Snake in the Box

We have a toy – no one seems to remember how we acquired it – which is a wooden box with a lid. When you slide it open, a snake pops out.

I don't have a pic of how it originally looked, but here are a couple of product photos from Amazon:

I recently painted our box. It now has a coiled snake on one side:

...and a mini snakes-and-ladders board on the other:

I used my recently-acquired set of acrylic markers which, despite being super cheap, are a delight to paint with.

I also painted another box and sent it to Tommy. It's a good gift for him because he's working on the snake-in-the-box problem in higher dimensions. His box has a coiled snake like mine, plus another snake traversing the edges of the box in a manner that respects the constraints of the problem.

When I went to the post office to mail it, they asked me what the box contains. I said it's a wooden toy, but in their customs form, they only had an "electronic toy" category – no "wooden toy" or even just "toy". A sign of the times! In the end, they filed it under "handicrafts".

Speaking of snakes, the other day at the Dhakuria Lakes, we saw an Oriental rat snake's unsuccessful attempt to hunt a rat. The snake climbed halfway up a tree and crept into a hollow in which the rat was hiding. But the rat jumped out just in time and scurried away, seemingly unharmed.

Fundamental Questions

Two perfectly-phrased questions (one of them rhetorical) that I came across online:

1. Last year Straits Times published this article (paywalled) with the somewhat clickbaity title: 'Wild boars likely to recolonise whole of Singapore in next decade: Study'.

The article was reposted on HardwareZone, where one forum member asked the all-important question:

means what? good or not good?

Straight down to brass tacks; a classic Singaporean response. I often think about it when reading academic papers or attending conferences.

2. Twitter user Zuvele, on her kindergarten-age nephew:

I asked my nephew if he could count to 1000. He said "no". I asked if he could count to 100 - yes. 200 - yes. Why not 1000? He looked at me wearily and said, "Who has the time?"

The Better-than-Average Effect

A 1981 study by psychologist Ola Svenson (PDF link) asked two groups of participants, Swedish and American, to compare their driving skills to those of their peers. They found that 93% of the US drivers and 69% of the Swedish drivers believed themselves to be more skillful than the median driver in their group.

Now it's theoretically possible for more than 50% of a group to be better than average at something, if average is taken to be the arithmetic mean. But the Svenson study was about the median, so the participants' beliefs can't possibly be true. This phenomenon, known as the better-than-average effect (BATE), has been replicated in multiple studies, and in many areas of life.

But I think this particular form of irrationality isn't limited to believing one is better than average. In my (completely anecdotal) experience, it also extends to some value-neutral domains (where there is no obvious better or worse), and at least one domain where I suspect most people believe they are worse than average.

Procrastination, I think, is an example of the latter. My guess is that most people believe they are worse than the median (i.e. that they procrastinate more than average).

I suppose you could turn it around and argue that this is just another manifestation of BATE (people think they are better than average at procrastinating). But that's just a matter of framing, and if it comes to that, most questions could be similarly flipped. For example, instead of asking drivers if they are more skillful than the median, you could ask if they are more likely than the median to cause an accident. In the second case, my guess is that most drivers would say they are less likely than the median.

BATE typically skews towards positive self-evaluations – in fact the article I linked to earlier defines it as "the tendency for people to perceive their abilities, attributes, and personality traits as superior compared with their average peer" – and procrastination is generally considered a negative trait. So if my hunch is right, procrastination is an exception: an example of a worse-than-average effect.

Now for the two value-neutral examples.

Consider the question of how strongly you feel versus how much you show. We might call this trait emotional demonstrativeness. For example, Chris Evert, in the passage I quoted in this post, was suggesting she was less emotionally demonstrative than Goolagong. If you did a survey, I reckon you'd find that most people think they are less demonstrative than the median.

Of course, how strongly you feel is a subjective state, so it's impossible to empirically compare emotional demonstrativeness. As I wrote in that post, "Presumably Goolagong reacted more vehemently than Evert did when she missed a volley. But perhaps Goolagong really did feel the disappointment more keenly – who can say?"

My other value-neutral example, however, is empirically testable. Variability in human attractiveness to mosquitoes can and has been studied and compared. Nevertheless, if you did a survey, I think most people would say they are more attractive to mosquitoes than the median.

