With some help from the TeX.SE community, I found out how to draw squiggly lines in Asymptote. The code for the squiggly lines can be found on that link to TeX.SE above. To the right is a sample output. The line is drawn with the default 45 degree angle. The circular curve is drawn with an 80 degree angle. The main reason I want to draw squiggly lines is to be able to use it to denote the singularity in a Carter-Penrose diagram, such as the one for Schwarzschild–de Sitter below.
I tried to teach differential geometry to a biophysicist friend yesterday (at his request; he doesn’t really need to know it, but he wanted to know how certain formulae commonly used in their literature came about). Rather surprisingly I hit an early snag. Hence the title of this entry.
Part of the problem was, as usual, my own folly. Since he is more interested in vector fields and tensor fields, I thought I can take a short cut and introduce notions more with a sheafy flavour. (At the end of the day, tangent and cotangent spaces are defined (rather circularly) as dual of each other, and each with a partial, hand-wavy description.) I certainly didn’t expect having to spend a large amount of time explaining the concept of the function.
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I’ve just discovered the software Asymptote, which purports to be the next generation replacement for MetaPost. A little bit of history: Donald Knuth wrote TeX. To produce pretty output, he wrote the software MetaFont, which is actually a general equation-solver/optimization/interpolation software that is well-adapted to generating fonts from some specified rules. It encapsulates the modern “a font is a program” paradigm. MetaPost was written based upon MetaFont to be a generic vector drawing format, using largely the same instructions as MetaFont. To produce a picture, one writes a program which describes what the picture looks like, with key points noted by coordinates; MetaPost then executes the program and interpolates the points to produce the final output.
Asymptote wants to be an improvement upon them. Like MetaPost, it uses LaTeX to generate labels and texts for inserting into the encapsulated-postscript output. The syntax is slightly changed: unlike MetaPost and MetaFont, Asymptote has a more C/C++ like language structure. But my favourite bit about the software is its native inclusion of 3D capabilities, which is immensely useful for making illustrations for harmonic analysis, general relativity, or differential geometry. (As an example, here I include something I made this evening at home. It is an illustration for the Klainerman-Machedon type bilinear estimates for the wave equation. Once I figured out the programming syntax, it only took 8 commands and a bit of high-school mathematics [to figure out the coordinate points] to compose the image.)
When talking about graphics software, I should of course mention PGF/TiKZ, which is by the same author of the popular LaTeX-beamer software for making PDF presentations. I like TiKZ. It has a decently clean syntax, and has very nice integration with LaTeX. (In particular, it has very nice integration with LaTeX-beamer, for obvious reasons.) Unfortunately, it suffers two problems. First is that it does not have native support of 3D drawing. For most people this is not a problem, but I do envision that a lot of the illustrations that I will draw will end up being in 3D. While there are some hacks to fake perspective/projection in TiKZ, it is a lot more convenient to just program the images in a language that supports 3D natively. The other problem is that TiKZ doesn’t always play well with DVI files. It prefers to be directly compiled into the PDF file. This is in part its design philosophy and in part limitations imposed by the DVI format.
(Asymptote should be able to be embedded directly into the LaTeX source; at least it is so stated in the user’s manual. The problem, of course, would be for collaborations and arXiv: because of the popularity of TiKZ/LaTeX-beamer, I can reasonably expect a collaborator or arXiv to have them installed on the computer. Asymptote, less so. So it will be wiser to stick with separate images from the document.)
There is a quite well-known joke that goes something like this: a mathematician, a physicist, and an engineer are each handed a little red rubber ball, and are told to find out its volume. The mathematician takes a caliper and measures the diameter of the ball, and uses the formula 
To continue my earlier rant, it appears that I am not the only one to think that modern mathematics education is misguided. It took Underwood Dudley (warning: a subscription to the AMS Notices is required to access the PDF file) 8 pages to get to it, but in the end, he comes to the same conclusion as I did, albeit from a different starting point. The joke above is meant to illustrate Professor Dudley’s main argument, that despite grandiose claims by the National Academy of Science or the National Research Council, a mathematics education does not provide a practical skill set that is necessary for most jobs. (I will come back to the emphasis on the word “practical” in a bit.) With many examples, Dudley’s essay illustrates a common fallacy, that mathematics is important to learn because frequently in life (especially at work) one will encounter situations which calls for computations beyond basic arithmetic.
