Monday morning / Friday afternoon thought inspiring blog

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.

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.

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.

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.

Recently I came across the free educational software package GCompris and it made me remember what my first experiences with the computer were like. I think I was about twelve when I first learned to use a computer at the house of my cousin. We started out on the Atari playing simple games and learning programming with GW-Basic. A bit later, when they got a personal computer, we learned a lot of english through more complex games such as Sim City, Sid Meier’s Civilization, Leisure Suit Larry and Monkey Island. Thus reasoning from my own experience there is no doubt in my mind that the computer can have a positive impact on the educational development of children. We did not spend our time exclusively behind the computer; it was only part of a broadly varied routine that included reading, playing table games, playing outside, playing sports, playing with Lego, etc.

The GCompris package, however, is aimed at children of a much younger age. I do not doubt that it can be very beneficial to the development of a child and, if were to have children one day, I would probably stimulate them to play with such games on the computer as well, but only as one activity amongst a whole range of non-computer related activities. I’m not sure that there are even children of such a young age that have the attention span to sit behind a computer all day, but if they do, I doubt that it is the best for their development into an adult person. Organising blocks on a screen requires, I believe, different skills than playing with them in real life. As always, some balance seems to be essential. Having said that, if you have a young child, you might want to check out GCompris.

If you are interested in the idea of using computers for verifying the correctness of mathematical proofs, you should check out the Mizar Project. Their language and software allows to express and relate mathematical statements such that proofs can be build in a formally consistent manner. It is all founded upon Tarski-Grothendieck set theory which is, if I understand it correctly, equivalent to axiomatic set theory and first-order logic. Their library currently includes quite a broad number of proofs, including one for Jordan curve theorem.

The attraction of such a system is of course the power that it can give us in finding proofs and checking them automatically. Unfortunately it seems that we have still some distance to go, before most of today’s mathematics can be manipulated and constructed as easily in this system as we currently do on paper and let humans fill in the logical gaps. I do not have real experience with the system, but I gather that the overhead still feels as too much for most people that would not mind putting a stronger and automatically verifiable foundation underneath the theorems they work with. A possible solution would be an interface that allows mathematicians to work with the mathematics at the level they are reasoning, and have the computer fill in the logic connections to and within the lower levels automatically. I’m not sure how feasible this is, but it would be a very significant development if we would obtain such an extension to the system.

Another project that definitively deserves mentioning is the Hilbert II project, initiated by Michael Meyling. This project specifically has the idea I described at the end of the previous paragraph as a goal. If you want to learn more about automatic theorem provers and formalised mathematics with computer programs, check out this talk of Freek Wiedijk or his website.

Nobody claims that the American continent has been ‘invented‘ by Columbus, or by whomever was first to set foot on the New World. He clearly discovered it, since it was already lying there when he sailed out and, as he was looking for a new route to Asia, he did not even aim to end up there thus only by chance did he learn of the new continent.

From this we conclude that there are at least two factors that influence our idea of what constitutes a discovery as opposed to an invention:

1. If something already exist and you learn about its existence by mere encounter, then we call it a discovery. An invention on the other hand is about creating something new, something that did not exist before.
2. Inventions do not happen by chance, since it requires thought and creativity. Furthermore, they tend to solve some specific problem; inventions have purpose. If something was encountered by chance, then in exactly that state, it can only be a discovery. On could say that Australia was discovered and then the British ‘invented’ the continent as a penal colony.

Based upon these observations, can we decide the following question:

Has mankind discovered or invented mathemathics?

Note that there is no definitive answer to this question. The problem is strongly related to the semantics of our language and completely ’solving it’ is not our intention. We take it for a philosophical question, one where our answer tells us more about how we view ourselves (mankind) and the universe we live in, rather than pertaining to some kind of absolute truth.

I used to be convinced that mathematics could only be called a discovery. The intricacy and beauty of all the constructs made me decide that it had to exist outside of human conciousness as well. Since we only came across something that was already there, it cannot be discovery.

Nowadays my conviction has weakend: maybe parts of mathematics really are ‘just an invention’. We try to solve problems and use mathematics for structuring our thoughts. Some problems can be solved in a multitude of ways and people working independently often have found their own but different solutions. All of our mathematics are ultimately based on the axiomatic framework we chose to work from, but even that framework is not universally agreed on by all.

We have barely touched the surface of the discussion, but because of time constraints I’m already going to conclude: mathematics has aspects of both invention and discovery. If we were ever to encounter an intelligent alien race, it would be nice to compare the mathematics they have built, if any, with ours. If it turns out that all concepts of developed and complex areas in mathematics translate one-to-one to each other, I would see that as an indication that those parts of our mathematics, at least in the context of this universe, have an instrinsic nature and thus can only be discovered, not invented.