Kolkata, where I grew up, and Singapore, where I live now, both have lots of mosquitoes, and over the years, I've heard many people say they are unusually attractive to them. I recently went hiking with a friend, and she said (unprompted) that she is less attractive to mosquitoes than average. As far as I can remember, it's the first time in my life that I've heard anyone say that.

ノスタルジア

I used to really like maths in high school, but for various reasons I didn't pursue it afterwards. Maths is still sufficiently a part of my life to be a blogpost category, but there were things – trigonometric identities, ways of solving differential equations – which I once had at my fingertips, but now have to painstakingly work out from first principles (if I can at all).

When I lived in Japan, I became halfway fluent in Japanese. After I left, I never made a sustained effort to keep in touch with the language, and now it makes me feel in equal parts sad, frustrated and stupid when I have to slowly parse a simple sentence to understand its meaning.

Sometime back Tommy wrote me an email which involved no maths, but where he used the phrase "Without prejudice or loss of generality". It took me straight back to combinatorics proofs (which I loved), and the wave of nostalgia hit me with surprising intensity, almost like a physical wave. More recently I had the same feeling at a European airport where all the announcements were in English, but suddenly and unexpectedly there was one in Japanese, asking Kanada-san to report to Gate No. ---. (I am not hiding the gate number; by the time I had translated the first part of the announcement in my head, I had missed it.)

Snatches of languages which I'm slowly forgetting seem to trigger a linguistic equivalent of the Proust effect.

* * *

The Brazilian footballer Philippe Coutinho scores a lot of goals with a trademark right-footed curling shot from just outside the penalty area. Last year he moved from Liverpool to Barcelona, and this week he scored just such a goal for his new team. A Liverpool supporter on Reddit wrote an unusually poignant comment: "it's like suddenly remembering that funny thing your ex used to do".

* * *

There are plenty of Japanese words which are said to have no equivalent in English (I have been guilty of invoking some of them myself). It amuses me therefore that the Japanese word for nostalgia, the title of this post, is simply a phonetic rendering of the English word: nosutarujia.

Tetris

Ultimately, entropy will prevail, but meanwhile we delight in small, fortuitous victories which buck the trend. Or to put it another way, you will eventually lose at Tetris, but until the stack grows too high, sometimes you get the satisfaction of just the right piece for just the right space.



Our food processor is from India, so it came with a Type D plug, and in the UK we had to use it with an adapter. At some point the plug broke. I decided to swap out the Type D plug for a UK Type G, thereby obviating the need for an adapter. Before I could buy a plug, we bought some darkroom equipment off eBay. It came with various odds and ends, including an old, solidly-constructed "Made in England" plug. I quickly united this unattached plug with our plugless food processor, and I am pleased to report that the two are now working in perfect harmony.

We also had a saucepan lid whose knob broke off. Nevertheless, I continued to use it for nearly a year. Lifting the knobless lid off the saucepan involved a complex manoeuvre: sliding a fork between the pan and the lid to lift up the lid slightly, then grabbing its raised rim with oven gloves to take it off. Many is the time I contemplated buying a new pan, but I disliked the idea of buying a pan-plus-lid when I really only needed a lid.

My flatmate recently got a pan for free with something else she bought. The first time she cooked in it, some of the teflon(?) coating peeled off, so she decided to discard the pan. This pan too had a lid, but of a different size. But when I took off the knob on its lid and tried it on the old lid, voila! It fit perfectly. Ah, the simple pleasure of being able to lift a saucepan lid at a moment's notice and with a minimum of effort.

I get more joy than I should, out of these types of incidents.

* * *

While writing this post, I got curious about whether a fast-enough player can theoretically play an infinite game of Tetris. I found out that in 1992, John Brzustowski set out to answer this very question in a Masters thesis in applied mathematics. He also conducted a survey where he asked Tetris players to give one piece of gameplay advice. The responses, like go proverbs, read like profound pieces of life advice:

•	Stay calm.
•	Don't wait for that perfect piece.
•	Pretend you are having sex.

Horsetail

My new hobby is nature-journalling, inspired by an online course I'm taking (free) and a book I bought this week (rather expensive). At present my kit contains only an HB pencil and an eraser, so the sketches are pretty bare-bones. But on the plus side, they only take a few minutes to complete.

Oliver Sacks offers three reasons to like horsetails: "their simplicity, their antiquity, and their mathematical elegance." It is said that the decreasing size of the segments inspired John Napier to invent logarithms.

