A foray into a new number system - an introduction to imaginary numbers

This is the introduction which I give to my first year mathematics class when they see imaginary numbers for the first time. I thought I'd type it up here as it's received good reactions the two times that I've introduced it in this format. Note that this probably isn't the canonical way to introduce complex numbers, but then most of my lectures don't necessarily take the normal route...

Complex Numbers, a philosophical detour

Before we get on to talking about imaginary numbers and complex numbers, let's try and break down our preconceptions about numbers in general.

We look at the world around us and see many things which we categorise. We see a computer, a piece of paper, we see other people, we see our hands. These are labels that we use to categorise the world, but these objects seem very physical and very real. We rarely question their existence, though if one wants to take the Cartesian view, we should also question the reality we are in. We are not going to go that far, but let's try and ask about the existence of numbers.

I have definitely seen five pieces of paper, but I have never seen a five. I've seen the number written down, but I can write down anything I want and it doesn't necessarily mean that it exists. I can write down a erga[oeiave21 but that doesn't suddenly bring a erga[oeiave21 into existence. How about a -5? I've definitely never seen a -5 though I understand perfectly well what it means. The integers seem to be very good ways of describing, or more specifically counting objects and the negative numbers are a good way of keeping track the transfer of objects from one place to another. I can also ask you to give me 3 coffees, and here I am really asking you to apply 3 as an operation to the object coffee. 3 is acting almost more like a verb than it is a noun. When I describe that there are 30 people in a class, I am really thinking about this as a description, or an adjective. So in the real world, somehow numbers feel like verbs and adjectives. I certainly wouldn't say that 'heavy' exists, but certainly a book which has been described as heavy does.

However, there is a world in which numbers really do seem to be more like nouns than they do in the world around us, and that is in the abstract world of mathematics. In the universe of equations, numbers somehow feel much more concrete and I can manipulate them and transmogrify them from one form to another using a set of mathematical operations which become more and more finely tuned and specialised as we learn more and more mathematics. I can take a 5 and I can apply a  the sin function to it to give me another number. I think of this rather as taking an object and putting it in a machine which turns it into another object. Here 5 is very real, but so is -5 and so is pi/4 and so are all the numbers that we've ever used. They are simply the objects which are manipulated by our mathematical machinery. Whether or not they exist as objects around us isn't very important for our use of them in the mathematical universe.

Incidentally, I have here separated the real universe from the more abstract, platonic, mathematical one, but it is fair to say that we have found mathematics as the best language with which to accurately describe the real universe. All of our models and precise descriptions of the universe are built using mathematics, and it acts as an incredible way of describing the laws of nature. Which came first, the mathematics or the universe? That is not a question I am going to get onto here, but it's certainly a profound one!

A foray into a new number system

OK, so we have a mathematical world of numbers and we can manipulate them. Thus, we should be perfectly happy to have some more ingredients in that world, that don't have such an obvious mapping to the things in the world around us. We will discover that actually they help us enormously in the things that we can do with the mathematical machinery. It's like having a powerful car but not the right fuel to really take it up to top speed. We are about to find out what that fuel is and push the limits of what our car can do!

Previously, if we set up a certain type of quadratic equation and plugged it into our machinery to find a solution, the machinery would jam and we wouldn't get an answer out. This was a real shame because it didn't seem to be that much more of a complicated equation than any other that we had studied. We are perfectly happy with solving an equation like:

x^2-1=0

You can plot the graph of y=x^2-1 and see that it equals 0 at two points x=1 and x=-1. That's fine, our mathematical machinery can deal with that fine, but when we ask to solve something so similar:

x^2+1=0

our traditional machinery comes juddering to a halt and we get an error message on the screen - you're not allowed to take the square root of a negative number, says our program. In fact when we plot the graph of y=x^2+1 it's clear that it doesn't cross the x axis, so it can't have a solution...can it? Maybe we're not looking hard enough. Maybe our machinery is fine, but we've just fed it the wrong fuel. In fact, we can find the solution just fine. The solutions are:

x=√-1 and -√-1

You might look at this and go "Absolutely not!" You can't take the square root of a negative number, but if you plug that into the equation, it works just fine and is a perfectly good solution. What is not true is that √-1 is like the normal numbers that we are used to using. In fact, let's give this solution a name. We'll call it i:

√-1=i

(Note that we are actually being mathematically sloppy here, but for a first pass, this will do - we can explain the subtleties later - in particular the domain of the square root is only the positive real numbers and thus we have to say what we mean by this function separately to deal with non integer powers on negative numbers.)

