What You Will

You can't see that Malvolio's cross-gartered stockings are bright yellow.

I have been lucky enough to recently come into possession of an Android phone. One of my favourite things about having such a clever phone is the Kindle app. I can put so many beautiful books in my pocket and there are many many many of them for which I don't even have to pay. A consequence of this is that this morning I (re)read Twelfth Night. I first read it in high school, inspired, I think, by Shakespeare in Love, which was our grade twelve English film study. (I know I wrote an essay comparing the film and the play, but I don't remember exactly why I chose that topic or if I read the play specifically for the essay.) I realised recently that I haven't touched Shakespeare since starting university and today I remembered just why that really is a pity.

Plays are fun to read and Shakespeare is just plain clever. I can fly from my cosy curled-up armchair spot to the Globe theatre; to the private showing at Candlemas 1602; to a performance in a modern theatre; to the director's chair at rehearsals; to the tech crew's  scaffolding and lighting board; to that half-enchanted land of Illyria where, ghost-like, I anxiously watch Viola extract herself from the horrible mess that Sir Toby and Fabian have taken it upon themselves to create. It's quite glorious. I daresay I miss some things and misinterpret others; but the stuff wasn't written, I don't think, so much to be analysed as to be enjoyed. Analysis will inevitably proceed from the enjoyment and some folk will carry that out in great detail. That is one good thing. Also a good thing is those of us left feeling awfully lucky that there's another play and another and another to be downloaded at the click of a button.

Learning/Understanding

(Two sides of the same coin)

Abstract

Mathematics, like poetry,
half-conceals truths
so abstract that,
were they made explicit,
the very substance of the universe
would burn.

You must learn what they
can express
to understand
what they can't.

 Lauren (aka SilverInkblot) of Autumn Brontide has been writing poetry in a series she calls "A Poetic Education". (The poemish thing above is part of my attempt at a response.)  My favourite so far is Half. The series is, I think, an interesting exploration of learning and schooling and education and how the three get tangled up together or end up (very sadly) excluding each other. I muddled around what she'd written a little, and she responded and expanded on her thoughts here. In consequence, I am thinking about education.

I read a few blogs about homeschooling, higher education and occasionally both. They're interesting, but I don't often do anything about them. I'm not a homeschooler. I'm not  a university professor. I guess I do a little bit of TA-ish work, but I've yet to discover a blog about those sorts of things! Despite that, I think it's worth reading those blogs.

Partly, they're just interesting. If something's interesting, I like to learn more about it. (Perhaps this is because I'm lucky in having received an education that by and large nurtured my love of learning, rather than crushing it.) Partly, I think they're useful even when I don't immediately act on them. Becoming aware of how education and learning work means that if I do end up making a decision about them, I have data to work with. And I'm sure I make minute day-to-day choices slightly differently when I make them against a broader background.

That is, I think awareness makes a difference and I think that an education -- even an education on education -- has to be built up over time. Which is why I think that saying something about it will almost always be better than saying nothing, even if the something has to be accompanied by and admission that it's a long way from being everything. I don't think the difficulties education has to overcome can go away overnight. But by engaging with them and thinking about them, we make change possible.

Trends and Community

Exams are finished! I'm not sure it's possible to adequately express my glee about this in writing. I am not an exam person. (Lest you think that redundant, let me assure you that I have on occasion met people who do handle exams with poise and grace.) Presently I will go home, see my family and laze around reading analysis textbooks. In the meantime, I am working at university, which is quite* cool.

The work I'm doing involves going way back through the archives of academic journals and sorting out which papers are relevant for one of my lecturer's projects. One of the side-effects is that I get to see a little bit of the flow of physics research over time.

It's weird to see the papers people were writing when I was still learning to read, all put together in a sort of conversation. It's no surprise that people were doing physics when I was a kid (I've certainly used papers from way before I was born before), but it is a little odd to find this community that I can never really know preserved in the pages (or pdf files) of the Am. J. Phys.

It's also interesting to see how different topics go in and out of vogue. There's always a certain amount of mechanics, quantum physics, electromagnetism and so on, but there are time periods when certain things crop up repeatedly and quite frequently. For instance there were a couple of years that saw a cluster of papers about the charge on a magnetised needle; during another phase it seemed quite fashionable to deal with surface tension problems. I think it's fascinating that such trends exist.

I'm not sure if these patterns have much direct practical application, but I do think they're useful in getting know physics. I recall once reading somewhere that before trying to participate in an internet forum, it can be helpful to 'lurk' and see how things are done there. I guess the academic equivalent is reading all the way back through the journal archives.
_____________________
*Where 'quite' means 'extraordinarily, but that seems potentially like an over-the-top response, so I'll just say quite,'

Thinking like a physicist

@mathematicsprof on Twitter recently tweeted a link to a page asking what it's like to understand advanced mathematics. There are a number of very interesting answers there, but one interested me particularly. I can't figure out if there's a way to link to it directly, but I'll quote it here:

From Jack Miller:

 

A two part question to determine if you "think like a mathematician," from Prof. Eugene Luks, Bucknell University, circa 1979.