Perhaps I should make the distinction clear here: I am not claiming that mathematics education is useless. I am just observing that fact of life that most people, going about their everyday lives, will very infrequently, if ever, encounter a situation that requires the finer understanding of mathematics beyond the middle school level (US; elementary school for East Asia). And therefore we, as scientists/mathematicians/educators/parents, should not oversell the learning of mathematics as something crucial to one’s future, and bully the kids into studying it. Or, in other words, while I think that familiarity, nay, fluency, with mathematical concepts is a requirement for a well-educated man, I do not consider erudition to be a requirement to lead a productive life in society.
The mathematical curriculum (and by extension the physical sciences and history and all other of the more academic classes in American schools) should not invent reasons to convince the students that mathematics is used in all facets of life, and hence important. The knowledge of how to change a flat tire is likely to be much more practical for the average Joe over his lifetime, than the knowledge of how to solve the quadratic equation. (Most of what you need to know to succeed in life, you learn in kindergarten anyway.) That people come to question what goes into the general education based on potential on-the-job utility is completely misguided. A general education for the populace should not be equated with vocational training. A general education should train the students in the ability to reason, to think soundly, to approach problems logically. A flexible mind that is open to new ideas and is capable of solving problems is an asset applicable to any job. By narrow-mindedly restricting one’s attention to the immediate and direct applications of classroom subjects, one runs the risk of missing the grander picture in which the whole is more than just the sum of its parts.
During the mathematicsarithmetic class one day in fourth grade, we (the students) were asked to convert the fraction 1/4 into a decimal representation. Other students put pencils to their scratch papers and started applying the rules of long division: 4 doesn’t go into 1, so we add a decimal point and find that 4 goes into 10 twice, etc. My hand was already up in the air before they finished drawing the long-division sign.
“Zero point two five,” I answered when the teacher acknowledged me.
“Oh that’s nice and fast,” he said, “and how did you get the answer?”
A look of confusion came over his face as I related how I did the computation without the rote-rules that he spent the class teaching us: “Well, four quarters make up a dollar, and each quarter is worth twenty-five pennies. So a quarter is 25 cents, or zero-point-two-five dollars.”
I was just shown Peter Gray’s article in Psychology Today with the preposterous premise that teaching less maths in school may actually improve student’s mathematical abilities. My knee-jerk response was to rise to the defence of the status quo. But after actually looking at what he wrote, and considering my own mathematical upbringing, I feel strongly compelled to agree with him (with some minor caveats). (more…)
Each profession has its own set of special language: some, like the medical profession, rely on long and precisely defined words, often with Latin, Greek, or German origins, to describe objects and events that we do not encounter on the everyday (I doubt anyone actually found the need to use pneumonoultramicroscopicsilicovolcanoconiosis in everyday conversation [it is a lung disease caused by inhalation of metallic dust]); some, like the legal profession, attach a preferred meaning to common everyday words (in addition to special terminology), and put certain heft on syntactical and grammatical specification to say exactly what they mean.
In the natural sciences, the tendency has been toward the coinage of words to refer to new objects or ideas. My guess is that this has to do with the earliest natural philosophers taking a cue from Adam and giving a name to everything they have not seen before. In mathematics (which, by the way, while may be arguably natural by certain definitions, is not a science), on the other hand, the tendency has been toward usurping everyday words for specialist purposes. (more…)
Besicovitch once said that “A mathematician’s reputation rests on the number of bad proofs he has given.” Of course, originating from someone educated in the Russian school, the word “bad” in the quote should probably be taken to mean “inelegant”. However, lack of beauty is certainly not the only possible deficiency in the quality of a published result. Cases abound where a proof is bad not in the aesthetics, but in something more fundamental. (more…)
Recently I sat on a panel of “research scientists” for Career Day at the New Jersey Governor’s School in the Sciences. During the discussion, I was pointedly asked, by a young woman who aspires to be a research mathematician, what “mathematicians do for research?” Much to my own dismay, I fumbled around for a short answer, but ended up giving a shpiel that is much too long for the occassion and which ended up mostly sound and fury, and probably confused everyone in the room except for myself.
Contrary to the impression I have just given, I have thought quite a bit on how to answer the question “what is it that you do?” I am even quite good at tailoring the answer to the audience. If that were the question asked, I would have told the audience of high school students that I study the long term behaviour of the universe using general relativity, that this falls in the realm of mathematics and not physics because it is patently impossible to recreate large scale structures of the universe, such as nebulae or black holes, in a laboratory on Earth, and so I study the mathematical consequences of the assumptions underlying general relativity. If time allowed or if I were further prompted, I would have told them about the propagation of waves and other fancy things from partial differential equations. (more…)
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