Speaking of simplicity, the notebook pictured here – A6 recycled-paper from Muji – is an absolute joy to use.

Leisure Deficit

“I seem to have banged on this year rather more than usual,” observes Alan Bennett in his latest collection of diaries, Keeping On Keeping On.

I came across this line in a book review I read last week and thought well, this is certainly not something I can say about myself. In 2017 I wrote fewer blogposts than any other year since this blog began.

Since I started my PhD in 2014, I have a little more free time than I did when I worked in a law firm. Funnily enough, this free time seems more 'crowded' than before. For some time I've been pondering why this is so, and I now have a theory which is as follows:

Let's say I have F hours of free time per day. Of that, I tend to spend some part (P) coming up with new projects (say P = F/8). The free time I would need to properly pursue all these projects (F*) is a function of P (say F* = 12P). F* − F is my leisure deficit: the gap between the free time I want and the free time I have. (At this point, you might pause to remark that I have a depressing habit of treating leisure like a resource to be exploited for maximum yield. You would be right.) Anyhow, for the (admittedly speculative and simplistic) values I used above, the leisure deficit turns out to be F/2. Which is to say, the more free time I have, the greater my leisure deficit.

Suppose as a finance lawyer, I had an average of 2 hours of free time on weekdays. Then F* (the free time needed) was 3 hours. Now I may have, say, 4 hours of free time, but F* is 6 hours, and the leisure deficit is 2 hours: twice as much as before. As with anything else, it's easier to see graphically:

I was thinking about a new year's resolution to spend more time working on my existing projects and less time coming up with new ones (my Japanese teacher once told me, in a periodic performance review, that one of my weaknesses is that I have too many hobbies). But I could also just make my peace with having some unfinished projects. In one of his essays, Montaigne, a kind of proto-blogger, wrote, "Let death take me planting my cabbages, indifferent to him and still more to my unfinished garden." Though it is not clear from the quote if Montaigne, like me, was wont to leaving projects unfinished simply because he got distracted by a new project; death is a more watertight excuse.

What fate awaits these unfinished projects? Some bide their time in cupboards, like the papier-mâché fruit-bowl which I made but still haven't painted. Others have only an incorporeal existence in my bookmarks folder. These include my abandoned attempts to learn Russian (Languages folder) and meditate every day (Psychology and meditation folder).

In case you're wondering, the parent folder is called Fitness because it started life as a collection of webpages on workouts and fitness plans. Later I subsumed some other folders under Fitness to keep things organised, and on the basis that they too promote a kind of fitness – mental fitness, if you will. The original bookmarks now live in the folder called Actual fitness. Or perhaps I should have called it: Fitness fitness.

Thankfully, some of the projects in the folder are still very much alive, like Knitting, which I learnt to do last month. Others, like this blog, are active, but get less attention than they deserve.

---

Edit: Since writing this post, I found out that the Swedish economist Staffan Linder also used the phrase 'leisure deficit' though, I believe, in a slightly different context. I will read his book later this month and update this note.

Goat Cupboard

A coat cupboard at LSE has an amusing modification made with a ball-point pen.

I have a soft spot for this one because it reminds me of (a) a modification which inspired one of my favourite sites on the internet, and (b) one of my favourite maths puzzles.

Day Length and Decision Theory

I am in currently Copenhagen where, 15 days after the solstice, there is no such thing as night. Daytime gives way to civil twilight, then nautical twilight – and that is the darkest it gets.

Growing up in Calcutta, I experienced relatively little seasonal variation in day-length. As far as I remember, it affected my life in only one respect: I was allowed to play in the streets in the afternoon on condition that I'd be back before dark, which meant I could stay out a little later in summer.

The longest day of the year in Calcutta is less than 3 hours longer than the shortest day. In London, where I now live, the difference is almost 9 hours. I sometimes wonder: if I had to choose between Calcutta and London based on day-length alone, which would I pick?

Graph made using data from ptaff.ca

Other things being equal, I prefer longer days. By moving to London, I gained about 350 daylight hours in summer, but gave up the same amount in winter.¹ The question is, does the loss offset the gain?