What is i? It's one of the solutions to the equation above. Plug it in, you'll see that it works. There's no funny business going on here. So what if it doesn't correspond to an object in the world you see around you, nor does -76 but we don't have a problem with using that number, do we? i stands for imaginary. So we call i the imaginary number, but in fact it is no more imaginary than most other numbers, it's just a little harder to understand it because we are used to things which represent a size. Numbers which are not imaginary are called real, but again, this name is probably not a very good name as -32 is in some sense no more "real" than is i.

Once you have defined this new type of number - a number that squares to a negative number, we open up so many new possibilities. Things that previously would have driven our machinery to a halt are now very easily accessible. With this new number we've just upgraded our mathematical machinery so that it can handle so many more problems than it could before.

It might seem that i wouldn't have anything to do with the real world and it's true that any measurement of the real world will give us one of the real numbers that we are used to, but using i makes many things much more natural than they would be without it. This is true in many many fields of science and so having i at hand is absolutely indispensable when you want to describe the real world. Very often given a question about the real world, it is much easier to take a detour into the world of these imaginary numbers which gives us a shortcut to an answer, than taking the long route using only the real numbers.

What we've shown here is that we can deal with √-1 but we can quite happily extend this to the square root of any negative number. From now on √-b where b is a positive number can simply be written as i√b. So i can be multiplied by real numbers. We can take 2 lots of i and 3 lots of i  and add them together to get 5 lots of i: 2i+3i=5i.

How about adding a real number onto this? Well it turns out that adding together 3+4i can't be simplified any more. You can't add apples and oranges and get pomegranates. What you end up with are a few apples and a few oranges, and that's as simple as it gets. In fact any number of this form, which has a real part, and an imaginary part is known as complex, and these complex numbers form a whole new domain which will supercharge our mathematical toolkit. We will see the power it gives us in the coming lectures...

Dunhuang to Jiayuguan to Zhangye to Beijing

I left you hanging as I failed miserably to steer my way through the cultural peculiarities of after-dinner etiquette in China. We were in Dunhang, in the West of Gansu province and I'd spent the day with a Chinese tour group, riding camels and hiking over sand dunes, seeing the spectacular grottos at the Mogao caves and being entirely overfed with lamb skewers and donkey meat.

I woke the next morning feeling distinctly shaky and entirely unable to face breakfast - I am still unsure whether this was due to an equine overdose, or simply the flu. The plan was to take a bus the six hours from Dunhuang to Jiayuguan, but this turned into one of the most trying journeys I've ever made. Sitting in the front of the minibus, I found my legs utterly stuck, rammed up against the plastic dash-board and with no wiggle room. The waves of nausea were already building and I could see my pallid face in the mirror, looking increasingly distressed. 

Other than a single stop, I sat for six hours in this agonising position, dehydrated and with spasms shooting through my legs and lower back, trapped with my hands gripping fiercely to the seat, trying desperately not to throw up, or by this point, explode from anywhere else. On arrival in Jiayuguan I was pouring with sweat from the effort of control and in a hallucinatory state of exhaustion, able only to speak in short sentences, and stare ahead semi-catatonically. It was hot and I was on the verge of collapse.

The plan was to go to the fort at Jiayuguan, but I knew that I was in no state to trip the silk road fantastic, or even to stand up for much longer, so I made my excuses and started along the road, struggling to hold onto my luggage, to find a hotel in which to collapse. After some failed attempts, I took a short taxi ride to a hotel with businessmen shouting and smoking furiously in the lobby, each one trying to speak louder and smoke more voluminously than the others. I paid the deposit and headed to my room to expel and then allow coma to descend. For the next four days I lived in a virtual dreamworld, oscillating slowly from the lobby of the hotel to my bed, buying bottles of water and attempting to eat the occasional pot of instant noodles, sleeping and feeling the endless hours of extreme nausea with podcast after podcast, and pondering whether this was on the verge of becoming a serious situation, alone, virtually unable to eat and wobbly on my feet during my lucid moments. A few moments came and went as I envisioned being found some days later, dessicated and wide eyed, and I pondered what was the appropriate action as I descended from the crest of lucidity.

It took four days to be able to leave the hotel and find something real to eat. My first meal of dumplings was taken with caution but I could feel the energy slowly seeping into my shakey muscles. Overall I lost some five kilos in my time in Gansu and I presume that most of this was lost in my four days of infirmity.

