Monday morning / Friday afternoon thought inspiring blog

What is the difference between a teacher and a student? Answer: a teacher is someone who has finished learning. Of course this is not strictly and generally true, but I do believe there is some merit to this thesis. Teachers should feel confident with the material they are teaching and thus feel no incentive for further study, at least not for the material they are teaching. Teachers have a rounded message and a clear vision of the field in question, whereas students do not want to impose their view on others but want to know the view of others. Exploring the views of others is an essential part of the learning process. Furthermore, a teacher must be convinced that his teachings are worth the effort to bring them across. Generally it takes far less time to absorb clearly presented ideas then it takes to present ideas clearly. That is why I currently still feel myself being more of a student than a teacher: though I like to help people understand ideas that I have mastered, I prefer learning new ideas of others. When people explicitly show interest I am very willing to explain them what I know about the subject, but teaching more at less at random seems to be a waste of energy considering that I could have gained far more by learning something new myself. This is why I doubt the future consistence in updating and even existence itself of this blog: it takes a lot of effort to put ideas into words and when it is more or less randomly directed there is very little pay-of for the effort. As long as I feel far more a student than a teacher, I believe this blog stands little chance of long term survival.

Last Monday and Friday were holidays and since most people I know were at home that day, including myself, I did not write anything. Today, once again, there will just be a link. I encourage you to check out some of the publications of the the Applied Geometry Lab at Caltech. If you are not a scientist and possibly even have trouble understanding the abstracts, a visit might still be worth it. Their papers have many excellent illustrations that are stimulating and interesting even to people outside science.

I do not think that many people actively belief in absolute happiness, but nontheless we might not always be concious of the fact that happiness and knowning how to value certain priviliges –closely related to experiencing happiness– are derived by our minds by comparing. To make this more explicit I sometimes say: ‘If it was not for the vallies, we could not see the hills’.

As a consequence, people who do not know what war means, that is how nasty it is, they also do not know peace and hence its value. Of course everyone knows what war and peace are, but we all do on different levels of detail and abstraction. Some of us have lived during the way, others were born close to it, either in time or proximity, and thus have gotten to know the adverse effects through the people that did live in the way and yet others learn about in school or on the television. Paradoxically, those who have the least knowledge of war and the atrocities that it brings, seem to hold the least value for peace and the corresponding priviliges and rights that are necessary to maintain peace.

In a society where nobody ever gets ill, nobody values their health. A similar statement applies to experiencing happiness. It is hard to be happy when you always get what you want or –even worse– when you do not know what it means to be denied something. This is why we should actually value, at least partly, even the nastier things that happen to us, because they teach us how to appreciate all the rest.

Playing is the best way to learn. Okay, maybe it is not the very best way, but it certainly ranks among the best. People who want to learn something should find an approach that they enjoy and, if possible, is even addictive to a certain degree. The way I learned some basics of the Japanese alfabets and characters, was through the RPG game Project LRNJ. I forgot most of it, since I have not practiced in years now, but I’m confident that I would pick it up pretty quickly if I were ever to start again.

I taught myself some Spanish and still actively try to maintain and improve that knowledge, applying the principle of enjoying the learning. That is by reading books, starting with translated works that I was confident to be interested in, and watching the Spanish news, movies and animated series dubbed in Spanish (Simpsons, Futurama, Family Guy) and eventually real Spanish programs, such as Los Serrano. Of course not everybody enjoys these things and it can be hard to get to the point, where you can actually enjoy it, so you do have to be persistent and have faith that your knowledge will improve to the level that you need to have fun with it.

So the next time you seriously want to learn something, you might want to try and actively seek a method that improves the chance that you will enjoy (part of) the learning process. Be creative with it!

Scientific knowledge should be widely and freely available. Of course it is nice that scientist can earn an extra buck by coauthoring a book and I’m not advocating that they should not be (financially) rewarded for their efforts, but making the work only available in a usually quite expensive, offline form prevents most works from reaching a sizeable audience and thus attain general usefulness that would be deserving of the quality of the content. Also, a lot of (introductory) knowledge is repeated over and over again in these books and while it can be useful to learn about a subject by yet another approach, most often there exist better treatments of the material elsewhere. Either the reader is already familiar enough with the material to even recognise casual errors in these sections or he would certainly be better off reading another text, that focusses on that particular subject.

