Aurora Borealis

February 8, 2012

A more personal post than usual today; if you dislike reading about other people’s holidays then you may wish to look away now!

We’ve been staying near Tromsø in the far North of Norway for the last week or so; it was my 30th birthday a couple of weeks ago, but at that point the days here were two hours long!  So we thought we’d delay the trip slightly.   Even though it is well inside the Arctic circle (which – I didn’t previously know this – delineates those Northern parts which have some period of 24-hour darkness in winter, and 24-hour light in summer),  Tromsø is quite a thriving little city, and it boasts various “most Northerly” things, such as university, brewery, botanic gardens, and (perhaps slightly less commendably) Burger King.  Not quite as cold as you might expect, by virtue of its coastal location, and the fact that it is situated right at the very Northern end of the Gulf Stream.  However, that is not to say it is not cold!  The mercury hasn’t risen above freezing since we’ve been here, and it has got as low as -19°C  on a couple of nights.

In fact this is one of the coldest and driest winters they’ve had here: usually a huge amount of snow falls over winter, but this year there has been almost none.  As a result the earth has been exposed to the full extent of the cold, meaning that the frost has permeated deep into the ground, and the spring which supplies the water to our cottage has partially frozen (finding and maintaining one’s own water supply is clearly rather more laborious here than in warmer climes).  So we have had little more than droplets of water coming from the taps.  Luckily there is a sauna in the house, so it is possible to mitigate the icy trickle which is our shower by heating oneself up as much is as bearable beforehand.

It’s a very beautiful place; quite mysterious, with magnificent mountains, a perpetually setting sun, frozen fjords and rivers, and (for some reason I have not yet worked out) occasionally steaming seas.  Perhaps most magical of all, however, and one of the main draws for us, is that Tromsø lies right in the centre of the Northern auroral zone .  This is the band between around 10 and 20 degrees of latitude from the magnetic North pole, to which charged particles from the sun are drawn by the Earth’s magnetic field.  These particles react with atoms in the atmosphere, in much the same way as electricity reacts with the gas in a neon tube, to create a wonderful spectacle known as the aurora borealis, or Northern lights (the Southern equivalent is known as the aurora australis).

I had seen the Northern lights once before as a teenager in the South of Scotland (when the solar wind is especially strong, the phenomenon occurs at lower latitudes).  I didn’t know what it was at the time, which I think added to the general sense of wonder – scientific explanations have a way of taming the awe of the inexplicable.  However, they were very faint, and I wasn’t quite prepared for quite how beautiful a full display would be.  It consists of mostly ghostly green lights, with occasional hints of red, blue and yellow (the colours depending on the atmospheric gases reacting).  They resemble giant flames, or luminous clouds of gas; constantly shifting and moving; often in a number of bands across the sky, but sometimes snaking into coils and spirals.  There is always a sense of rippling and flickering, and when they are at their peak they seem to burn fiercely, flitting quickly in and out of existence.

Unsurprisingly, there have been some quite outlandish sources attributed to them by different native peoples of polar regions.  It seems to vary from region to region whether they were thought of as having positive or negative connotations; whether they were gifts from a benign god, or bad omens and symbols of celestial displeasure.    They are known in Scotland (especially Orkney and Shetland, where they are relatively common) as the “Merry Dancers”, which seems quite appropriate.  Slightly less understandable is the traditional French term, “chavres dansantes”, which translates as “dancing goats”.  The Inuit attached spiritual significance to them, believing them to be images of their ancestors.  What these ancestors were interpreted as doing seems to have varied from tribe to tribe, with some believing them to be simply sending messages through dance, and others perceiving them to be playing football with a walrus skull.   Still more inverted this latter myth, and saw walruses playing football with a human skull!

Other odd interpretations come from different tribes of native North Americans.  Most of these seem to be based on the assumption that the lights were from large fires in the North, but there any trace of common sense seems to end.  For example: the Makah Indians of Washington State thought that there was a tribe of dwarfs living in the far North, who were “half the length of a canoe paddle and so strong they caught whales with their hands”.  These dwarfs boiled whale blubber over the fires which caused the lights.  On the other hand, the Menominee Indians of Wisconsin thought the fires were from torches used by friendly giants to spear fish at night.

