Butterflys in my bathroom

It has not been a particularly successful weekend for me plumbing-wise.

The butterfly flapped its wings.

My bathroom basin had a dripping tap, so I surmised that the tap needed a new washer. So, off I went to the local hardware store to buy said item, and returned to unscrew the top of the tap to fit the new washer. Unfortunately, the tap thread was completely frozen and no amount of fiddling about would shift it; I recall my plumber telling me that this tap had a problem several years ago. Never mind, I didn't like the taps anyway, so I decided to replace both taps on the basin, which would involve unscrewing the taps from below. So I applied some penetrating oil, and then went off to a large out-of-town hardware store where they had a good selection of taps, found what I wanted, and returned to fit them to my basin. I turned off the water at the main supply, and proceeded to use my under-the-sink wrench to loosen the nut connecting the water pipe to the tap, but rapidly discovered that the wrench's handle was far too short (and thin!) to apply sufficient torque to do the job, nor was it possible to attach anything to the handle to increase the torque. Off I went to the large hardware store again to buy a better under-the-sink wrench, which did the job once I inserted my hammer in its jaws to apply sufficient torque. Now all I needed to do was to detach the tap itself from the basin, so I applied the wrench again but found that the nut attaching the tap to the basin was frozen in place. To unfreeze it I reasoned that I could jiggle the tap backwards and forwards, using wrenches simultaneously above and below the basin. Jiggle, jiggle, wrench, curse, WRENCH ... crack!! The basin was now in several pieces, with cracks radiating from the tap that I had been working on. WTF happened? Oh well, the basin would now need to be replaced as well so I started to pull away the loose section behind the tap, and immediately cut my thumb by trapping it between the edges of two broken basin pieces. It was a deep cut about 2cm long and there was blood everywhere, so I had to retire for a while to mend my thumb. Later on, I returned to the basin to discover that the reason it had broken was that the part of the tap that passed through the basin had a square cross section, and the hole in the basin that it passed through was also square. Great!! Now I know that wrenching the tap around backwards and forwards was guaranteed to break the basin. Anyway, the tap was now free of the basin because I had smashed the basin, but there was enough of the basin left intact that I could still use it pending its replacement, so I put a new tap on the free end of the water pipe and gracefully dangled it over the broken edge of the basin. I turned the main water supply back on, so now I had a tap that didn't drip, but a basin that needed to be replaced.

The butterfly flapped its wings, and the puff of air developed into a gust of wind.

OK, so now I needed a new bashroom basin. I picked up a piece of the broken basin in order to use it as a colour swatch to match its nice pale blue colour to a new basin in the large out-of-town hardware store. Off I went to the store to find that all of their basins were white. Oh no! At first I assumed that they put the white basins on display and held a selection of coloured ones in the store room, but I rapidly discovered that they had only white. This was a large store, so this was as good as things would get for me. I realised with dawning horror that if I wanted a colour matched bathroom suite then I would need to replace the entire suite. I looked around nervously at the prices of whole suites, and realising that I would have to have it fitted professionally I mentally added in that cost as well. The total cost would not be less than £1000 in round figures. Perhaps I'll get used to having a broken bathroom basin! Maybe I could pass it off as an interesting new art form, and put a First Aid kit by the side of it for people to staunch their wounds when they cut themselves! No, that won't work. I'll have to spend loadsamoney on fixing this problem.

The butterfly flapped its wings, the puff of air developed into a gust of wind, and the gust of wind developed into a howling storm.

I should have got a plumber in to fix my dripping tap, but personal pride took over and made me attempt to do the job myself.

By the way, I have now found some specialist suppliers who sell discontinued coloured bathroom suites, so maybe I could replace only the basin, but I'll have a think about things to decide whether I might as well buy a whole suite anyway

NanoArt 2007 voting

I blogged a couple of times before about NanoArt 2007, see here (background information on the competition) and here (my 5-frame animation competition entry).

The NanoArt 2007 voting is now open here (I don't know why the year 2006 appears in this link!) until 31 March 2008.

Here are the voting steps (you can go straight to vote for my entry here):

1. Click on the album thumbnail to open the album.
2. Click on the image thumbnail to view the image.
3. Click on the number of stars you would like to rate this image.

My entry is different from the others because it is an animated GIF that you view here; this link is quoted beneath the static GIF that is displayed on the NanoArt 2007 competition web page. Unfortunately, the static GIF looks really boring, so I don't have much hope that many people will discover that my entry is actually an animation. Never mind, at least I tried.