Most people are thought to be loss averse: the pain we experience if we lose £100 is more than the pleasure of winning £100, and in general, bad things have more impact than good things. The psychologist Paul Rozin illustrated this beautifully: "a single cockroach will completely wreck the appeal of a bowl of cherries, but a cherry will do nothing at all for a bowl of cockroaches."²

But when it comes to day length, I am not quite sure what I prefer. Sometimes I lean towards more variation, sometimes towards less. Perhaps this means I am indifferent!

A question for the reader: How much variation seems optimal to you? No variation (12-hour days all year, like at the equator), extreme variation (6 months of darkness and 6 months of light, like at the poles), or somewhere in between?

---

1.	In reality the gain is not exactly equal to the loss, but let's pretend it is, for the sake of simplicity.
2.	As quoted in Thinking, Fast and Slow by Daniel Kahneman.

Jupiter–Venus Conjunction

Venus and Jupiter, the two brightest objects in the night sky (excluding of course the Moon), were very close tonight – just a third of a degree apart (for reference, the full Moon is about half a degree in diameter). We will not see a closer Jupiter–Venus conjunction until 2039.

I positioned myself at a spot whence, I calculated, I would see the two planets set over the Shard. But my trigonometric efforts were in vain; clouds obscured the view while the sky was still quite bright and the planets some distance above the horizon (at the upper left in the photo below). Oh well, I'll try again tomorrow.

Whenever I see Venus in the evening sky, I think of these lines from Terrapin Station by the Grateful Dead:

Counting stars by candlelight / All are dim but one is bright / The spiral light of Venus / Rising first and shining best...

The American Pound

An American professor teaches my Philosophy of Economics course. An incident from Monday's class:

Professor:	On the y-axis we have pound benefit to farmers. [tries to draw a pound symbol, fails, tries again, draws an even worse one]
Class:	[laughter]
Professor:	I... I don't know how to draw a pound.
Class:	[more laughter]
Professor:	I'll practise, I'll practise.

Below is a photo of his effort.

The Roti Algorithm

Rotis (by which I mean the thin, circular rotis cooked without oil, called ruti in Bangla) are made by rolling atta (unleavened wholegrain wheat) dough into thin circular disks, which are then cooked over a dry tawa.

Cooking one roti at a time is inefficient because firstly, you waste time placing each roti on the tawa and then taking it off, and secondly, by the time the last roti is cooked, the first one is no longer warm. It is possible to cook several rotis on the tawa simultaneously, but to be properly cooked, each side of each roti must be in contact with the tawa for roughly the same period of time. Like most experienced roti-makers, my mother achieves this by flipping the rotis in a complex sequence. But when I asked her, she could not tell me what exact sequence she follows, because the technique, born of long experience, comes naturally to her.

So I gave some thought to the problem, and came up with an algorithm for cooking any number of rotis on the tawa at the same time, thus bringing advanced techniques within the grasp of even the most hapless roti-noob.

The algorithm is best illustrated by a flowchart:

A couple of definitions which are used in the flowchart or later in this post:

"Lowest Side" means the lower side of the bottom roti in the stack (i.e. the side in contact with the tawa).

"T" means the time taken to cook the Lowest Side, and is counted from the instant of completion of the most recent flip of all the rotis in the stack.

Some notes on the algorithm:

•	As it is inconvenient to flip or remove rotis which are in the middle or bottom of the stack, I designed the algorithm so that (a) the flip operation only involves flipping either the top roti or the whole stack, and (b) once a roti cooked on both sides, the next action always brings it to the top of the stack whence it can easily be removed.
•	On high heat, the Lowest Side gets cooked before it can conduct much heat to the roti above it. So while the Lowest Side is being cooked, the states of any rotis above the bottom roti are not significantly affected.
•	To make multiple rotis at once, the dough has to be fairly dry so that the rotis in the stack do not stick to each other. If they're still sticky, it helps to spread a thin film of dry atta on each roti.
•	It is rarely necessary and never practical to cook more than 5-6 rotis at a time, because the time thereby saved is offset by the difficulty of flipping several rotis at once. But the algorithm works for any number of rotis.

The algorithm is actually easier to master than the flowchart might suggest. Once you get the hang of it, the next step is obvious even without referring to the flowchart.

To give you an idea of how the algorithm works, I made a short animated video (0:36) for 6 rotis. Each side of each roti is represented by a black rectangle, which turns red when it is cooked. T in the video is 1.5 seconds, which is shorter than the actual time it takes to cook one side of a roti in real life.