I suppose these outlandish explanations go some way to conveying the mysteriousness of the lights…you could probably see whatever you wanted in them if you looked hard enough.  There are also records of strange phenomena occurring when the lights appear, such as whistling and crackling sounds in the sky, and interference with electrical devices: there are even reports of battery-powered radios working without batteries during particularly strong events.  Our radio would make such odd sounds when they were active that we had to switch it off.

Here in Tromsø the northern lights are just a part of life – in the winter months they appear roughly every other clear night to some extent, so seeing them during a trip here is really just a function of the weather.  We’ve been very lucky in this respect, and have seen them 4 nights out of 7, sometimes quite faintly, but once or twice utterly spectacularly.  So, if you want to see the Northern Lights, and live in Europe, I highly recommend coming to Tromsø.  Just check the weather forecast, and, as soon as there is sure to be a clear spell, hop on a flight.  Just be sure to bring some thermal underwear, and lots of money (a pint of beer costs about £9.50)!

You may also want to look at this very useful website, which collates all the possible data you might need to predict a display, such as geomagnetic activity (if you are looking at this post sometime soon after I have written it, notice the great activity around the 24th/25th January 2012 – the lights were visible as far South as London at that point), the current location of the auroral zone, and a 360° webcam of the skies above Tromsø.

Below are some photos.


The Higgs Boson

January 11, 2012

I am beginning to attract some religious conspiracy theorists…I think I’d better change the subject!

So…deep breath.  I’m going to attempt to explain this whole Higgs boson thing which the news keeps going on about, and which, seeing as it is supposedly one of the most important things ever, I’ve been meaning for a while to actually try to properly understand.  Usual disclaimers: I am not a physicist (in fact whether or not I’m even a proper mathematician is arguable) and I am writing this mainly as a motivation to increase my own understanding.  However, my theory is that, unless an expert is a supremely good communicator, it is often easier to gain a basic understanding of a complex subject from another interested layperson (as they know exactly how you feel).  Certainly I would have liked someone else to have written something like this to save me the effort!

I think we have all heard about the search for the Higgs boson by the people at CERN.  Probably, if you’re still reading this, you have also, like me, wondered exactly what this boson is, what it does, and why it matters so much.  And probably you have some vague notion that it is a particle which “gives other particles mass”.  That is the point I shall start from.

But first, a question – why are things the size they are?  Sounds a bit vague and philosophical, I know.  But the size of an object is determined by the size of the molecules which make it up, which are in turn determined by the size of their constituent atoms.  Atoms consist of a nucleus made up of protons and neutrons, surrounded by orbiting electrons.  And the size of an atom is determined by the sizes of the orbits of its electrons.  But the size of electrons’ orbits depends on the mass of the electron!  So in order to find an answer to why things are the size they are, we need to address the question of why an electron has the mass it does.  And while we’re at it, we may as well ask why other elementary particles have the mass they do…for example, why do photons have no mass at all?

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Christmas, Hanukkah, and Fibonacci

January 4, 2012

Happy New Year everyone!  As usual I’m way overdue adding to this blog, and intend to write a proper post shortly on one of the subjects that have recently piqued my interest.  However, in the meantime, I can’t resist quickly pointing out something which came to my notice today, courtesy of my daily email from arXiv; a quirky little article that neatly ties together 3 of the things I have written about recently: calendars, religion, and number sequences (in particular, Fibonacci numbers).

The paper in question is a very short one, attributed to one Shalosh B. Ekhad.  A quick search reveals that this “person” is in fact a computer belonging to a mathematician called Doron Zeilberger, who is quite well known in mathematical circles for his love of computers, and for not being entirely serious all of the time.  However, I will humour him by writing as if it was indeed the entity named Shalosh B. Ekhad who wrote this article.

So, Ekhad became interested in how often the Jewish holiday of Hanukkah coincides with Christmas, and began to run computer searches (which I suppose equates to just thinking about it, if you are actually a computer).    For the non-Jews reading this (I had to look it up myself) Hanukkah is an 8-day holiday which begins on the 25th day of the month of Kislev in the Hebrew Calendar.   The Hebrew calendar is an example of a lunisolar system, in that it takes into account the relative motions of both the sun and the moon. Each year consists of 12 lunar months of either 29 or 30 days, apart from leap years which have an extra month.   Leap years occur 7 times every 19 years; if we think of the first year of a 19-year cycle as being year 1, then the leap years are years 3, 6, 8, 11, 14, 17, and 19.