I managed on that fourth day to make my way the short distance to the famous pass at Jiayuguan, the Westernmost outpost of the Great Wall and a very important historical site at the intersection of trade and military power. The sun was beating down as my muscles slowly de-enervated, and I pressed on to look around the structure which I had seen in many photos before. 

It is a feature of many Chinese historical sites that the balance of preservation and reconstruction that the West has become used to is not present. The idea, very often, of simply keeping what is left in as good condition as possible is replaced with the rebuilding of the structure to its once present glory. This is true at many of the most touristically popular parts of the Great Wall and it is true at the Fort at Jiayuguan. In East Asia this practice is not new, as within Japan as well, the modus operandi for temples was to rebuild them every 50 years or so, and so there are actually very few temples which retain any of their original pieces. It is hard to quibble with this from a utilitarian point of view, but certainly from the Western idea of heritage and the importance of the actual bricks and mortar being the substrates of history, rather than the mere facade, it is quite strange to walk around such a rebuilt fort. I hadn't taken this into account before I visited it, but as I saw parts covered in scaffolding and with fresh concrete being mixed to keep the perfect picture of presentation fresh, I became somewhat despondent that I was merely visiting the Disney version of the great fort at Jiyuguan. This view it seems is rather a narrow-minded one and in fact the live-action roleplay of traditional Gansu costumes, archery, artwork and music have the potential to make the experience rather magical, if one can only get over the prejudice that the walls themselves are not Ming dynasty.
I left the fort and headed back for a last night in Jiayuguan, trying to regain as much strength as I could before the next train ride.

One of the main reasons that I chose Gansu as my destination was because of the amazing landforms I'd seen in photos of the South of Gansu, on the border with Qinghai, at Zhangye. These coloured striations in the landscape looked too magical to be real, and so I figured I would just have to check them out for myself, so I took another train, through the day to Zhangye, arriving there in the late afternoon. Zhangye is a small town of some 1.2 million people, one of the smallest cities I've been to in China, but for such a small place (in Chinese terms) it has quite a vibrancy. The taxi system in Zhangye doesn't work as in all other Chinese cities I've visited, but rather the taxi driver will take as many passengers independently as can fit into the cab. The pricing system was not something that I ever quite understood. 

Into my cab came three young Chinese travellers, all living in Xiamen but hailing from all corners of the country. Just as I didn't know where I was going, nor did they and so we decided to find a hotel all together. I had never attempted haggling at a Chinese hotel, but the group quickly taught me that it's the done thing, and managed to get a hotel where we were paying a third the price of what I had payed up to this point. Having dumped our bags we headed out for dinner and a stroll around the city. We made our way to the drum tower, a monument which you will find near the heart of any Chinese city and found ourselves walking through the central parks and meeting points in the heat of Zhangye. 

Family, and the link between the older generation and the younger one is a pivotal part of life in China. Parents are most often both working and rather than leaving a baby or toddler in daycare, the grandparents are usually left in charge. You will find, walking through any city in the mornings and the evenings people in their 60s and upwards pushing prams and holding young kids, playing with them and talking amongst themselves. There seems to be an enormous joy in this connection of the generations which is something which only seems to happen sporadically in most European societies that I've spent time in (though I think that the Mediterranean cultures seem to have more of this link). 

In the evenings in any park in any Chinese city I've ever been to you will see grandparents and grandchildren out playing, along with group dancing with the elderly, all in unison with flag, or swords, or fans, or bats and balls, perfectly synchronised to music blasting out of crackly speakers. It's interesting that the atmosphere feels warm and joyful yet there tends to be little smiling from the dancers, and it seems that this routine is taken simply as the evening exercise, rather than a moment of taking pleasure in the music, though it may well be that the pleasure simply isn't shown explicitly.

I took this video some years back in Beijing as couples danced at the Temple of Heaven in the heart of Beijing:

We sat down in the park and watched the dancers, the parents and grandparents and children, and they watched us back. I would be surprised if there were more than a handful of Westerners in this city at any one time and so the attention I got was pretty intense, though the stares felt more those of intrigue than suspicion. I sat there as a little girl, perhaps five or six, stared at me, then walked up slowly, put her hand on my arm, and her other to my beard and stared into my eyes, transfixed. It was a lovely moment and her mother watched on, smiling and amused at her precocious daughter who simply wanted to discover. I spoke with her and her friends for a few minutes, though they were a little too shy to speak much, and my Beijing accent was seemingly not easy for those speakers of the local dialect to understand. The particularly intrigued girl was the one on the right here.
We spent a good hour in the park, watching and being watched and chatting occasionally with locals who would wander over to see what I was up to there.

