OK, ok, I might as well throw up some entertainment. How about some RenÃ© Laloux to tide you through the weekend (or must I call it the fin-de-semaine?) — like Gandahar (1988):
(Part 1 of 8, so it seems.)
Other projects have been hanging heavy on me, so it looks like I won’t be blagging over the weekend. I have a healthy stack of draft posts, including the next installment in my series building towards supersymmetry, along with some feel-good bites on innumeracy, Conservapaedia and so forth (which are partly material reposted from elsewhere). With luck, I’ll get a couple of those up early next week.
To fill the gap and help convince myself anybody will miss the melodious sound of my textual voice, I added a “Reader Favorites” list to the sidebar. The entries which qualified as “favorites” score highly on a confidential, nonlinear function of visits, links and comments.
Ever noticed a painted yellow line in the parking lot around many supermarkets and retail stores? The magic yellow line emits a signal that causes carts to stop dead in their tracks, preventing carts from leaving the parking lot.
Now you can build your own portable yellow line — with up to a 20 foot range. Need I say more? Hint: it works inside the store.
Somebody going by the handle “Orthonormal Basis of Evil” has posted instructions for building this device to the Web site Instructables. There are eleven steps between the first act of assembly and the assault upon retail America. Most interesting from a math-and-physics standpoint are isolating the locking signal, which requires a Fast Fourier Transform (FFT), and recreating the locking signal, which is a nice exercise in electromagnetism. Essentially, the lesson is that a current in a wire creates a magnetic field, varying the current creates a fluctuating field, and these fluctuations can propagate through space to carry energy and influence electric charges elsewhere, possibly inducing new electric currents following the lead of the first. Here, the transmitting wire is either buried beneath the parking lot or carried on the person, and the “elsewhere” being influenced is inside the locking apparatus of the shopping cart’s front wheel.
(I may be mistaken here, but I think this is actually a case of near-field induction, rather than the typical far-field plane waves one often studies.)
Installing the device on its human user is an exercise in cybernetics, which “Orthonormal Basis of Evil” has chosen to illustrate in a way a little different from the electromagnetism textbooks I grew up with:
Continue reading Shopping Cart EMP
My group theory teacher, Prof. Daniel Freedman, had some interesting professorial habits. When invoking some bit of background knowledge with which we were all supposed to have been familiar, he would say, “As you learned in high school. . . .” Typically, this would make a lecture sound a bit like the following:
“To finish the proof, note that we’re taking the trace of a product of matrices. As you learned in high school, the trace is invariant under cyclic permutations. . . .”
Prof. Freedman also said “seventeen” for “zero” from time to time. After working out a long series of mathematical expressions on the blackboard, showing that this and that cancel so that the overall result should be nothing, with the students alternating their glances between the board and their notes, he would complete the equation and proclaim, “Equals seventeen!” At which point, all the students look up and wonder, momentarily, what they just missed.
“Here, we’re summing over the indices of an antisymmetric tensor, so by exchanging i and j here and relabeling there, we can show that the quantity has to equal the negative of itself. The contraction of the tensor is therefore, as you learned in high school — seventeen!”
One day, I managed to best his line. I realized that the formula currently on the board had to work out to one, not zero, so when he wrote the equals sign, paused and turned to the class with an inquiring eye, I quickly raised my hand and said, “Eighteen!”
Incidentally, truly simple topics like Euler’s formula and trigonometric identities were supposed to have been learned in middle or elementary school.
Today, we’ll talk about one of the things Prof. Freedman said we should have covered in high school: the rotation matrices for two- and three-dimensional rotations. This will give us the quantitative, symbolic tools necessary to talk about commutativy and non-commutativity, the topic we explored in an earlier post.
Continue reading Rotation Matrices
I’ve blagged before about physics videos available on the Web. I just heard about SciTalks, a site for gathering and organizing links to videos on scientific and technical topics. This was something I recall Eric and I wishing for just a few weeks ago, so Iâ€™m glad to see somebody trying to make it happen! Hopefully, people will pick up this idea and run with it, so that science-themed videos will actually be easy to find. It’ll be, like, totally sweet: with freely available “multimedia” resources, we’ll finally live up to the hopes we had in, I dunno, 1995 or thereabouts.