As you might expect, this means that dates of the Hebrew calendar vary quite wildly in relation to those of our strictly solar Gregorian calendar.  In particular it means that the pattern of years in which Christmas falls within the Hanukkah period is highly unpredictable.  What Ekhad found was that it will happen in 27% of the years of this (3rd) millenium, with this figure falling for subsequent milleniums, until the 9th millenium, when it will stop happening altogether until at least the year 20000 AD.  So far, so mildly diverting.  Much more interesting is the following observation: during the period when the gaps between  between the years 1801 and 7390, that is, in the time-span in which the gaps between years in which Christmas falls within Hanukkah are relatively small, the number of years making up these gaps are always Fibonacci numbers!  (I talked about Fibonacci numbers in this post).  In particular they are always either 2, 3, 5 or 8.  Ekhad then goes on to point out that exactly the same phenomenon occurs for years in which Christmas falls within Sukkot, another Jewish holiday lasting 7 days.

This seems quite incredible…perhaps slightly less so for those who are used to Fibonacci numbers cropping up in the most unexpected places, but it cries out for explanation all the same.  So what is going on here?  Well, the regular occurrence of gaps of 2 and 3 years between these special Christmas-in-Hanukkah years surely has something to do with the fact that the number of non-leap years between leap years in the Hebrew calendar is always either 2 or 3.  The sequence of gaps between leap years (3,2,3,3,3,2,3) would also go some way to explaining the occurrence of 5 and 8 as well, as both of these numbers can be made from sums of consecutive numbers from this sequence.  But then so can 6,9,10,13….

So there’s more to it than that.  Any insights?


Publishing/perishing

November 17, 2011

A minor celebration today…I’ve finally had my first paper accepted for publication.  As a measure of how long and convoluted this process has been, consider the fact that this is the very same* paper that I discussed in this post.  That was written in October 2010, and was titled “A year’s work, lessons learnt”.  Which means, according to my calculations, that the time between beginning work on this project, and actually getting something published has been over 2 years!

Of course, most of this time was not spent actually doing anything related to that particular paper.  In fact, the majority of the time was spent waiting for referees to get round to reading the thing.  Actually “waiting” is the wrong word, as I have come to realise that the best strategy when submitting papers to journals is not to wait, but to completely put it out of your mind (unfortunately this doesn’t help when you then have to revise it months later), and perhaps set some kind of reminder to get in touch with the editor one year in the future and ask exactly what is going on.  I currently have two other papers “under review”, one of which has been “with editor” (I assume this to mean that the editor hasn’t got around to actually looking at it, let alone passing it to a referee) since May, and the other which, perhaps thankfully, I have no way of knowing what is happening with.

There is intermittent hand-wringing about the peer-review system in mathematical circles, and in academia in general.  Like exams, and job interviews, it seems to be grudgingly accepted to be the least bad form of evaluation.  Recently Timothy Gowers raised the possibility of an alternative system on his blog, which led to much fevered debate (I have just noted that I am at least the the seventh blog to have linked to that particular post, so it is safe to assume the debate sparked by it stretches much further than that particular lengthy list of comments!).

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11:11:11 11/11/11

November 11, 2011

I thought I should celebrate the once-in-a-century fact that all of our time/date digits are the same this morning by actually writing another post within a week of the last one (something I don’t think I’ve yet managed on this blog). Of course, the pedants among you might think it improper not to include the “20” in “2011”, but if we were to include those two digits then this occurrence will only ever have happened once before – on this day in 1111 – and will never happen again!  Which is a bit upsetting.