The next day we got up early and went to find a driver for the day. Again, I realised that while I have a lot of practice with haggling, I should leave this to the masters and so we quickly got an amazing price for a driver to take us to the two sites that we wanted to visit. The first, was the Danxia geological landform and was the place I'd seen in so many surreal and over-photoshopped images but had caught my imagination from the first time I'd come across it.

It was hot, and the air was dry and we were at high altitude. I was still on my way to recovery from my flu and so the day was utterly exhausting, but truly spectacular in terms of the things we saw.

We drove the hour or so to the landforms and took the bus from the entrance into the park. The landscape is as surreal as it is magical, with the most vivid striations of colours I've ever seen. The land looks like dough mix which his been piled up, folded over and kneaded, so that the layers of colour remain perfectly separated but randomly stratified as far as the eye can see. The variations of colour are remarkable, but not at all constant, and we would go from a region with dark browns and virtual blacks to those with layers of what seemed like browns and blues, greys and vivid yellows.

The bus took us from one lookout spot to another, but I struggled soon to have the energy to climb the hills to the highest points, having to stop regularly to catch my breath. We spent a couple of hours hiking up to the lookout points to see the incredible landscapes, and while mine are not quite as vivid as the photoshopped ones you will see online, it was still a spectacular vision.



Having spent the morning in the sweltering dry air we headed off again, getting some noodles by a roadside restaurant and headed off for another two hour drive to the Mati si (Ma-Tee-Ss) temples, closer to the mountains bordering Qinghai province. These are Buddhist monasteries built directly into the Qian Lian hills. The journey there is spectacular as the snow covered mountains loom ahead and you pass through miles upon miles of potholed roads in the arable farmland which takes up most of the usable space in this part of Gansu.

I have tried to sum up my experience at these spectacular temples, but fail dismally given my lack of photography or ability to pay enough attention given that by this point I was just about running on empty, just about able to drag my still weary legs around. This site here gives a far better description than I can muster. 

I did get a couple of shots, of the temples and from the temples looking out into the beautiful valley which stretches out below the mountains.

Anyway, to draw things to a close, this part of China is generally off the major tourist trail but is well worth exploring. It holds some exquisite countryside, some wonderful food, some very important historical sites, many friendly people and some cities which are very different from those you will find on the East coast. That being said, without being able to speak Chinese, this would neither be an easy journey, nor, probably a particularly fun one, but if you have any chance to explore here with someone who does speak the language, I would highly recommend it.

Getting back to Zhangye, I turned in for my last night before heading back, ready to wake at 5am for my train back to Beijing. The train ride was long (getting on for 30 hours), and punctuated by lots of Chinese businessmen who wanted to drink with me, but thankfully I seemed to escape these invitations without insulting too many people. On arriving back into Beijing where summer had fully arrived with 40 degree temperatures and beautiful afternoon storms, I dove straight into the conference and was back to work...

There will, at some stage soon be a few photos from Beijing, and possibly a few from a recent trip to Slovenia to a wonderful conference on chaos and non-linear dynamics.

I am about to start teaching again and will be teaching both a first year maths course and a string theory course, for the first time. Both of these I am looking forward to a lot and my new routine is something which, after some six weeks on the road, is wonderfully appealing. It's going to be non-stop for the next three months but it should be a good deal of fun too!

Of getting lost in Gansu

The mildly alliterative title will have to be taken as artistic license, though my mind did fade into a strange netherworld for a few days during my journey as will become clear soon enough...

It has been almost six months since my last blog post and so much has been happening in the mean time. Summer has come and gone in Cape Town and we are left with a hodge-podge of beautifully chill but sunny days where the blue sky looks to be photoshopped and rainy days where the mountain is hidden behind sheaths of cloud which race down the mountain, laughing at all umbrellas in their path.

Toady is a rainy one, but I am sat in my local cafe, O'ways, which serves wonderfully prepared Chinese teas and satisfyingly homely vegan food to a host of regulars who nod and smile as we see each other on sweltering Saturday mornings as well as the rain-soaked ones. Today I have a steaming pot of pu'er tea sitting on a candle and the egg-timer about to tell me that it has reached the perfect brew. Memories of sitting in a tea-house in Kunming, chatting with the owner for an entire day and tasting some of the most amazing pu'er teas I've ever been lucky enough to sample flood back.