This is why I am hoping that projects such as Scholarpedia, MIT’s OpenCourseWare and many other initiatives to make scientific knowledge and teaching material generally available, will gain traction in the coming years. Wikipedia is of course another fine example of a great body of knowledge, but their particular approach of building scientific knowledge does seem to have some problems. I would not want them to stop their efforts, but parallel initiatives, that explore other ways of making scientific knowledge widely available, mostly trying to solve the problem of accountability and credibility, are to be followed closely and be given a deserving piece of the total attention.

If you never have heard of Chladni plates and patters, check out the following two links and the subject: Chladni plates and Chladni patterns for a violin. The last link provides some insight on the matter, whereas the first only shows you some intriguing pictures.

The patterns are related to the eigenmodes of the geometry an each corresponds to a harmonic function over the geomtry at hand. The relation of these pattern to the properties of the geometry makes them an important tool to scientists that want computer algorithms to ‘understand’ geometry. Though awareness and experimentation is on the rise, we are far from a full understanding how to use these magical patterns to their full potential.

Google confirms it: playing beats winning. That is to say, playing is more important than winning. I think that most people know this at least subconsciously and behave accordingly. The essence of playing, I believe, is winning from yourself, improving your previous efforts, applying new tactics or strategy and feeling more accomplished than the last time you played. Sometimes this personal improvement can be measured directly by beating some opponent, but never comes the satisfaction from the mere act of winning, at least normally it should not. How meaningful is a win if the opponent never really was a match to begin with or if it was accomplished by ‘bending the rules’? The few rare people that actually enjoy winning, just for winning-sake are no fun to play with. Luckily this group of people is actually very small and I’m not sure I really know people who are like that.

I thus conclude, that to get real satisfaction from playing the risk of losing once in a while is an integral part of the equation. So it turns out that playing is not only about winning, but also about losing.

Sorry, only two links today: check out the Mertens and Möbius functions. The former is built upon the latter. Its structure is surprising and almost remeniscent of random behaviour, but completely random it is not. I just find it mesmerizing.

Möbius function

Mertens function

The Mertens functions is closely related to the world famous Riemann zeta function. This latter function is conjectured to have a certain property, but while few people doubt the veracity of the statement, nobody in over a century has been able to find a proof.

A captcha is intented to verify that at the other end of an interaction with a webform, it really is a human being, with good intentions, instead of a program run by a spammer that is submitting the data.

Not all captchas, however, are equally good as some can easily be solved by a computer program. To learn more, for instance about what constitutes a hard to solve captcha for computers, check out the website of PWNtcha - capture decoder.

Picture yourself a ball with hairs attached to the surface. You are given the task to comb the hairs on the surface in such a manner that there are no “bald spots” or “opposing combings”, that means that no two adjacent hairs should point in opposite or “radically different” directions. Can you do it?

Too bad for those who thought that they can: mathematics tell us that this is impossible. This result is popularly known as the hairy ball theorem. The details of the mathmatical statement are naturally more intricate than the explanation I gave in the previous paragraph. In more mathematical terms is says that a smoothly varying vector field over the surface of a three-dimensional sphere, has to have at least one point where the field is 0. In terms of the hairs on the ball, the only solution to have a ‘smooth combing‘, thus without places where the hairs have radical different directions, is to have some place where the hair lenghts are zero.

Another consequence of this theorem applies to wind blowing over the surface of the earth. Either there have to be places where there is no wind or at some places the wind would be blowing in opposite directions, directly adjacent to each other. This corresponds nicely to what we know about hurricanes and tornados, that in their centre — the eye — there is no wind. Even if we were to imagine one big flow of wind going over the earth’s surface in a direction along the equator, then still there would be areas without wind at the poles.

So what about the hairs on a four-dimensional sphere? It turns out that those can be combed smoothly over the entire surface. That means that owners of four-dimensional pets have a much easier time preparing their furry companion for the pet show than the people on earth with their three-dimensional pets.

Note that a pet ‘without holes’ is mathematically equivalent to a sphere, thus the result also applies to objects that are topologically the same as a sphere. Topologically equivalent means that the object can be elastically deformed from one object into the other without connecting the surface with itself or breaking the object. The most popular example of topological equivalence is that of a coffee cup and a doughnut. Now you can laugh at the joke: A topologist is one who doesn’t know the difference between a doughnut and a coffee cup.