Here is a question for you: what day of the week will it be the next time this happens, on 11/11/2111?  This is the precisely the kind of question that some “idiot savants” are famously good at answering very quickly.  How could they possibly do this, in their heads, in a matter of seconds?  It seems very mysterious, until you give it some thought. Not that I have!  But John Conway, a highly esteemed mathematician who needs no introduction to any other mathematicians who may be reading this (non-mathematicians might possibly unwittingly know of him through his creation the Game of Life…if you can remember back to the dark old days of Windows 95, this was actually included as a “game” along with Minesweeper et al.)*, has actually invented a method of giving the day of the week on any given date.  Why?  I don’t know.  Perhaps he was bored of competing with mere mortals and decided to take on the savants.

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Schrödinger’s cat

November 8, 2011

I usually read novels in bed, as my brain tends to be too tired to take in any more information for the day.  So the fact that this is the third post I am starting with a reference to a popular science book makes me think that perhaps I have not been working hard enough…

The book in question is The Emperor’s New Mind, by Roger Penrose.  The main thesis of this wonderful book is, apparently (and in a very small nutshell), that the mind does not work like a computer*.  However, I am currently about 3/4 of the way through it, and this has not yet been touched upon!  Rather, over 400 pages or so, Penrose has valiantly attempted to explain Turing machines, classical mechanics, relativity, quantum theory and cosmology to the interested (and, one must assume, quite dedicated) layperson.  I can only assume that all this is going to coalesce into a grand theory of Mind, but it does so far seem like quite an ambitious project.  Having tried to achieve this kind of comprehensive introduction to even the smallest of mathematical subjects myself in previous posts (you might have noticed that I have long since given up trying to do this), I have great respect for Penrose’s tenacity.  I find that the problem with this type of enterprise lies in trying to tread the line between being impenetrable to non-mathematicians, and boring for mathematicians.   While The Emperor’s New Mind is a great book, I think it is safe to say that it probably falls on the former side of this line; it is perhaps not entirely suitable for bedtime reading.

Anyway, I have just been reading Penrose’s take on the maltreated feline of this post’s title, and it got me thinking, so I thought I would discuss it.  The cat in question is a paradox which Erwin Schrödinger came up with in order to show the absurdity of trying to apply quantum theory at the classical physical level (that is, the everyday world with which we interact, as opposed to the exceedingly odd quantum level of subatomic particles).   This is, of course, a massive and complex subject, and I will only provide the merest of scrapes of its surface!  If you happen to be a pedantic physicist, then please do comment on any inaccuracies in what follows.

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Mathematicians and religion

September 26, 2011

Where was I?  Well, last week* we established, among other oddities, that diclofenac is bad for the religion of Zoroastrianism.  That post didn’t really have anything to do with mathematics (although I did at least attempt to tenuously link it chaos theory), so I will make up for it by at least mentioning some mathematicians this week, if not actual mathematics.  However, I will stick with the topic of religion for the time being.

This is partly inspired by a book I’ve just read: Galileo’s Daughter, by Dava Sobel.  It doesn’t really match up to Longitude, but is a good read nonetheless.  It is really about the life and work of Galileo Galilei, although Sobel gives us the hard science and history in a more easily digestible form, by interweaving commentary on his relationship with his daughter.  She seems to have been a quite extraordinary woman: sent to a convent at age thirteen due to her illegitimacy (and hence lack of marriage prospects), she spent her whole life in extreme poverty within those walls,  but still managed to be a doctor, playwright, composer, musician and prolific correspondent in the little time she had which wasn’t dedicated to prayer, labour and general suffering.

Anyway, one of the things which struck me most about Galileo’s life was his relationship with the all-powerful Catholic church at this time.  He was a very devout Catholic: publicly, of course (claiming Catholicism is, after all, preferable to torture and painful death), but more surprisingly, given the utter ignorance and persecution he suffered at the hands of the Inquisition, he remained privately devoted to the church.  He even said, near the end of his life:

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How conkers created Israel, cats cause wars, and painkillers are bad for religion

November 22, 2010

I’ve recently been reading a book called Chaos, by James Gleick.  It is a nice, easy-to-read overview of chaos theory in all its forms.  Chaos theory is not really a proper mathematical field, more of an ideology, which has applications in all walks of life.   The phrase seems to be bandied about less these days, perhaps because the ideas have become so accepted that it is no longer considered a theory, but just “how things are”.  It takes the form of turbulence, entropy and unpredictability; it has great influence on the weather, the traffic, the stock markets…indeed it is hard to imagine how science worked before the notion of chaos.  As one physicist in the book puts it:

“Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predicability”.