I am now back in Cape Town after a considerable time in distant lands, and I will be away for another two weeks before racing headlong into teaching towards the end of July. To be honest while the travel has been as satisfying and inspiring as always, and the conference I was at in Beijing was productive and fascinating, I am very much looking forward to meeting the new first year cohorts whom I will be trying my best to get through the slings and arrows of MAM1000, renowned as perhaps the scariest course in the university, though in reality I know that every one of the 200 plus student in front of me will have the capacity to ace it given the right push. MAM1000 I have taught before but I will also be teaching honours string theory this year which is going to be a lot of work, but should also be very rewarding.

Before all this starts I will be taking off to Slovenia for a one week school and a one week conference on non-linear dynamics which looks like it's going to be utterly fascinating. On my way to the venue in Maribor, I will pass through Ljubljana which holds powerful and positive memories for me and an overnight stay there will allow for some important moments of reflection.

Right, I have gone way off topic, so back to the last month.

Back in 2008, just before the Olympics in Beijing, I went far West, into the heart of Northern China to one of the holiest mountains of the Daoist religion, Kong Tong Shan, just on the outskirts of the city of Pingliang in Gansu province to hunt for an eclipse. The ensuing, somewhat farcicle adventures were chronicled here.

I was rather fascinated by Gansu on that short trip and vowed to return some day to explore this province which is not, in general, on the tourist trail. The people, on my first trip there, I found to be quiet and reflective, and a lot calmer than their Eastern counterparts, the landscapes are desolate and dry, and the culture is a mix of Han Chinese with strong influences from the Western Muslim minorities and a history built in large part on the trade along the silk road.

This time I had a little longer to explore, and with only a handfull of city names in my head I booked a train to Lanzhou, the capital of Gansu province, a city known for its pollution, for the fact that it straddles the Yellow River and for its beef noodles.

The train ride from Beijing was 18 hours and so I booked myself into a hard sleeper, which means that you have one of 6 bunks in an open compartment with no barrier between you and the corridor where many a sociology major could happily pen their thesis on the comings and goings off the Chinese traveler.  The beds are thin and hard and just long enough that my feet poke into the aisle, but I tend to be able to drift in and out of sleep on these far from ideal mattresses.

The journey was a trial by fire, being thrown headlong into so many of the habits of this land which are both trying and fascinating for the foreigner. Within a few minutes of heading off, a middle-aged woman in my compartment had taken out her phone and started playing her music full volume. while in the UK, it is likely that nobody would say anything at first, everyone else in the cabin would roll their eyes at each other and make tutting motions until a consensus had been built that this wasn't on, at which point the bravest might, possibly, say something.

I searched around to make eye contact with the others in the near vicinity, waiting for the moment where we would all agree, by silent majority that this behaviour was not acceptable and someone would say anything, but nobody looked up from their games, or from staring out of the window, so I simply rolled my eyes to myself and tried to block out the music.

I took this lack of response to mean that nobody minded the music, and reprimanded myself for judging everybody by my particular yardstick of what I considered good behaviour. This same thing happened frequently on my journey, with people playing phone games, TV shows and music out loud to the train, bus, hotel lobby or restaurant, with nobody showing any sense of disapproval.

I asked several Chinese friends on getting back if my appraisal had been correct and the Chinese simply didn't think of this as rude behaviour, and was surprised by the response. It turns out that most people on the train would have been pained by the blasting music, thinking that the woman was very rude and, as is so often used as the excuse, or perhaps the blame, uneducated. However, it simply isn't in the Chinese culture (a term which means far more than does the term British culture - see On China, the best book I have ever read on the history of this country) to alter the hierarchy and poke ones head above to dictate what should and shouldn't be done. In this statement I am both misunderstanding and misrepresenting, but this is roughly the conclusion I have made from speaking with a good number of friends on this subject.

We raced on West, and eventually the music came to an end. Half hour speeches would echo from the train tannoy every now and then, lecturing us on the benefits of drinking milk. Both men and women would hack their phlegm loudly and dramatically and spit either into a corner of the compartment, or, if it was within easy reach, the bin, and smoke would waft through the cabin from those on their way to chain smoke in the spaces between the carriages.

The beginning of this journey may seem negative, but this was really a period of acclimatization for me, remembering those habits which I someone grew more accustomed to when I lived in Beijing and had to re-evaluate in terms of how I would deal with them. In the end one simply learns to live with them for, if you allowed them to bother you, you would be constantly on the offensive and constantly getting into confused arguments with those who didn't think that their habits had anything to do with you. I take this as a lesson in patience and a study into another culture and eventually the zen descends and I can live with these practices.

As night drew on and I drifted in and out of sleep, attempting to meditate on the snores of the men and women around me, I listened to podcast after podcast, which to be honest kept me sane through a lot of my time on this trip.