Part I: You're in a room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on the floor.  How do you make hot water for tea?
Answer to Part I: Put tea kettle on stove, turn on burner, heat until water boils.

Part II: Next, you're in another room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on a table.  How do you make hot water for tea?
Non-mathematician's answer to Part II: Put tea kettle on stove, turn on burner, heat until water boils.
Mathematician's answer to Part II: Put the tea kettle on the floor. 

Why?  Because a solution to any new problem is elegantly complete when it can be reduced to a previously demonstrated case.

 This might be why I'm studying physics more than maths. I can see why putting the kettle on the floor solves the problem rather elegantly - I think it's a nicer solution than the "non-mathematician's answer" up there - but it's not how I would solve the problem. Isn't it obvious that the table is negligible in this situation, so that Part II is reduced to Part I?

Mathematicians aren't, I think, supposed to say things are negligible. Assuming that the table is negligible isn't rigourous. It does, however, get to the right solution without (explicitly, at least) going via the floor. It's still elegant (if you can get over the idea of neglecting the table) and it takes less effort.

Perhaps it's related to the idea that physics is not so much about working out how to describe some given bit of the universe as it is about working out which bits of the universe we can describe and doing so. This is usually expressed in terms of finding symmetries, from what I've seen and heard. Here, I think the system is invariant under the introduction of the table, which is a symmetry.

There are probably other ways of solving the problem, too. I think it's a very interesting exercise in how people think!

With Recurring Kittens

One.

I wrote my first exam for this semester yesterday. I think it went reasonably well; it was oddly satisfying to throw out the thirty-plus pages of notes I'd generated over the last few days. (Not the actual notes I made in class - I keep those. But I tossed the duplicates that ensured the class notes did actually (mostly) transfer to my memory.) There is now some danger that I begin to feel that having worked so hard for that course, I needn't worry about the others. Which would be foolish. To combat this assumption, I'm going to try to finish a section of Quantum Mechanics between writing each take here. (I'm stealing the general idea from here.)

Two.

I came across this writing competition yesterday, which asks for writing inspired by Benjamin Franklin's quote "If you would persuade you must appeal to interest rather than intellect." Now, physics is indubitably awesome because it allows us to harness the power of mathematics to understand the nature of the universe. It's pretty cool that way. But part of the fun - especially, perhaps, when it comes to revision - is things like my Q.M. lecturer's comment that there are physics-loving kittens who cry every time students try to explain the uncertainty principle without explicitly stating that it's fundamental. Doesn't sound plausible, you say? Well neither does quantum mechanics.

I don't think those two ideas are properly linked up there. If I figure out how to link them properly, I might have a competition entry.

Three.

I am going to use part three to write about the idea of determinism in classical mechanics so that I can refer to that idea in part four. It's pretty interesting for it's own sake too, though. Determinism starts with the sort of idea that if I know that a train leaves the station at two o'clock and travels at a constant speed of 60 km/h (I have no idea how fast trains actually travel) in a constant direction, I can tell you exactly where it will be at three o'clock. Of course, the train needn't travel at a constant speed in a constant direction, so I could  be quite wrong. When it comes to Newton's laws of mechanics, however, the only thing that can cause a particle to change its state of motion is some kind of external force (that's basically Newton's first law). So if I know all the particles and all the forces they can exert, I can work out everything that will happen. This raises some rather interesting questions about free will, since although practically nobody could know what every single particle in the universe is doing, the idea that it's theoretically possible is rather creepy. Quantum mechanics saves the day here, though: it turns out that even theoretically it's not possible to know everything about even one particle. Of course, that's rather weird in its own right.

Four.

This painting is awesome.

Il castello di Bentheim (Jacob Van Ruisdael)

I don't think you can explain why a painting is awesome by describing particles that fly around colliding and absorbing one another. They can be deterministic classical particles or random quantum particles, but they don't explain things like beauty. Or truth. (You can talk about my perception of truth in terms of particles in my brain, perhaps, but not about truth itself.) And they're not supposed to. That's why physics isn't metaphysics. It's kind of obvious in some ways and awfully hard to hold onto in others. Part of the appeal of physics (apart from the kittens) is trying to understand things. We understand more and more stuff, at a more and more fundamental level, as we go on, but at some point, in some directions, it has to stop working. Which is just as well, on the whole, but can be a little disappointing in the moment.