Six Days in Kyoto

A friend of mine who was travelling to continental Europe for the first time recently asked me for advice on how to plan her trip, specifically, whether to cover lots of cities, or visit fewer cities but spend more time in each. As someone who likes to travel but is perenially constrained by holiday and budget limitations, this is a problem to which I have devoted much thought. And my musings have led me to come up with the Urban Travel Enjoyment Curve (UTEC).

The exact shape of the curve depends on a number of factors including personal preferences, the city in question, and the expense of travelling to and living in the city, but the form of the curve (for me, and – I believe – for most people) is something like this:

If, for instance, I misguidedly planned a trip to Paris where I would only get to spend 30 minutes in the city, my enjoyment would actually be negative. The time, money and effort expended in travelling to Paris and back would outweigh any enjoyment I might derive from my 30-minute visit. So for small values of t, UTEC lies below the t axis, i.e. enjoyment is negative.

At a slightly higher value of t, perhaps around 30 hours, UTEC intersects the t axis. This is the point where I would be indifferent between going and not going for the trip.

Initially, each unit of time spent in the city produces increased enjoyment, and the rate of increase is positive (i.e. UTEC is convex downward). Then it reaches an inflection point, after which enjoyment still increases but the rate of increase is negative (i.e. UTEC is concave downward). Finally, after I have had my fill of the city, my enjoyment peaks and then goes downhill because of mounting hotel costs, foregone opportunities to visit other places, and so on.

If only we had a formula for each city's UTEC, trip-planning would be reduced to a simple optimisation problem. Since we don't, we have to rely on guesswork, which means UTEC – like most things on this blog – is mildly interesting but ultimately useless.

One trip where I got my planning absolutely spot-on was in the summer of 2011, when I was living and working in Tokyo.

For the six-day Golden Week holiday, I toyed with the idea of visiting several cities, but eventually I decided to spend all six days in Kyoto. Kyoto has dozens of wonderful things to see and neighbourhoods to explore. Besides, it would be ironic to do a whirlwind tour of, of all things, Zen temples and rock gardens built for contemplation.

With the luxury of six full days at my disposal, I took in the sights at a leisurely pace, spoke to many people I met, and slept in parks when I tired of walking. Some of my nicest memories are of places I would probably have left out of a tighter itinerary. Here are three such stories.

A Chance Encounter

For reasons not entirely clear to me, Japanese visitors to Kyoto sometimes like to dress up as geishas and stroll around Minami Higashiyama.

Many photos of "geishas" taken by foreign tourists are in fact of tourists in costume; real geisha are elusive and few.

But late one evening in Gion, returning to my hostel after dinner, I unexpectedly saw a real geisha. We crossed each other in a narrow alley; I hadn't noticed her until she was five feet away. It seemed disrespectful to take a photo, and in any case, I was too awestruck to do anything.

Friends have asked me how I knew she was a real geisha. There was something about her walk, her poise and the way she wore her clothes; I saw her and Knew.

Philosophy and Tortoises

My favourite walk in Kyoto is Tetsugaku no Michi (Philosopher's Path), a pedestrian path along a cherry-tree-lined canal. The path goes past temples, shrines and back-gardens, and is named after a Zen philosopher who used it for daily meditation.

It was a pleasant walk, but more than that, it was fun because I found myself ascribing philosophical implications to the most mundane things I saw or heard: the overheard conversations of other walkers, a spider ensnaring a butterfly, a swimming tortoise.

Actually I might have missed the tortoise, if not for the two Japanese guys who were observing it with interest.

The tortoise was trying to swim across the canal but the current impeded it, and though it had nearly made it to the other bank, the little guy was evidently at the end of its strength. One time it clamped its jaws onto a reed, but the reed broke off and was swept away with the current. Finally, after a supreme effort, the tortoise safely reached the bank. The Japanese guys looked at me delightedly and said, "Yokatta!" Which means something like "(S)he did it!"

The Cutest Thing I Ever Saw

Shimogamo Jinja is not as spectacular or historically significant as some other shrines in Kyoto, but I wanted to visit it because behind the shrine lies a sacred grove known as Tadasu no Mori (the Forest Where Lies Cannot Be Concealed). Not surprisingly, it is a popular spot for settling disputes and for first dates.

But I had an unexpected treat in store. There was some kind of competition going on, and the courtyard in front of the shrine was filled with young kendo trainees.