By 5am the cabin was abuzz again and I helped myself to a breakfast as it flew by on one of the trolleys, the woman pushing it shouting out the daily specials for anyone who happened not to have already been woken by the general buzz of those grunting at top volume on their phones and the tannoy advertisements for some other government-advised health practice.

Around six thirty I found myself in Lanzhou, bleary eyed and ready to explore after finding a hotel not far from the station. On this occasion I had booked in advance but in general I would simply turn up in a city and find somewhere to stay on the spot. I dumped my bags, had a shower and headed off down the street. The noise and smell of a street in China is something which makes me feel truly at home. I would never want, or indeed be able, to live in China again, though I am fascinated and enamoured by the country in many ways, but the pollution and general pace of life in Chinese cities is, I am sure, enough to knock decades off one's life. That said, walking through a Chinese city, exploring the dumpling stands and the noodle shops, marveling at the styles and smiling back at the stares, brings a certain calm detachment which I don't experience anywhere else.

I made my way towards the Yellow River and found myself in Lanzhou's Waterwheel Garden, still early enough in the morning to find old men practicing Tai Chi and women walking around backwards vigorously slapping their arms to promote both mental fitness and good circulation.

Image from http://www.topchinatravel.com/pic/city/lanzhou/attractions/Waterwheel-Park-7.jpg

I found a tea garden in the park where I went for the next couple of days, sitting, reading and watching the world go by and drinking litres and litres of a combination of green tea, goji berries and miscellaneous herbs and fruits which seemed to be a local speciality and good, either for my period pains or possibly for renal problems - I could never quite figure this one out.

Lanzhou is also famed for its beef noodles, which can be found all over the city.  The noodles themselves are hand-pulled and thrown into a rich beef stock with chunks of meat, plenty of chilli, coriander and cabbage, and with a miscellany of fermented vegetables on the side. While nothing compared to Vietnamese Pho in terms of subtlety and fragrance, this is a hearty, very tasty dish which is a perfect way to satisfy a stomach fresh off an 18 hour train ride. The fact that the noodles are pulled in front of you, as the soup is boiling adds to the experience and the freshness of this very famous dish. (From Austin Guidry)

My days in Lanzhou were both a time to acclimatize, to get my Chinese back up to speed, which happened far quicker than I had imagined, and to plan the next part of my journey. I decided to head far West after this, to Dunhuang, perhaps the most famous part of Gansu province, with the Gobi desert at its feet and the Mogao grottos nearby.

Another overnight train ride took me a further 1000km away from Beijing and right to the most distant corner of Gansu, close to the intersection with Xinjiang and Qinghai provinces. (From TravelChinaGuide.com):

On the way I met a couple from Guizhou, who, on seeing that I had nothing planned and was just going to wing it as I arrived there, persuaded me to join the tour they were taking. Having been on one Chinese tour bus before, I was reticent, but also realised that this was probably going to be the most convenient and cheapest way to see the sights.

Arriving in Dunhuang we were picked up by a minibus with 20 or so rather bemused Chinese folk from all over the country, trying to figure out what I was doing there. To be honest I was also trying to figure out what I was doing there, but having traveled enough in China one realises that sometimes you just go along for the ride and don't ask too many questions. We started off by heading to the desert, and the famous Crescent Lake, built around a small oasis and with sand dunes hiding it from prying eyes. A short camel ride across the dunes took us up to see the views of the desert, stretching into the distance.


The site is very touristy indeed, but still impressive and the Crescent Lake feels like so many other sites in China, recently rebuilt and without the authenticity of a Roman or Norman ruin. I think that again my perspective of seeing many partially-destroyed but real historical sites through my life colour my view of such reconstructions and I feel that I have to remove my prejudice somewhat when I come across such Chinese versions of historical accuracy. I can see positives to both attitudes but do find myself less moved by a newly reconstructed model of an historical piece of architecture.

After a few hours in the desert we were fully dessicated, utterly shaken by the camel rides, and ready to move on. The bus took us next to the Mogao caves, a series of hundreds of grottos built into the side of a small mountain and each one holding the Buddhist iconography, statuary and relics of families who have, over the centuries made their offerings to Buddha for a happier, richer and more peaceful life. Unfortunately photography is prohibited within the caves but there are plenty of impressive pictures to be found here.