Five.

 It is a mark of something, I'm sure, that I've quite lost track of my reading list. Probably the amount of work involved in a final year maths/physics course. I don't actually know if I've read (well, finished) anything (that's a book) since Silver on the Tree, although I suspect I haven't. And I don't remember exactly when I read that, so that it's an altogether sorry state of affairs. However, in my efforts to do something meaningful and productive that does not involve calculating the probability amplitude function for a particle on a ring for the umpteenth time, I realised that my average reading rate for the year is still a book a week. And I definitely read more academic papers than I used to, which ought to count for something. Perhaps not altogether a sorry state of affairs, then. But it's rather odd to say that I can't remember the last time I finished reading a book.

Six.

Do you still remember the physics-loving kittens? I suggested that such a thing might be implausible, but through the wonders of Google image search and The Particle Zoo, I have now found such kittens. Behold:

You can click the image for more quarky quirky physics humour with kittens.

If you have not encountered The Particle Zoo before, it's where you go to buy a universe in a box. No, seriously.

Seven.

And that's that. I did not study a section of quantum mechanics between each post, because I couldn't make up my mind about how to define a section. So I did some studying and some writing and it all got mixed up together. As long as I don't start drawing kittens in my exam paper, I think that's okay. I missed the boat for keeping quantum mechanics off the blog a while ago.

Feminine(?) Role Models

Preface: Apparently the people at The Aperiodical are not as great as I think they are. You should make your own judgement, though. (In my undenied naivete, I think it's cool that somebody mentioned me on a podcast. Two, actually.)

 So recently some people at Michigan university did some research finding that feminine maths and science role models are not inspiring. I've been wondering for a little while why the article bothers me. Perhaps it's because I exhibit most of the characteristics that they say put girls off studying STEM (science, technology, engineering, mathematics, if I recall correctly) subjects. Which is, you know, kind of sad. I would not like to think that by wearing pretty skirts and getting good marks I send out a message that other people can't . . . can't what? Do mathematics? Understand physics? That sounds plain ridiculous.


Part of it is that the study was actually fairly limited -- it only looked a middle school girls. More significantly, from what I understand, 'feminine' here means something more like 'glamourous' than, say, 'womanly'. (The introduction to the paper proper refers to "pink-laptop-toting ‘‘Computer Engineer Barbie,’’" for instance.) That's not my immediate interpretation of feminine, but perhaps it's other people's. (If you're reading this, I'd love to know what you think.) I can believe that most girls don't think they have the potential to be the real-life version of Computer Engineer Barbie. I for one don't even want to. Does that make me unfeminine?

I like pretty dresses and I have a formidable collection of hair ribbons, but I can't be bothered to go through a fifteen minute make-up routine every morning. I own pretty high heeled sandals, but I don't wear them often. For one thing they're impractical for running up to the Physics department on the second (I think that's third in the USA, where people count funny) floor of Science block. For another, I'd feel overdressed when I arrived amidst the rest of the jeans-and-sneaker-clad students. If that's unfeminine, then I guess all those conclusions about STEM-successful women being unfeminine are probably true. Only I'd be inclined to think that most women are rather unfeminine then. (In fact this is semi-almost-implied in both the press release above and the paper proper.) Motherhood certainly doesn't sound feminine by this metric, which strikes me as rather odd.

If feminine means putting a fair amount of time and effort into satisfying certain perceptions of beauty, then being feminine means having less time to be good at anything else. It also means that anyone with a different idea of what beauty is will rather dislike being labelled as feminine. It wouldn't surprise me that many women who are successful in any number of other fields qualify as 'unfeminine'. On the other hand if being feminine is simply being typically female, I don't think it's nearly as likely to scare girls away from maths and science. (Of course STEM-successful women may still be perceived as unfeminine, which is not cool, but not something I really want to get into here.)

I do think the study is interesting and even useful, but I don't like the lack of definition of 'feminine'. It seems to mean something different every time I see it, until I find myself wondering how I've ended up in a world where it's referred to as counter-stereotypical for a woman to be feminine.

Perhaps I'm overthinking things, but that does strike me as a little odd.

Redefining Survival Mode

Every once in a while one is forced to go into survival mode, I think. It's simply not possible to do everything that one would like to, or even to do most of it altogether properly. So one ends up doing an okay job of the critical stuff until things improve. For instance, I may or may not have recently been heard to say 'I don't understand that, but it doesn't matter -- it's not in the exam.' It's not that I don't care about understanding things. It's that I have a fairly good argument for understanding the things that are in the exam first.