In other news, Terence Tao writes about “ultrafilters, nonstandard analysis, and epsilon management,” making me feel that I was not the only one to gripe and grouse during the epsilon-delta part of undergraduate analysis. Tao also mentions the idea of a “highly rational number,” that is, a rational number p / q where p and q are both limited in magnitude (for some slightly technical definition of “limited”). This nomenclature opens all sorts of possibilities: for example, I’m curious if one could restrict the whole numbers to a new group, the “wholesome numbers,” which have large family values.
I wouldn’t bring up the topic in this context if I didn’t have a video to show. Ladies and gentlemen, Tom Lehrer:
Via Kevin Beck I just learned that Sal Cordova, famous (in some circles) for rank dishonesty and general lack of mathematical aptitude, has been claiming that Lagrangian mechanics was inspired by Intelligent Design. For those who are not au courant with physics, Lagrangian mechanics is an alternative take on the classical physics — think billiard balls, pendulums, planets orbiting the Sun — studied by Newton. The enterprise is named for Joseph-Louis Lagrange, who along with Euler and others laid the groundwork. It’s equivalent to Newton’s F = ma approach, but more convenient for some problems, and because it talks about the same physics in a different way, it provides a different and useful starting point for developing new theories. (For example, Barton Zwiebach’s First Course in String Theory generalizes the Lagrangian description of zero-dimensional objects, particles, to invent a theory of one-dimensional objects, strings. This is much easier to do in a Lagrangian rather than a Newtonian formalism.) Phenomena in relativity and quantum field theory are also often studied via a Lagrangian approach.
Many people are familiar with basic characteristics of light. We know, for instance, that light travels in straight lines; when light bounces off a mirror, the angle of incidence equals the angle of reflection; light can be spread out or focused together using lenses; and so forth. When we study optics, we can derive all these disparate facts from a very simple, central premise: when traveling from point A to point B, a light ray takes that path for which the travel time is a minimum. (A more precise statement is that the physical path taken by the light ray is such that a small perturbation to the path does not significantly change the travel time; this is connected to the calculus idea that the slope of a curve at a minimum or maximum is zero. For our purposes, we won’t have to worry about these details.) If there’s nothing in the way to change the light’s propagation speed, or if the material through which the light travels is uniform, then the path of minimum time is a straight line. Requiring that the light go from A down to a mirror and bounce back up to B means — I leave the geometry as an easy exercise to the interested reader — that the angles of incidence and reflection will be equal.
Lagrangian mechanics takes a similar approach, taking the idea of a “minimum principle” and applying it not to light, but to the motion of matter — balls, planets, frightened cats and so forth. Instead of calculating the travel time, as we did with light, we consider the energy of the moving objects; more precisely, our calculations involve the difference between kinetic and potential energies. The “Lagrangian” for classical problems — remember, we can generalize the ideas later — is the difference between the kinetic energy and the potential, and we find the path through space an object will take by adding up, or integrating, the Lagrangian along all possible paths. The physical path, the one the object really follows, is the one whose total Lagrangian, or “action,” is a minimum.
Now, what in blazes does any of this have to do with Intelligent Design?
Continue reading Lagrangian Mechanics is Intelligent Design?
Today we will advance our coverage toward quantum mechanics by looking at an unusual feature of daily life. We’ll be looking at an aspect of the world which doesn’t quite behave as expected; though it won’t be as counterintuitive as, say, the Heisenberg uncertainty relations, it does tend to make people blink a few times and say, “That’s not — well, I guess it is right.” Furthermore, poking into this area will motivate the development of some mathematical tools which will remarkably simplify our study of symmetry in quantum physics.
Fortunately, then, I found an assistant to help me with the demonstrations. Please welcome my fellow physics enthusiast, here on an academic scholarship after a rough-and-tumble life in Bear City:
Continue reading Rotation and Commutation
Because this is, of course, what everyone ought to do with a computational paper, we’ve put our code online, so you can check our calculations, or use these methods on your own data, without having to implement them from scratch. I trust that I will no longer have to referee papers where people use GnuPlot to draw lines on log-log graphs, as though that meant something, and that in five to ten years even science journalists and editors of Wired will begin to get the message.