Check out the home page of Ron Fedkiw at Stanford University. It has many nices movies and they are all the result of scientific research in some sense.

You might never have realised it, but most of the special effects that are digitally added to movies require some hefty mathematics. How do they visualise a large cruise ship hitting a bridge or Godzilla’s tail taking out part of a building? They simulate the event and calculate what approximately happens. There is one big difference though between simulations for Hollywood and those outside: in Hollywood they only need the results to look accurate to the eye of the average person in the movie theater, whereas others usually want real physical accuracy. Hollywood can thus use other, in particular faster, methods for their simulations.

Note that not all effects are purely digitally added. A lot of images come from real simulations only with small scale models. Digital simulation, however, is getting more popular because of the flexibility it gives us. With current advances in computer hardware and the techniques developed by people like Ron Fedkiw, the simulations are getting faster and faster. This causes a shift to less simulations with scale models and more with the computer. For instance, the computer simulation allows to generate a sequence from any camera viewpoint even ones that are impossible in the physical world. Also, when the creative director want to try a slightly different animation, it is easier to just tweak a few simulation parameters and recalculate the result then to rebuild a whole physical setup and redo the experiment.

From the thesis of Dafna Talmor, Well-Spaced Points for Numerical Methods (1997):

It is reasonable to conjecture that every well-spaced point set can be tetrahedralized to form a good aspect ratio mesh. However, the Delaunay tetrahedralization fails to oblige us in that respect. Currently, it is unknown if such a tetrahedralization algorithm exists, or if there exists a counter-example: a well-spaced point set with the property that none of its tetrahedralizations is of bounded aspect ratio.

This is probably hard to understand for most of you due to lack of familiarity with the subject. Let me try to provide a translation:

1. There are two kinds of point sets, namely those that are well-spaced and those that are not. Intuitively well-spacedness means that the points are distributed evenly over space. No two points are too close together and there are no gaps. In a crate of oranges the points formed by the centres of all oranges are well-spaced.

2. Delaunay is a criterion for completely triangulating a point set. Applying Delaunay we get a triangulation of any collection of points. When we apply Delaunay to a two-dimensional point set (points in a plane) then, in the case of well-spaced points, we get a triangulation that consists only of ‘voluminous’ triangles, that means triangles that are not arbitrarily flat. In three dimensions, however, a Delaunay triangulation of well-spaced points can contain highly flat tetrahedra (three-dimensional triangles; a pyramid with a triangle instead of a square as bottom face).

3. The Delaunay triangulation is not the only triangulation that is possible. In fact, the possible number of triangulations for a point set is usually very, very large. In the Delaunay triangulation of well-spaced points in 3D only relatively few tetrahedra are flat and with some manipulation they can often be removed by slightly changing the connectivity of the points. If that does not work then points can be moved around a little or even points can be added to remove the flat tetrahedron.

4. What Talmor remarks, and what I recently came across myself, is that is unknown if it is always possible to remove the flat tetrahedra from a Delaunay triangulation of well-spaced points by changing the connectivity. The huge number of possible triangulations makes it unfeasible to try them all. There are two possibilities:

1. For some well-spaced point sets there is just no triangulation possible that does not contain a flat tetrahedron.
2. Every well-spaced point set has at least one triangulation without flat tetrahedra. Talmor says that this option is ‘reasonable to conjecture‘ and I concur. However, if such a triangulation is always possible, that leaves us hanging with one big and obvious question:

Is there are algorithm that produces a triangulation without flat tetrahedra, without trying all possible combinations of triangulations?

That sure is a vexing issue.

I imagine that at this point some of you still do not understand the gist of the question. If I find time, I will add some pictures to illustrate some of the concepts.

My interest in this problem is not to solve the issue all together. I’m just interested in creating good triangulations, without flat tetrahedra that is. What I would like to have is an algorithm that for a well-spaced point set produces the triangulation in which the flattest tetrahedron, is more voluminous than the flattest tetrahedra of all other triangulations. Otherwise fomulated: if we measure the quality of a triangulation by the flattest tetrahedron in contains, then we want to find the triangulation with the highest quality that is possible for a particular point set.

It is obvious that Talmor’s problem and my problem relate: assuming that a triangulation without flat tetrahedra always exists, then the optimal triangulation would surely offer an example of such a triangulation.

update: I recently discovered that in the paper Sliver Exudation (2000) it is proven that a triangulation without slivers does indeed exist, if the points have a ‘nice enough’ distribution to begin with. I’m not sure whether their sliver exudation algorithm can actually find the (approximate) optimal triangulation. At least it can find a triangulation without slivers, though the theoretical guarantueed quality is very small.