However, I've noticed that my definition of what exactly constitutes this survival mode seems to be changing. Survival mode used to mean sitting and staring tiredly at my cup of tea instead of getting on with my work. Now I sit and stare tiredly at my cup of tea, contemplating the curl of the tea velocity field and wondering if heat transfer would affect that velocity field. And what would the scalar heat field superimposed (as a colour map, say) on the velocity field look like? And if heat flow was treated as a vector? I may even pull up something like Mathematica or Gnuplot to get an idea of what those things would look like. All this still instead of the work I'm supposed to do, although I won't take it any further than that.

It still feels like survival mode, but it seems qualitatively different from the other kind of survival mode. I'm not quite sure where it comes from. Perhaps it's the increased 'mental fitness' after a couple of years of university maths and science; a gradual change in the way I think about things that's just highlighted by the fact that I'm survival mode-ing. Perhaps half a dozen things. I'd need to create an ensemble of identical systems and observe their evolution over time to be sure.

Whatever the cause, I think it's a kind of interesting phenomenon. I should probably stop staring tiredly at the computer screen and go figure out how to use a Cornu spiral, though.

This describes Fresnel diffraction. I'm still figuring out how.

I like bullet points

• It's somehow less intimidating to write a bunch of not necessarily related points out of whatever vaguely interesting soup is floating around in my head than to pound out a nice set of linky paragraphs with a common theme. Let's not talk about why I might intimidate myself about posting to my own little blog in this corner of the internet. Test season and rationality do not have a high level of overlap.
• Let's not talk about test season either. I'd rather tell you that I started listening to the Math/Maths podcast and it's really awesome. Also, it makes me feel like I should actually remember to write about mathsy physicsy stuff on here more often. I thin we can call that a double win. The podcast assumes you know a little bit about maths (or math, for the Americans), but it certainly doesn't expect you to be at research level in anything. I like listening to something with a bit of meat to it without feeling like I've bitten off more than I can chew!
• I'm not sure whether or not that was a mixed metaphor.
• I've finished the first two of my final year courses! Our computational physics courses are largely based around actually writing code, so there's no final theory exam. The general consensus is that continuous assessment is actually more work than otherwise (I write a three hour theory exam for my 16 credit theoretical courses; I wrote a four hour final practical test for an 8 credit computational course), but it's lovely to be finished already!



Hydrogen molecule ion orbitals.

We get to make pretty pictures in comp. phys. too. Like this one, showing the electron orbitals (where the electron is most likely to occur) of a hydrogen molecule ion. This one was done in Mathematica, which is a wonderful tool for crunching through maths that's technically doable but not very much fun. Also, it draws pretty pictures.

(I haven't taken the care with formatting that I would in a proper report, so if there's anything odd about the image, that's probably why! You can look up molecular orbitals or the linear combination of atomic orbitals model if you're really interested in seeing the science done properly.)

• We spent some time yesterday trying to get Mathematica to draw rank two tensors for us, before realising that it was a rather silly idea. A rank zero tensor is just a scalar, or point on the number line, so it's pretty easy to understand. A rank one tensor is a vector (in R³), which you can visualise as an arrow in three dimensional space. A rank two tensor maps one three dimensional vector to another, which means, as far as I understand it, that you'd need a nine-dimensional blackboard to draw it out. Unfortunately (or perhaps fortunately), Mathematica doesn't have a Plot9D command. I can't imagine why not.
• One of the best parts about getting this far into my degree is that as a class we both know each other well enough and are sufficiently interested in physics that between lectures we (sometimes!) do stuff like trying to draw (potentially impossible) things in Mathematica or work out the details of a proof we glossed over in class. See also: hitting 'random' repeatedly on xkcd, and looking at graphs showing that the exponential growth rate for yoghurt is higher than that of gingerbeer or sourdough by a ridiculous amount.
• Are there physicsy versions of things like Math/Maths and Aperiodical? I can find stuff about science-in-general or maths-in-particular easily enough, perhaps because I already know where some of it is, but physics-in-particular doesn't seem to be very well represented. I don't know if maths gets more attention on it's own because it's sometimes excluded from 'science', if it's just considered more awesome than physics or if I just happen to have stumbled upon the online maths community and have yet to discover the physicsy* analogue.
 

*I have now used 'physicsy' three times. This makes it a real word. To quote the estimable Lewis Carroll (in The Hunting of the Snark) "I have said it thrice: // What I tell you three times is true."

For Us

Very Good Friday by ~eldersprite on deviantART

The thing about Good Friday is that we did -- we do, even -- all the very horriblest things to Him, but He will turn around and say "I love you. Be blessed." He reaches not just past, but through the pride, the selfishness, the ugliness of sin and offers Himself to us.