At this point my Chinese was stretched to breaking point as the two hour tour, in which we visited some two dozen caves, was led by a Chinese guide. Right now I am fluent in basic Chinese but there is a sharp cutoff whereby anything with specialised vocabulary tends to be beyond me. I got the general idea of the use of the caves as family-owned places of worship, but the details were sadly lost on me. In contrast to the Crescent Lake, the Mogao Caves have been very well preserved in their original form, and the enormous statues and incredibly detailed illustrations are spectacular, even without the benefit of knowing the significance of every image or the meaning of the particular hand configurations which one Buddha might have over another.

We stayed in Dunhuang overnight and then next morning started on a six hour minibus trip to Jiayuguan. This was the beginning of some of the physically hardest few days I've ever had and, as we started off on our journey and my body started shaking and waves of nausea started overcoming me, it was clear that I was coming down with something. I was sat in the front seat of the bus, my knees up against the dashboard and completely unable to move, trying to stave off the cramps of being stuck in such a confined space while attempting not to let anything escape from any orifice of my body. Six hours later, unable to eat, and white as a sheet, though thankfully having not embarrassed myself with any unexpected eruptions, we arrived into Jiayuguan. I was shaken from the stress of having to concentrate on every part of my body throughout the journey, and as everyone else started off on the way to the Jiayuguan fort, I slipped away and tried to find a hotel. I hadn't been able to eat since the night before when the Chinese couple I'd met on the train had taken me for the Dunhuang speciality of donkey meat noodles and lamb kebabs.

During the meal the night before we had gone to a number of random restaurants to sample different dishes. At each restaurant the husband had insisted on paying. I had protested, pleading to be able to at least pay at the next place. He of course had agreed, and then reneged on the promise as soon as it came to pay the bill, physically pushing my wallet away. In the last restaurant I got up as the pile of kebabs diminished and headed to the counter to pay the bill. He rushed up, just as I had given the money to check the bill and looked alarmingly at the amount, embarrassed that it was the largest bill of the evening. As we went back to the table, the mood changed and I realised that in trying to be fair, I had really made an enormous faux pas. In attempting equality, I had managed to humiliate this man in front of his wife, and in front of the owners of the restaurant. I have always seen the game played whereby everyone tries to pay the bill after a meal in any restaurant in China, and have often played along, but I realised, all too late on this occasion that it really wasn't a game, and that as a traveler in this man's country, I really should have graciously accepted his payment for everything. The dour look on his face, in contrast to his previous constant shouts of 'gan bei' and beaming smile of being in the company of a foreigner in such a distant place, was enough to make me understand quite how serious a mistake I had made.

On leaving the restaurant he leapt across the street to one of the many stands with vast mounds of the local dried fruit on sale and bought me two enormous bags of apricots and figs, smiling as he handed them to me, showing that with this gift, all was restored and again we could be friends.

Though I have been coming to China for almost a decade now I still find myself in these embarrassing situations, still unable to know quite how to play the game, or indeed whether I should be playing the game at all or not. Thankfully on this occasion the balance was restored with the gift of dried fruit, but I am sure that I have done things which have humiliated others with a simple action or word that I have unknowlingly let slip. I'm sure that I will be coming back to China for years to come and am certain that I will learn new lessons each time I do.

Right, for now that will have to do, as I find myself quickly descending into delirium in Jiayuguan and without a hotel to stay in. I shall continue this as soon as possible and recall the tales of being stuck, unable to eat or get out of bed in the middle of the middle of China, of walking around in a daze in the fort by the Westernmost edge of the Great Wall, of the surreal landforms at Danxia near Zhangye and of my subsequent thoughts as I returned to Beijing.

Of light and ice over Cape Town

(With apologies for the strange alignment. Blogger is playing funny games with me today).

I thought I'd do a bit of a summary of the atmospheric effects I've seen over the last few months as I will be giving a talk tomorrow at Northumbria University on atmospheric optics and it seemed an opportune moment. In the six months so far in Cape Town I've seen some of the most amazing atmospheric optics I've seen anywhere in the world. With such a combination of weather systems in such a small space, this really isn't very surprising, but it's always a wonder to see them interacting in real time.

The Table Cloth is a sight which can be seen so regularly that it's easy to forget quite how unfamiliar to most such a phenomenon must be. When conditions are right, a layer of cloud can be seen literally pouring off the top of table mountain. It's a spectacular sight and the speed, coupled with the diaphanous nature of it is quite beautiful: 

I took this photo from Lion's head on a wonderful full moon hike up to the (almost) summit. This spectacular view can be seen just an hour's walk and a five minute drive from the city centre.