Update (17 January 2008):

It is fascinating to watch the votes accumulate. At each stage you can see the number of votes cast thus far and their average rating, so if you take a peek often enough you can deduce each vote as it is cast, provided that the total number of votes is not so large that there are rounding errors in the average rating.

There is obviously someone who is trying to "spike" my entry by voting with a rating 0/5, whilst all the other votes that I have received have a high rating. How pathetic is that?!

Although I am curious to observe the pattern of voting, it is not that important to me what score I get because I have already achieved my goal which was to create an animated nanoartform. I would be interested to hear of any prior examples of this artform.

Interpreting mathematics

In the 5th January 2008 issue of New Scientist there is a short article by Mark Buchanan entitled "When it's time to sit back and think again", which discusses some of the results reported in an arXiv paper entitled "Symbolic manipulators affect mathematical mindsets" at http://arxiv.org/abs/0712.1187.

The phenomenon that is discussed in the paper is the tendency for people to switch off their brain when they use symbolic algebra programs (the paper specifically singles out Mathematica), but the problem is more widespread than this because it occurs with basic 4-function calculators (e.g. do I divide by 1.1 or multiply by 0.9?) or with advanced numerical software (e.g. why do the eigenvectors come out completely different for trivially different data?). This causes people to drop down into a calculational mode where they act merely as operators of the software/hardware, whilst not bothering to form a higher-level interpretation of what they are calculating (e.g. its physical interpretation).

This is like the difference between a worm's eye-view (e.g. low-level calculational mode) and a bird's eye-view (e.g. high-level physical interpretation mode). It is like the difference between having a local serialised view of each part of the problem that you are solving or a global parallelised view of the whole problem. It is like the difference between being a calculator or a visionary.

I know of people who are calculators but who can't see the grand picture, and who usually cannot communicate with anyone other than like-minded calculators. I know of people who are visionaries but who can't express their ideas in enough detail to carry them out, and who are highly articulate but whose apparent lack of rigour really annoys the ace calculators. I know of very few people who are both calculators and visionaries, but these people are really interesting to know.

The education system trains people to produce standard solutions to problems, so that everyone calculates using the same language. It is relatively easy to teach people to rote-learn standard procedures, and to then test them on this knowledge in exams. It is much less easy to teach people the skills that are needed to relate these calculations to the rest of the world, or so one would think.

My approach to counteracting the tendency to drop down into a calculational mode of thinking is to visualise what I am calculating; I try to avoid doing calculations that I can't visualise. When I draw a picture of what I want to calculate then the calculation itself follows almost automatically, and calculational subtleties (e.g. the epsilons and deltas) are easily resolved by referring back to the picture. I would go so far as to say that if I can't draw a picture then I don't understand what I am calculating.

I was amused to see that the principal example cited in the paper http://arxiv.org/abs/0712.1187 was one in which several students struggled to use Mathematica to evaluate the following integral (I have omitted various constants):

Integrate[x^2 Sin[x]^2, {x, -Infinity, Infinity}]

This is a clear example of students in calculational mode, who have adopted the worm's eye view of the problem as just being a calculation. They tried feeding the integral to Mathematica in various different ways, but without success. What they do not do was to diagnose their problem by simply visualising what they were calculating; it is not even necessary to know the physics behind this integral.

Using the same tool (i.e. Mathematica) that the students were using in their attempts to evaluate the integral, here is a plot of the integrand over a finite interval.

Normally, an integrand that is simple as this would not need to be plotted out explicitly because its behaviour is obvious from its structure, i.e. an x^2 factor that diverges times a Sin[x]^2 factor that oscillates between 0 and 1. Nevertheless, in this case I did plot it out as part of my ingrained habit of visualising calculations using Mathematica. The students should have been doing this as well, so I presume that they had not been very well tutored in their use of Mathematica. Had the students attempted even a rudimentary visualisation then they would have immediately realised that the limits of the integral they were trying to evaluate could not possibly be infinite.

As the visualisation habit becomes part of your way of working, you eventually reach a point where the solution of some problems comprises visualisation followed by calculation. There always remains a set of "difficult" problems for which your current set of visualisation techniques is inadequate, in which case you have to use pure calculation to get to a solution. But then you should be on the lookout for ways to capture the essence of your solution in a new visualisation technique.

Wouldn't it be nice if there was a standard set of visualisation techniques that you could use alongside the existing set of calculational techniques? If this set of visualisation techniques was carefully designed then it would be just as rigorous (and teachable) as standard calculational techniques. It would be a very interesting exercise to reformulate existing material using such a visual language; for instance, I made an attempt to do this sort of thing for the topology of the SO(3) rotation group here.