While we were still sinners, Christ died for us. -- Romans 5:8

Quantum mechanics is mind bending, but it's got nothing on this. Relativity is beautiful, but it's got nothing on this. Because the guy who dreamed up the universe whose description would be the source of so much joy and beauty and incredibly hard work? That guy chose to be incarnated inside the universe He'd created and to take the very hardest route out. For me. For you. For us.

There are not enough words to describe the awe (although many people have done a better job than me!). Better, though, to try and fall short than to refuse to try. A little like the parable of the talents, perhaps.

It turns out brick walls do exist

If you insist on verifying the existence of boundaries by repeatedly banging your head on them, you will end up bruised. That's just hearsay, of course. It's most certainly unrelated to the fact that I haven't found time to write here -- or 'most anywhere that's not a report to be handed in for marks -- in the last two months or so. Most certainly unrelated.

Having said that, I'm  sure it won't seem out of the way to ramble a little about how running out of time relates to number systems. I've been hooking a few ideas together and while none of this is rigourous or even necessarily true, I do think it's interesting. The thing about time is that it has to be continuous, kind of like the real numbers. If we allow it to be discrete -- like the integers, say -- we end up with Zeno's paradox:

Suppose Achilles is chasing a tortoise. In the first moment of his chase, we can say that he covers half the distance to the tortoise. In the next moment, he covers half the remaining distance. In the third moment, half of what is left then. Achilles always needs to cover half of the distance left before he can reach the tortoise, but he can continue like this indefinitely without actually catching up. So, says Zeno, it is impossible for Achilles to catch up with the tortoise.

The flaw in this argument is the assumption that time is discretised. It's modelled using integers: moment 1, moment 2, moment 3. We can get a better description of time by using real numbers: between time 3 and and time 4 is time 3.5. Between 3.5 and 4 is 3.75. Between 3.75 and 4 -- well, I could go on forever, which is how Achilles manages to catch the tortoise. (What he does with it next is still up for debate.) However, despite time's going on forever, I still can't manage to get everything I want done.

I can sort of explain that by looking at a mathematical kind of density. Suppose S is a set of numbers. If I can pick any two real numbers and find a third number that's between them and in the set S, then I can say that 'S is dense in the reals'. The rational numbers (numbers that can be written as fractions) are dense in the reals, for instance, but the integers are not. If I pick 1/2 and 1/4, I can't find an integer between them. I can find a rational number between them: 1/3 is perhaps the most obvious. (If you think about it, being dense in the reals actually implies that S has an infinite number of members between any two real numbers.) It seems to me that my perception, or perhaps my experience, is not dense in time. There may, in some sense, be an infinite amount of time between now and tomorrow morning, but I'm certainly not going to get an infinite amount of stuff done in it! In the same way, there are infinitely many (real) numbers between 0 and 100, but only 101 integers (or 99 on the open interval).

I guess I could link in things like response time here too or the fact that movies, which are obviously finite, give the impression of being continuous simply by being more densely packed than our perception. Or I could ditch the biology and go play Mouse Guard with the rest of the family. Mmm.

Numerical Analysis on a Calculator

Today in my differential equations class, we ended up trying to solve the equation x=cos x. And couldn't. Well, I'm not sure everyone was trying very hard, but I had scribbled trig identities all over my page without making any progress whatsoever. Eventually the lecturer took pity on us (or decided that giving us any longer was a waste of his time, perhaps!) and told us to take out our calculators. Oh! So all that analytical fiddling wasn't helpful. It's quite easy to solve the equation numerically on a pretty standard scientific calculator, though, and quite a pretty technique too, I think. I thought I'd share it.

Pick any number you like. If you pick it close to the right answer, the process will be a little quicker, but it doesn't really matter. From the picture below (drawn with Gnuplot), 1 seems like a reasonable starting point.

Now you put cos(1) into your calculator. You don't get 1 back out, so that's not your solution. But if you take cos of that answer, and then cos of that answer and so on for a little while (you can probably do something like just hitting '=' over and over) the answer eventually stops changing. You've found a value which - at least to the accuracy your calculator displays - is it's own cosine. That's the solution. Quite neat! (I think that method is the numerical form of Picard integration, but I could be quite wrong.)

Having started playing with Gnuplot, I don't want to stop, so before I go back to work, why don't I show you a picture of the probability densities we're calculating for the particle-in-a-box problem? It's pretty, at least.

Probabilities of finding an electron at different points in an infinite potential well for five different energy states. (I hope.)

Details

Despite appearance, I am neither dead nor gone. I'm landing. Working out how to cook for two. Working out how four people with three cars can share a single garage. Hoping that the university internet will be a little more reliable once the registration hordes have faded.