Scores of toddlers wearing balloons on their helmets and bashing each other over the head with sticks – if there is a cuter sight than that, I am yet to see it.

2 + 2 = 5

When I was in first year of college, a senior who was preparing for a competitive exam asked me to help him with his maths. For each hour of tuition he would treat me to biryani, which I considered an excellent deal.

In our second lesson, I explained how to solve quadratic equations by factoring by inspection.

Then I derived for him the general solution for ax² + bx + c = 0, the quadratic formula which in Indian textbooks is often called Sridhar Acharya's formula after the 9th century Indian mathematician who described a general method for solving quadratic equations:

I advised my friend that if a and c are integers with relatively few factors, factoring by inspection is quicker, so he should try it first before using the quadratic formula. He rejected this suggestion with an argument of such staggering irrationality and misplaced patriotic pride, that I could think of nothing to say in reply: "When there is a perfectly good method discovered by an Indian, why should I use another method?"

The incident came to mind because I noticed an amusing photo on the Guardian (online edition) front page today:

The teacher(?) is factoring a quadratic polynomial 2x² + 5x + 2, but the expression in the second line is incorrect: (2x + 2)(x + 1) actually equals 2x² + 4x + 2. The second line should read (2x + 1)(x + 2).

And yeah, there also should be a full-stop after 'charts'.

The 1/27720 Sheppey Asparagus

It is said that Jiro Ono, perhaps the finest sushi chef in the world and a fanatical perfectionist, always uses his own palm to measure the quantity of rice for each piece of sushi, because an assistant's palm would introduce inconsistency.

My approach to cooking is far less exact. Onlookers are often alarmed to note that I don't bother to measure out spices and condiments, preferring instead to pour apparently arbitrary quantities from the containers directly into the cooking pot.

However, following the introduction of a new cutting board in our kitchen (a standard Sainsbury's cutting board, which Anasua engraved with a laser from her lab), guests can be sure that their food will have evenly-chopped ingredients.

This asparagus tip, for example, is 1/27720 of a Sheppey. As Douglas Adams fans may know, a Sheppey is a unit of length, defined as the closest distance at which sheep remain picturesque.

Below left is a photo of sheep grazing on snow on a Himalayan hillside, taken at ~1 Sheppey. Below right is a sheep seen on a day walk in Kent at a distance of much less than a Sheppey.

Dashy Writing

My work notebook is quite organised (once I left it in a partner's office and she returned it to me, saying she guessed it was mine because of all the tables and numbered lists). But sometimes I retrospectively clutter the pages with doodles or recreational maths. Here is a page where I worked on a practical problem of geometrical optics.

 

Unfortunately, my handwriting in my work notebook tends to be slipshod. I suspect it is not quite at the stage where, as suggested in the early 20th century Lessons in Dashy Writing, it can be a promising ladder by which I rise in the world.

 

Sunset on the Grand Canal

Yesterday, after dinner in Venice and with nothing better to do, I applied myself to the problem of finding the perfect spot on the Grand Canal to watch the sunset.

My objective was to find a spot whence the setting sun can be seen reflected in the water of the canal. This is not as easy as it may seem: unlike, say, a west-facing seashore from where you can always see the sun go down over the water, the Grand Canal curves this way and that.

It is possible to find the right place by trial-and-error. But this would involve a lot of walking about (I speak from experience) and is not guaranteed to produce results: the Grand Canal is nearly 4 km long, and not all sections have fondamenti (canal-side paths). Alternatively, you could take a vaporetto (water-bus) ride along the canal, but (a) by the time you realise that you are at the right place, the boat may have taken a turn, or (b) when you are at the right place, the sun may be too high or have already set, or (c) more aggressive tourists may have occupied the best windows. The other option of course is to hire a gondola, but private water transport involves more cash than I am willing to shell out, and is therefore outside the scope of this discussion.

So I did some maths.

The Grand Canal is shaped like an inverted S.

Image created by NASA, used with permission.

If you modelled a mathematical function to approximate its shape, tangents to the curve would make all possible angles to the horizontal. For a given solar azimuth angle, there are at least two spots on the canal where the tangent is parallel to the perpendicular projection on the surface of the vector from your position to the sun.