Among several important take-home points, I found the following particularly amusing:
Continue reading Don’t Make Baby Gauss Cry
[VIDEO REMOVED FROM GOOGLE’S ARCHIVES]
This is the lecture in which Feynman presents an example I have appropriated before, concerning the necessity of knowing math before being able to do science, and how popularizations of physics often fail because they leave out the mathematics.
Feynman’s example goes like this: I can say that when a planet travels in its orbit, a line from the planet to the Sun sweeps out equal areas in equal times. I can also say that the force pulling on the planet is always directed toward the Sun. Both of these statements require a little math — “equal areas,” “equal times” — but it’s not really math, not a kind to give the layman heebie-jeebies. Given some time for elaboration, one could translate both of these statements into “layman language.” However, one cannot explain in lay terminology why the two statements are equivalent.
Continue reading Missing No More
This Friday, for your viewing entertainment, the panda gnomes which keep the bits flowing through the tubes and stop the Blagnet from unraveling present the Beatles in 1964, performing “The Most Lamentable Comedy and Most Cruel Tragedy of Pyramus and Thisbe.”
Hah! And you thought the only thing the Beatles had to do with Shakespeare was the BBC production of King Lear which John piped into the background of “I am the Walrus.”
So Tara Smith writes about her trip to the Creation Museum, and she says they watched a movie about angels where the chairs vibrated and the seats squirted water any time when the movie made a really stupid claim, and down in the comment thread CJ compared this to the “feelies” of Brave New World, and somehow I felt a song coming on:
Continue reading The Soma Song
One of the many notable entries in this fortnight’s Circle is Steven Novella’s piece on the purported autism-mercury link (hint, hint: there isn’t one, Robert F. Kennedy and Tom Tancredo not withstanding). Dr. Novella also has two good posts on Michael Egnor‘s recent torrid affair with dualism, so if you’d rather get your materialism fix from a Yale University neurologist instead of a physics buff who never learns to lay off the sriracha sauce, there you go.
I’d like to take this moment to give special thanks to all the people who have added Science After Sunclipse to their regular web-surfing experience. In particular, I’ve noticed this little site appearing on some blagrolls in very august company, which makes me happy indeed.
While I’m drawing the diagrams for my next actual math post, why not enjoy a video mashup set to a Tom Lehrer tune?
Forget the future of citizen journalism — this is what online video is all about.
A few posts ago, I mentioned the model of network growth by preferential attachment. This is a big enough topic in network theory that it’s worth poking in detail. I discussed some of the subject’s history in this paper (1.2 Mb PDF, plus presentation), which they tell me passed some stage of review and will appear in the print volume of the conference proceedings, i.e., an overpriced Springer book nobody will actually buy. But in addition to learning the names and dates involved in the story, we would like to play around with the ideas ourselves.
The other day, I realized how easy this would be, and now that I’ve actually presented this to a classroom full of people, I’d like to write about it. Today, I’ll present a Python program for growing a network by the “rich get richer” effect known as preferential attachment.
Continue reading Preferential Attachment in Python
I first tried writing a story when I was in the fifth grade. I kept going through the rest of elementary school, on into middle school and eventually joined the staff of Virgil I. Grissom High’s literary magazine. Sometime in that stretch, I wrote a story about discovering an abandoned alien city on Mars; it felt like a novel to me, but it was probably only about ten thousand words long. My bits and scraps of prose got good reviews from the literary magazine and the PTA. After long enough pleasing that audience, you inevitably wonder if you’ve got anything to say at all.
How, I wondered, does a writer tell that they’ve “arrived”? What’s the best and truest sign that you’ve “made it”?
Remember, I’m the blogger. I make the decisions.
10. The Fifth Element — sure, it’s over the top, but that just means it cleared a high bar (and Milla Jovovich is an excellent addition to any Periodic Table).
9. Metropolis — at least as recently restored, it’s deep, complicated, character-driven and visually hardcore.
8. Forbidden Planet — Monsters! Monsters from the id! Nobody can be trusted, except maybe Robbie the Robot, ’cause he’s programmed with the Three Laws.
7. Brazil — there’s a reason paperwork labeled “Form 27B-6” suddenly started appearing in the MIT Student Services Center a few years ago.
6. Ghost in the Shell — the perfect mixture of philosophy and explosions.
Continue reading Meme-o-Rama: Top 10 SF Movies