Lectures start on Monday morning. In preparation, I have worked out my timetable. I haven't worked out my lecture venues yet. Or labelled my files. Or actually registered for the extra elective which is making my timetable look as scary as it does. (There are six timetable blocks, running 07:45 - 17:30, Monday to Friday. I have  courses in five of those blocks.) I most certainly haven't looked over last years notes so that I'm ready to pick up where we left off, or more than half thought about doing that so-lauded preparatory reading for any of my courses.

Probably, none of that is necessary. Certainly, it's more important that I know how to work the kitchen of the house we're staying in, that I've got the best route from the fridge to the supermarket and back worked out, that I've met the lovely people who are helping our church youth group get organised again, that I'm working out how to do the things I do 'just for fun', but which are really rather important to my sanity.

So tonight I'm going to work out how to make handwritten file tags look just as snazzy as computer-designed ones. I'm going to work on the Baltic myth I'm retelling for a deviantART contest/project. (I don't expect to win, but I want to enter anyway.) I'll have a go at cooking for one, since my sister and partner-in-crimeooking is going out.

Nothing fancy; nothing tragic; nothing exciting, perhaps. But life is taking shape. Sometimes the drama is in the details.

Here and Back Again

Sometimes life just turns itself inside out. I suspect this happens rather a lot, but right now it's happening so that I notice. See, on Saturday morning I'm flying up to my university town. In some senses, this is the standard 'moving out' box checked in the great big growing up checklist. But -

I'm actually flying back to the city I've lived in for the last two years. It's just that (except for my sister) the rest of the family aren't coming with. I'm going to end up in an area I know far better than the one I'm living in now. I'm going back to my very own bed and my desk and my books. Oh glorious books! That may seem a little ridiculous when you know that in the room I'm currently sharing with my sister, I use more shelf space for books than for clothes, but I really am looking forward to being able to pull out Brewster's Dictionary of Phrase and Fable whenever I want it or to check a detail from my biography of Lewis Carroll just as soon as I get home*. I will miss my family, but there's a strong tug from the other side too.

On the same note, I'm going back to people I already know and a bunch of systems I know how to work. Sure, some stuff will be new (we'll have to do our own banking - ick), but a lot of it won't be. And a lot of what I'm expected to struggle with seems more fun than intimidating. One girl asked me how we would manage cooking and I very nearly assured her that my parents would be  fine without us. (In my defense, I was trying to keep track of three conversations at once just then.) I'm sure we'll make grocery shopping mistakes, but I'm looking forward to 'Maritzburg supermarkets where things are organised the way they should be (not that I'm biased or anything).

I don't think the leaving/coming home paradox is at all unique to me - everybody (well, 'most everybody) grows up and moves on. I do think I have a slightly unusual logistical twist on it though! And since I won't have a new city to explore, I might need a list of books to occupy myself with until the deluge of coursework is keeping me busy. If making lists of books to read for fun ranks that high on my priority list, life really can't be that bad. I'm looking forward to seeing what 2012 sem. 1 has to throw at me^†.

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*I realise I could check details online, but that's just not the same.
†Yep, semester one. Our school year matches the calendar year. (Possibly it's different because summer holidays run over Dec/Jan in the southern hemisphere.)

Seven Quick Takes

One
  The thing about living in a UCT+2 timezone is that by the time I see people posting about Friday, it's well into Saturday for me. Or maybe it's just a problem because I follow a lot of international American blogs. Or maybe it's just that I'm not organised enough. At any rate, the last couple of weekends have seen me scrolling through my feedreader saying 'Oh, I guess it's too late to write one of those 7QT posts'. But it looks like I might've pulled it off this time.

Two
  It's not that I'm a perfectionist or anything. I completely accept that nothing will be done 100% right. I'd just like everything to be within a few standard deviations of correct. 3-sigma certainty, for example, is 99.7%. A 0.3% error in the posting time would give me, um, less than four and a half minutes into Saturday when I could post. See? Totally not perfectionist about stuff. Not whatsoever.

Three
  The holidays are too long. Now, people will object if I advocate for more term time, but I think it would be cool if we could take a couple of weeks from the long summer/winter holidays and tack them onto the ridiculously short mid-semester breaks. It might be nice to have a more balanced kind of year, but it would definitely be nice if I didn't have time to realise that the sensible thing to do with my curriculum might be to pick up Operations Research and Numerical Methods and drop Real Analysis and Algebraic Structure. I think it's worth struggling to take the courses I'll most enjoy, though.

Four
 It's not that the other modules are horrid, but it feels like switching waffles and ice-cream for macaroni and cheese. Macaroni cheese is yummy and good for you, but, well, it's not waffles and ice-cream.

Better than macaroni cheese? [Photo by Michael Kwan]

 It's kind of hard to justify that, though. The applied maths courses would probably open more doors for me in terms of postgrad studies and the fact that they don't clash with my required physics courses is probably also significant. The pure maths courses look like more fun. And given that I don't really know what I want to do next year, I can't base my decisions on that. But it's not worth stressing about till I'm back on campus. So really, the holidays should just be shorter.