My maths not being advanced enough to come up with such a function, I cheated a little. I used an online tool to calculate the azimuthal angle of the sun. Then, with the aid of a ruler, I approximated tangents to the curve and determined the points where the angles matched. A spot between the San Stae and Ca' d’Oro vaporetto stops looked promising. So today, for about half an hour up to the predicted time of sunset, and armed with a ticket which gives me unlimited vaporetto rides, I embarked on a succession of rides back and forth between these two stations.

The photos are not out of the ordinary: it wasn’t an especially spectacular sunset, and there were no buildings with interesting silhouettes. But it was fun to do the maths.

Minimalism

I have finally fixed the dice. (This is a random post generator link on the sidebar; it had vanished after I switched to the new template.) My achievement of the weekend.

It is one month since v3.0 was unveiled. I am happy with the new look, though I can’t say the initial goal of minimalism has been realised. But for lovers of minimalism, here are three of my favourite anecdotes on the subject.

1. Spartans

The Spartans were renowned for being men of few words. Indeed, Spartan is almost a synonym for minimalist, and the word laconic derives from Laconia, the principal region of the Spartan state.

Herodotus (The Histories, Book 3.46) tells us that when the banished Samians reached Sparta, they came before the magistrates and, as was customary, made a long speech to show the greatness of their need. But the Spartans answered that they had forgotten the beginning of the speech and could make nothing of the remainder. After this the Samians came a second time with a sack, and said nothing but this: “The sack wants flour.” The Spartans replied that they need not have said “the sack”; however, they resolved to give them aid.

2. Physicists

Wikipedia informs us that British physicist Paul Dirac was notoriously taciturn. After a lecture he gave at the University of Toronto, a member of the audience remarked that he hadn’t understood part of a derivation. There followed a long and increasingly awkward silence. When the host finally prodded him to respond, Dirac said, “That was a statement, not a question.”

3. Mathematicians

My high-school book on number theory had the story of how Mersenne’s conjecture was demonstrated to be false. In 1644 the French monk Marin Mersenne stated that the numbers 2ⁿ – 1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n < 257. Mersenne admitted that he had not tested all the numbers, but owing to the notorious difficulty of integer factorisation, his conjecture went unverified for two and a half centuries.

In 1903 F. N. Cole made a presentation to the American Mathematical Society with the rather bland title, On the Factorisation of Large Numbers. Cole’s ‘lecture’ went thus. He approached the chalkboard and in complete silence proceeded to raise 2 to the power of 67. He then carefully subtracted 1, arriving at 147,573,952,589,676,412,927. Cole then moved to the other side of the board, wrote 193,707,721 × 761,838,257,287, and worked through the multiplication in longhand. The two results were equal. Cole returned to his seat, not having uttered a word during his hour-long presentation.

It is said that this is the only lecture in the history of the AMS where the audience applauded.

If I were Cole, I would have arranged for T-shirts to be sold outside the lecture venue:
M₆₇ is composite.

Truck on Lake Road, Kolkata

Are you a struggling mathematician? Do you find you are unable to prove theorems? Or worse still, when you do prove them, do you find that your proofs are unsound?

If so, help is at hand.

Subway Phonaesthetics

Out of all the subway systems I have travelled on (Calcutta, Delhi, London, Istanbul, Tokyo, Kyoto), I think Tokyo Metro has by far the nicest sounding announcements. My favourite is the announcement for Aoyama-itchōme. I love the gliding vowels of Aoyama flowing into the geminate ch of itchōme, the ‘long vowel’ ō and the gentle, abbreviated me.

I made a recording today, and if it doesn’t sound as nice as I just made it out to be, you can blame it on the audio recording on my camera.

This is the romaji text of the announcement:

Tsugi wa Aoyama-itchōme. Aoyama-itchōme desu. Norikae no go annai desu. Hanzōmon-sen, Toei Ōedo-sen wa onorikae kudasai.

A literal translation would be as follows:

The next one is Aoyama-itchōme. It’s Aoyama-itchōme. Transfer information: please change here for the Hanzōmon line and the Toei Ōedo line.

Of course, the translation fails to convey some information, especially the level of politeness and formality expressed in the Japanese announcement. Politeness can translate in interesting ways. There is a (possibly apocryphal) story about a Japanese maths professor who used to tell his students, “Please let n be an integer.”

A ploy to encourage reader participation: What is your favourite subway announcement? And do you like it for the sound, the associations, or something else?