I live just below Devil's peak (one of the main peaks of the mountain), and while I can't see the cloth lying this flat from my balcony, I can still see some wonderful cloud formations due to the airflow patters around the peaks. I came back home a couple of months back and raced home, seeing some incredible lenticular clouds forming over my apartment block.  These are somewhat warped lenticular clouds, but the soft edges and dense centre are defining features of this formation. From my office alone I regularly see wonderful cloud shadows. These are shadows of clouds cast on layers of mist and fog, and quite often the mountain itself casts this shadow as the sun sets behind it. I took this from my office a few weeks back and you can see the lines of crepuscular rays along with the shadow of the peak cast on the thin layer of fog below

We've had some lovely halos on campus too, this one perhaps the most perfectly placed around Jameson hall, the main auditorium on the Upper Campus of the university: 

On a trip down to the Cape Of Good Hope just before Christmas I was treated to another amazing display, with one of the clearest halos I've ever seen. The cirrus clouds absolutely filling the sky from horizon to horizon made their appearance inevitable. As always I pointed these out to passersby, with varying levels of enthusiasm in return.

Sun Dogs have so far been rather elusive, but this is mostly because at dusk, when you are most likely to see them, the sun is hidden behind the mountain from where I am.

I've seen a couple of Green Flashes but haven't managed to capture anything very clear so far. I shall be looking at a new lens in the coming months and that should help with getting a lovely crisp image of the flash.

Anyway, there have been plenty more amazing sights but these have been the atmospheric highlights so far. In the New Year I'm hoping to get out to really remote areas to see some dark skies for a bit of astrophotography and will update as soon as I do.

Of accents and onion rings

I find myself in the North of England for the first time in years, surrounded by accents which I can't quite get a handle on and an England that feels somehow familiar yet foreign. Because I come back so infrequently, I tend only to see Oxford and London and so my memory of the beautiful diversity of this country fades. It is lovely to be reminded of this again, and having spent Christmas in Swansea having had a scenic drive through Southern Wales, I'm getting a good dose of this for the first time in far too long. While I have traveled the world, I feel that I've neglected seeing much of the UK. I don't really understand the idea of patriotism, yet I still find coming 'home' to the UK a rather pleasant sensation, even if I am coming here from a city which I am quickly coming to feel very comfortable in.

I took the train up from King's Cross this morning, racing through beautiful countryside and being blessed by the Angel of the North as we tore up through York, Darlington and Gateshead and arriving finally in Newcastle. I'm staying with an airbnb host, a woman in her 60s who has a lot of travelers staying with her, often working in the university, as I am. She was arriving home later so I found a barber's shop to give my beard a much needed trim. I realised immediately how hard I was going to have to concentrate to understand the accent up here. Many things have happened to my English as I've lived on different continents. While native English speakers think such an idea very strange, I've had perhaps a dozen non-native speakers ask me if I'm Irish of late. I think that my accent has simply been softened by being around so many different accents over the years and I've definitely simplified my grammar in order to make myself better understood. Living in China also meant that the way that I asked questions changed due to the nature of questions in Chinese. I will often ask a question as a statement with an upward intonation at the end (where a question particle would go in Chinese). This has confused people greatly over the years, most notably Italians for some reason.

I popped into a cafe to read a little before going to my host and was greeted my a woman behind the counter whose words I truly didn't understand at all. Luckily, having bought coffee on a number of previous occasions I knew the protocol but I am a little thrown by not being able to understand my own countryfolk.

I took a quiet stroll through the streets of the city centre as it grew dark and have so far been very impressed by the beautiful architecture around this part of the city. I jumped on a bus and took it towards Wallsend, going over the Tyne and past Northumbria University, where I will spend the next two weeks in the mathematics department. Hopefully I will be able to talk soon about what I will be doing there, but truth be told, I'm not entirely sure myself yet.

While I feel very safe in Cape Town, it is part of the way of things there to know where you can and can't go at what time of the day or night, so I found myself asking my host whether it was ok to walk around the area at night. She replied that while the locals might swear quite a bit, they mean no harm and with that I pottered off to a shop to buy some groceries for the coming days. I was greeted by a friendly shopkeeper, perhaps of Turkish descent, who regaled me with tales of the enormous onion rings he and his wife were presented with in Thailand. Come tomorrow if I am lucky he may be able to show me the pictures on his phone, comparing said onion rings with a large glass of coke. I shall certainly be back as to miss such an opportunity would surely be a crime.

For now I have a little work to be getting on with and so will dive into it before I fall asleep, in a new bed, in a new city. The familiarity of this novelty is an interesting paradox.