Five
 When I'm trying not to stress (for whatever collection of reasons) I read. A lot. In the last few days I've read Northanger Abbey as well as all five books in Rick Riordan's Percy Jackson and the Lightning Thief series. Riordan is awesome, but I couldn't help noticing that Percy dreams an awful lot in those books. Dreams are used as a really cool plot device, but given the number of throwaway comments the guy makes about his past dreams, he must have way more non-plot-related dreams than plot related ones. But he says he dreams way more at camp (where the action happens) than elsewhere. The epistemo-temporal maths doesn't work out. (It's still better than Harry Potter, where there are forty kids per year, but six or seven hundred in the school . . .) Despite such things, I love the books.

Six
 No, I don't think epistemo-temporal is actually a word. But let's pretend and use our etymological detective skills, yes? Epistemo from the Greek word ἐπιστήμη (epistēmē), meaning "knowledge". Temporal from the Latin root tempor- meaning "time". That is, the knowledge Percy gains by dreaming doesn't seem to tie up with the amount of time he spends dreaming. And I feel totally justified in mixing Latin and Greek, since Riordan does it all the time, although we hardly noticed until it became the premise for the new Heroes of Olympus series.

Next on my (re)reading list (anyone who writes about classical mythology set today with a steampunk edge has to be pretty awesome, right?)

Seven
 There is a blog called Faraday's Cage is where you put Schroedinger's Cat.

 It reminds me of why engineering is awesome, as well as why it frustrated me. It makes me think it doesn't really matter that much which bunch of cool courses you take for your undergrad degree: you can still shift around a bit more later if you're willing to work. It's pretty cool if you're interested in stuff like Physics/Maths/Engineering/Science Education/Gifted Child Education/Homeschooling/Cute Fluffy Animals*. I saw it featured here and it's part of the reason The Lost Hero is still on the to-be-read list (rather than the currently-reading list).
___
*Actually, it's pretty cool even if you aren't, but you probably have to like some of them to enjoy it.

Mathematics

I want to unpeel mathematics
and hand it to you
on a plate of curiosities.

I want to find the gravel
you brushed off your knees
three years ago and tell you
"These are seeds. We could
plant them together if you liked."

I want to fly
three hundred million metres per second
(that is, to be massless)
by exploiting the nature of
multidimensionality and

I want you to come with.



But my wings grow tired
just imagining
and I can't
find fertile soil.

When I curl into an armchair
with my maths book and tea,
saying, "You should try it sometime,"

I don't expect you to listen.

---

In general, I don't particularly like depressing/sad poetry. Sadness, it seems to me, is not an end in itself (which is not to say it's without value). Poems can successfully take sadness and use it to another end (this reflects life, I think), but I would propose

wallowing ≠ art.

This is rather wallow-y. However, I told myself very firmly when I started writing here that I was not to have expectations of art. So I argued with myself a bit:

   This is rubbish.
   It isn't! It's true!
   Well, it's certainly not factual and I don't see it uncovering the intrinsic nature of reality.
   Didn't you like the part about maths being like flying even a little?
   Okay, that wasn't too bad, but people won't get it.
   How do you know what people will get? You're not people. And the second part is also good. The emotion is universal, but the context is specific.
   Fine then. I'm not saying it's bad, but it's not good enough. You know that's not the whole story.
   Yeah, it's not the whole story, but you haven't lived the whole story yet. How do you expect to write it?
   You can't write it yet.
   Please?
   No.
   What if I guilt you about never writing blog posts?
   What?
   It's been five days since the last post. Can't I put this up?
   No.
   Oh come o-o-o-n.
   Fine then. Make a fool of yourself. But put up a disclaimer saying I had nothing to do with it.

And then I posted it, against my [better/worse] judgment, together with the transcript as a disclaimer.

Music – Poetry

Recently, I have been learning (or trying to learn) to play classical guitar. I had piano lessons years ago, so I have some grasp of basic musical theory, but everything is still pretty new. I can see that the music wants me to play an F and a D together, but I end up staring at the fretboard wondering how to play both of those at the same time. It forces me to slow down and think about which notes I'm playing; gives me a chance to wonder why.

---

The first poetry I remember enjoying is the dwarves' songs at the beginning of Tolkien's The Hobbit. I read them more as poems than as songs, but they carry music in their wording and are generally delightful. They did not, however, give me much cause to think about what poetry is all about, since they follow every convention of structure and metre and rhyme that I understood at the time. Those, it seemed to me, were poems. The newfangled free verse stuff that my English book went on about was not.

