the Colbert Report so it doesn’t really come as a surprise that the Daily Show is also one of my favourite shows. And as I really enjoyed Stephen Colbert being interviewed at Harvard’s Kennedy School of Government it was to be expected that I would enjoy Jon Stewart being interviewed by Bill Moyers. As much as I love those two shows on Comedy Central it is also rather nice to see Jon and Stephen talking about their work.

I think it’s something that I just enjoy in general; people talking about their work, be it comedians, writers or scientists, it’s really cool to get to find out more about their work, about how they do it, getting a sneak peak into their mind. Incidentally this is pretty much the only thing I liked about the documentary Fermat’s Last Theorem: mathematicians talking about their work, other than that there wasn’t much in that tv documentary (rather thin on the maths content side, but then again it’s supposed to be accessible – the book on the other hand was much better).

Anyway, sorry for the quick detour, back to what we were talking about; thank PBS and see Bill Moyers interviewing Jon Stewart here.

Tribute to Contour Maps of Death

First off you’ll want to check out Foxy’s post called Contour Maps of Death

to know what this is all about.

The main reason why I post this tribute to his post is that the code he included to create this type of cool plot is for Mathematica and I unfortunately only have Maple and Matlab. So I set out to translate his code so I could create these plots too. I found a simple enough way to do it in Matlab and this is the code of the m-file

a = 3;

for x = -a:0.015:a
for y = -a:0.015:a
z = x+y*i;
f = z*(z-(1+i))*(z-2)*(z-2.5*i);
g = abs(f);
frac = g – fix(g);
h = fix(10*frac);
h2 = mod(h,2);
if h2 == 1
plot(x,y,’.k’,'MarkerSize’,4.5)
hold on;
end
end
end

I haven’t figured out a way to do the same thing with Maple. I believe the approach would have to be slightly different. Still, if you use either Matlab or Mathematica you now have the code to create these cool plots yourself.

On a related note check out Foxy’s blog FoxMaths where you can find plenty of really cool entries on math, be it experimental or else.

Computational Geometry and South Park

Okay, I won’t even pretend that I understand what the medial axis is, nor will I pretend that I know much about Computational Geometry. But if you’ve got a math presentation that involves South Park in one way or another I just have to check it out – and so do you.

First let me credit Ernie’s 3D Pancakes where I found out about Mesecina – computational geometry you can see. Check out the long demo talking about the medial axis and this crazy program called Mesecina which can even be downloaded, though so far you can only get the development version but apparently a release version with documentation will be available soon.

I might download it once the exams are over in case I have some time for crazy mathperiments.

2nd revision update

I started this post on Sunday but then my computer started acting up and since then I was too busy procrastinating to finish it.

Now that the Easter holidays are over ’tis time for another revision update.

We have two more weeks of revision left before the exams start and we certainly feel less ready than we did this time last year, but it’s hopefully nothing that good work can’t fix.

So let’s see, where do we stand and what’s our battle-plan for the next two weeks.

Vector Calculus and Differential Equations

Let’s do the decent thing and put both methods courses together. Tomorrow we’ll pick up a number of books from the library which includes cool things like Schaum’s Outline, on such beautiful subjects as Vector Analysis, Differential Equations and Fourier Analysis. Mainly to have more questions to practise with. After all these courses boil down to being able to use the methods and the more practise one has the better.

Probability and Statistics II

Even after having done most problem sheets I still find the course awful. I should be able to do most things but I can’t yet say that I feel confident. I’ll try to distill the essential part of the course and ensure that I really understand this (as far as is possible given the annoying subject matter).

Numerical Analysis

I used to feel good about this course but I’ve had to review my views after glancing at some past papers. I expected the exam to be mainly about applying the material but unfortunately being able to prove most of the main results is also a very important aspect so I’ll have to study these bits as well. But apart from this I don’t see any difficulties.

Analysis II and Complex Analysis

Still a bit rusty on Analysis II but overall I’m feeling all right about it. Complex Analysis on the other hand used to be awful until last Saturday when I’ve spent the time I had to wait at the airport, etc. to really go over the notes and I’m really starting to get it. So much so that I disagree with one part of a proof and don’t understand why some of the non-examinable proofs are non-examinable. Now I need to let this new found knowledge sink in to be able to apply it to typical exam questions.

Algebra II and Rings and Fields

I’ve done very little on these courses for very simple reasons. On one hand these are some of the last exams so we still have plenty of time for revision and on the other hand we do understand the material really well. For Algebra II we simply need to start doing a few questions to get in the groove again and for Rings and Fields we need to memorise all the proofs as the exam is unfortunately really big on bookwork (at least this is our conjecture after seeing the past papers).

Yesterday I’ve changed my overall revision approach. I used to try to cover absolutely everything to have a shot at acing a few exams but the approach turned out to be too boring to keep up. Instead I’ve decided to look at the big picture and start with those things that I find more difficult. I might still get around to cover almost everything but it’s definitely more interesting to tackle the more difficult bits first rather than covering every set of lecture notes from beginning to end. Ah well, we’ll see how well this new approach will go.

Shall we abolish pencil and paper arithmetic?

Lately there have been a number of posts discussing the state of contemporary elementary maths teaching. While I certainly agree that Reform Math is bound to be a complete failure I’m not sure back to basics is the best alternative. One thing I’ve certainly despised about school is the endless hours we’ve spent wasted on practising long division. Of course I can still do long division but I highly doubt that it made me a better mathematician. Talking of old school algorithms my father was taught to take square roots using paper and pencil. Years ago I learned that algorithm on a whim and used it to calculate square roots mentally to keep me busy on my way to work.