Poor physicists! Always having their work eliminated by something or other.  Luckily this doesn’t happen in mathematics.  Chaos in mathematics is studied in the form of dynamical systems,  in which small perturbations in initial conditions can have a dramatic long-term effect.  This sensitivity is known in popular culture as the “butterfly effect”, from a paper by Edward Lorenz – a pioneer of chaos theory – titled: Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

Lorenz was  a meteorologist, and first noticed chaotic effects whilst running weather simulations.  Weather is notoriously chaotic (see, for example, long-term forecasts by the Met Office for evidence of this), and one day, whilst trying to restart a simulation where he had left off, he fed in data which had been output from the middle of a previous session.  He noticed that the outcome was wildly different from his previous results, a consequence of the computer having rounded his output to what he had thought was an insignificantly fewer number of decimal points.    Gleick goes into weather patterns in some depth, as well as delving into such interesting topics as the fractal – and by implication, infinite – nature of coastlines (the closer you get the more little “bays” there are), and the chaotic behaviour a human heart displays while fibrillating (basically what a defibrillator does is to reset a chaotic system with a massive jolt of electricity).*

But I am not actually going to talk about chaos theory today.  Well, not quite.  Instead I am going to share a few odd and interesting freakonomics-style chains of events I’ve learnt about recently.  They all involve seemingly insignificant things – conkers, diclofenac and a cat parasite, to be precise  – which have (arguably) had a huge impact on world events.   In that sense you could possibly claim that this was some kind of chaos in practice.  But that would be quite a tenuous way to try and link it with what I’ve written so far, so I won’t.

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All models are wrong (or: lies, damned lies and statistics)*

November 10, 2010

The statistician is seen with a certain amount of disdain (or possibly sympathy) by their pure mathematical brethren. And it is with that firmly in mind that I (as a fledgling statistician) take the reins of this worthy blog.

We have some idea of what mathematics is from Adam’s posts; but what is statistics?  Statistics is applied maths with uncertainty. In statistics mathematical techniques are used to model and quantify our uncertainty about reality. Modelling climate change, predicting the outcome of elections, wrecking the financial system and ensuring the casino always wins: statistics is everywhere. And uncertainty is the key to statistics.

In order to get across an understanding of what uncertainty is I will try to describe some of the different kinds we face and how statistics deals with them.  The five levels in the following taxonomy lie on a continuum running from complete certainty to complete uncertainty, and provide a means of measuring the range and limitations of statistics in different situations.** The further we go along this continuum the less effective statistics is at prediction and inference, and many problems in statistics and quantitative social sciences like economics come from not recognising just how far along the continuum we are.

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How google does what it does

October 29, 2010

This post is inspired by a friend of mine, who very late one night recently made a valiant attempt (given the circumstances) to explain to me how the google website-ranking system works.  I was surprised to hear that at its heart it is a simple – although quite clever – application of linear algebra.  A couple of days later, in one of those funny occurrences of suddenly encountering the same concept/thing multiple times after having spent a lifetime never hearing about it, I actually came across a very similar piece of theory in my research, and read a bit more about it.  So I thought I’d explain it for the interested among you.  However, I am swiftly learning that attempting to write an entire introduction to a mathematical subject in one blog post is a foolish thing to do!  Usually what seems to happen is that I spend the entire post writing about some fundamental aspect of the subject, and then give up and explain the rest very quickly and inadequately.  So I’m afraid I’m going to have to assume you know basic linear algebra…if not, then you might want to stop reading now.

Now, other than indexing websites and providing a portal through which to access them, clearly the most crucial aspect of a search engine is the ordering system it uses to list the sites.  There needs to be some way of assigning an “importance score” to each webpage, such that the ones which people are most likely to want view come first.  Arguably the sole reason google are as successful as they are is a very effective method of doing this invented by Larry Page while he was at university, conveniently called PageRank.  The system uses the links to a page to determine its score, and crucially, it measures not just the number of these links but their “quality”;  that is, it assigns higher importance to links coming from pages which themselves have a high score.

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