It worried me that the great fount of wisdom that was Comprehensive English Practice: Grade 6 said (or seemed to say) that breaking writing up onto lots more lines than were needed made it poetry. (It doesn't, of course, but what does make it poetry is rather subtler.) Somewhere in the middle of that textbook is Seamus Heaney's Storm on the Island. I don't think it's exactly a difficult poem. It works at face value; it is not the kind of poem that describes a ship without mentioning the ship and is not in fact about a ship at all. It is quite a complicated poem, because it largely describes what isn't there. (Which is rather the point.) It hits a balance point that eleven-year-old me had to think about, but could understand.

---

I am learning to play an étude by Dionisio Aguado. It is not, in the grand scheme of things, very difficult at all, but it is quite a challenge for me. I am pleased when I work out how to play another bar and delighted when I can pick up the patterns of the music. Oh, this is the same chord, but with the A an octave lower and I keep playing this sequence because it's the broken-chord A minor triad and the piece is written in A minor! I can't find a melody line in my étude, but when I have to slow down to think about it, I can find patterns in it. And then, hopefully, I can take those patterns back to the sweep of the music played as quickly and flowingly as it should be, so that I can see the weave without forgetting the warp and the weft.

---

Sometimes art is at least as much thinking as feeling.

Recursive Wanting

recursive adj.: related to learning joined up writing for the second time (not really)

The topic of things people want is a rather thorny one, I think. After all, I may want to eat an entire slab of chocolate, but doing so will not actually make me much happier. Whereas I want to do things like write blog posts and practice guitar and what's more, doing those things makes me happier. So wanting something is not necessarily much of a recommendation, but it's probably worth being aware of.

Sometimes I think I want things that I don't, in fact want. This is confusing. What seems to be going on is a kind of recursive wanting. I don't want x, but I wish I did and since the two things are so similar, I mistake the one for the other and put a fair amount of effort into achieving x. Then I'm not satisfied.

For instance, consider TV. Most people seem to like watching TV, or at least watching some version of TV shows on their computers. I would submit that people really want to watch TV when they at least think about skipping their homework while they watch the newest episode of their favourite show. At any rate, once the homework or whatever other prioritized tasks they may have are finished, they watch the show. There's also a rarer sort of person who wants to want to watch TV, because so much enjoyment seems to be derived from it and it's generally what people do. It might not occur to these people to skip the homework in favour of the new show. In fact, even after finishing the homework, they might decide to finish up some other project before watching the show. And even once they're actually watching, they could plausibly get distracted by something else. It's not that the TV show isn't any good, but it isn't actually what they wanted.

I'm not actually sure exactly how they could've got what they wanted. When I catch myself doing things like that, I get frustrated because I'm not even sure exactly what it is I want. When I don't catch myself, I get frustrated because nothing seems to work the way it's supposed to. At least with the former, there's the possibility of doing something about it! Even if something is recognising that actually, you can't have everything you want.

So, vague goal-like entity set: start paying attention to how recursively I want things.

The Axiom of Choice

I can't think of a single good reason to blog about this, except that it makes me happy. That's good enough, right?

So, axioms. Axioms, if you didn't know, are the basic statements that we accept without proof. Logically, maths starts with a handful of axioms which are used to prove everything else. Well, almost. Kurt Godel showed that some things can neither be proved or disproved, which is where things get interesting. No matter what axioms you start with, there will either be inconsistencies, or ideas that can't be shown to be true or false.

Mathematicians have used different sets of axioms over the course of history, getting more and more precise. (Round and edible is not an incorrect definition of an orange, but it describes an apple too; round, edible and citrus is better, but still includes lemons.) The system that's most often used currently used is the creation of mathematicians Zermelo and Fraenkel. Their original system is abbreviated ZF, but the one used now is called ZFC. That's Zermelo-Fraenkel plus the axiom of choice.

The axiom of the choice is one of those things that can't be proved either way using ZF (some very smart people did the work to show that it can't be shown to be true or untrue), and it's been added to the basic set of axioms, like I added 'citrus' to my list of things that describe an orange. I like seeing how maths grows like that. It's a simple enough idea: it says that if I have a bunch of identical things, I can pick one without specifying which one to pick. It seems intuitive enough, but it has some weird consequences.

Particularly, it leads to the Banach-Tarski theorem. The Banach-Tarski theorem is so weird that it's usually called the Banach-Tarski paradox. It says that if you have a ball, you can chop it up into a finite number of pieces and then reassemble those piece to form two balls, each the size of the original.

See? Weird. Despite that, the axiom of choice has survived controversy to become the kind of axiom that is assumed to be assumed. And that is the power of sheer awesome at work in a mathematics near you.

Also, there's a band called Axiom of Choice. That's cool.