Anyway, the point of this post is that while looking for pencil and paper algorithms for taking square roots (as I have since forgotten how to do this) I came across a paper entitled: Let’s Abolish Pencil-and-Paper Arithmetic which makes a point that mental arithmetic should be taught rather than pencil and paper methods and that there’s nothing wrong with using calculators.

Talking of people who dislike the use of calculators reminds me of a substitute teacher we once had for A-level maths who seemed almost violently opposed to calculators but loved slide rules so much that he wasted parts of a lesson to show us what they were and how to use them (as our generation had been so unfortunate to grow up without them). I mean, seriously, both calculators and slide rules are tools to facilitate computation so grow up and embrace the enemy.

Besides calculators are so old fashioned already. The new way to corrupt our youth are computer algebra systems. One of my first year computing lecturers might have been right as he said that at some point in the future these systems would become as commonplace as calculators nowadays and it could mean that the necessity to practice taking derivatives, etc. would vanish. I doubt it will happen soon but he’s got a point. Let’s face it, so far I’ve used Maple, Matlab and GAP and I have no intention to forgo their use in the future. Often when working on something or playing with a particular problem one may want to check a number of cases or just collect some data to get a feeling for what is happening. In such cases where one looks for patterns in the answers the actual computations are rather unimportant and it is only sensible to automate this task.

Then again, don’t take anything I say too seriously. After all I’m the kind of guy who’d favour getting rid of a number of compulsory mathematical methods courses and introduce at least one decent set theory and mathematical logic course in the standard undergraduate curriculum (and I would place it in the first year).

Number Gossip

Check out this site which tells you everything you ever wanted to know about whichever number you enter (as long as it’s less than 10′000). Sounds like a pretty fun idea.

It also reminded me of the book The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, by Keith Devlin. It’s a highly enjoyable book that tries to explain how humans acquired the ability to do higher maths (ie. things like Abstract Algebra, Analysis, etc. rather than numeracy which has been more thoroughly researched).

1st revision update

It’s been some time since I’ve last wrote anything but I was busy revising – after all that’s what the Easter holidays are all about.

Almost one month ago I sketched an outline of how well revision should go. Now I’ve had two weeks of continuous revision and I guess it’s time for an update.

Analysis II

I did go over Analysis I first and I’ve gone through the lecture notes for Analysis II and Complex Analysis. As I’ve said Analysis II shouldn’t be too much of a problem and I stand by this statement, though I had to realise just how much I’ve forgotten or more precisely how rusty my skills have gotten. So the most important thing for this half-unit is to practice a bunch of questions and get a feel for it again.

Algebra II

I went really crazy and revised all my first year Algebra notes before looking at Algebra II. This really shouldn’t be a problem. I still knew most of it and what can I say, it’s my baby, I just dig this stuff.

Vector Calculus

Goodness, have I forgotten a lot or have I forgotten more?! Still, in principle it’s not too bad but I really need to practice this stuff. Problem sheets and exams could be enough but I’ll borrow Schaum’s outline: Vector Analysis when I’m back in London and do plenty of those questions as well. Still, I still assert that the course is trivial – unless there are some really insane questions on the exam.

Probability and Statistics II

I don’t know if I still think that this course sucks the most. I aim to do 2 hours a day just going through problem sheets and worked examples and I really start to get the hang of it. Though I still think this exam might turn out to be one of the hardest. For the record the hardest calculus questions and craziest integration tricks I’ve come across in the second year were all in this course – go figure.

Numerical Analysis

So far I’ve done 1/3 of the problem sheets and covered maybe half the notes, mainly because I take it easy and spread it out to keep the material alive as the exam is going to be the last. Though so far things look pretty bright. The hardest part is Gradients and Hessians but a bit of practice should fix whatever difficulty we might encounter.

Complex Analysis

Going over the lecture notes and doing the penultimate problem sheet made me think that we are totally screwed. Granted I probably overreacted. I literally will have to slowly work through the entire course again to get a feel for this stuff – for I never got it. I blame all the boring repetitions of past material it successfully drowned out the more difficult new stuff. But once I work through it from start to finish I should get it.

Rings and Fields

This was pretty laid back as it was fresh in my memory. Though I was surprised to find myself realising quite a lot of subtle facts that I hadn’t really fully appreciated so far. So I’m really chuffed. Though today I worked through a past paper and glanced at a few others and I’m not too impressed. The past papers are perfect for rote memorization. Every question asked for a few definitions and then a number of standard proofs. Basically 99% bookwork. I’d be kind of disappointed if the exam turned out to be that way. The progress tests were definitely more interesting. It’d be sad if the conceptually most difficult course could be aced by memorising the lecture notes without understanding any of the material.

Differential Equations

Working through the lecture notes I find that I have indeed been correct and the relevant material fits on less than 10 sheets of paper. Go figure. As with Vector Calculus the important bit is practising many standard questions. After all it’s a methods course. In some ways I think it’s actually easier than Vector Calculus. The notes were much worse but a lot of the methods are actually piss easy.

To summarise; things aren’t looking too bad but it’s definitely more work than last year – though I believe it is realistic to aim for higher marks than last year (would be rather neat).

PS: One can tell that I’m working more than last year simply by realising that I’m getting up for revision on most mornings – I mean I, of all people, get up before 12. Go figure.