No Abacus Handy? Use your hands.

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Suppose you want to do some math, but you don’t have an abacus handy. Oh, the horror! What do you do?
No problem! Your hands make a *great* two-digit soroban-type abacus. The four beads on the lower deck are your four fingers; the bead on the upper deck is your thumb, as illustrated in this diagram (with apologies for my terrible artwork):
hand-abacus.jpg
So the numbers from one to nine look like:
finger-numbers.jpg
To get two digits, you use your right hand for the ones, and your left for the tens. So, for example, let’s look at a simple addition:
finger-add.jpg
Once you know the abacus, doing this with your hands is pretty simple. It’s definitely a limited technique, since you can’t get past one hundred without using your toes, but it’s a nifty trick, and it’s easy to teach kids to do this for working out math problems. If you’ve got a kid who’s a tactile thinker, it’s amazing how much learning to do this can help them. I’ve seen kids who do paper math with many digits by working out subparts of the problem using this style of finger-abacus.
There’s actually a whole Korean teaching method for math called something like chisan-bop. From what I understand, they build up on this quite a bit, to be able to do much more complicated stuff than just two-digit addition, but I haven’t been able to find an english textbook on chisan-bop. All the english texts basically show what I just did above: the two-digit abacus on the fingers.

Second Law Slop from Granville Sewell

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A reader sent me a link to an article by that inimatable genius of the intelligent design community, Granville Sewell. (As much as I hate to admit it, Sewell is a professor of mathematics at Texas A&M. I don’t know what his professional specialty is, but if his work in that area is anything like the dreck he produces in defense of ID, then it’s shocking that he got a faculty position, much less tenure.) Sewell wrote *yet another* one of those horrible “second law of thermodynamics” papers and submitted it *as an opinion piece* to a math journal (“The Mathematical Intelligencer”). It was, needless to say, not received well by people who actually care about quality math, and he was roundly flamed in letters in the following issue. The paper that I’m looking at is [his *defense* to criticisms in the original paper.](http://www.iscid.org/papers/Sewell_EvolutionThermodynamics_012304.pdf)
As one might expect from one of the ICSID guys, it’s a sloppy rehash of the same-old creationist arguments – it’s mainly the same old creationist thermodynamic crap, mixed with a bit of big numbers, and a little dose of obfuscatory mathematics.

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Friday Random Ten, Oct 6

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It’s friday again, which means I get to bore you with my bizarre taste in music.
1. **Spock’s Beard, “A Guy Named Sid”.** SB is a fantastic neo-prog band, one of my favorites. This is a track off the first album after their long-time lead singer/songwriter left the band. It’s definitely a big change in sound for them, but they’re still excellent.
2. **King Crimson, “The King Crimson Barber Shop”**. A few years ago, my wife bought me a set of special KC reissues. The reissue of “Three of a Perfect Pair” included a bunch of extra tracks – remixes of “Sleepless”, extended versions of some of the instrumental stuff – and an *extremely* silly barbershop quartet sung by the guys.
3. **Marillion, “Berlin”**. Marillion is a progressive rock band that’s been around for quite a while. This is off of *their* first album after their long-time lyricist/lead singer left. They didn’t do quite as well as SB did after the singer left: the first couple of albums with the new singer just didn’t stand up to the older stuff. (Although they did eventually find their feet again; they’re recent work is fantastic.) This is probably the best track off of the first album with the new lead singer.
4. **Scott Vestal and the BG’98 Band, “Home Sweet Home”**. Scott Vestal is one of the best banjoists in the world today. He plays everything from very traditional Scruggs-style bluegrass to incredibly out-there purely improvised jazz. In the late 90s, he recorded a series of yearly albums of mostly traditional instrumental bluegrass. It’s really pretty cool to see just what a bunch of really talented guys can do with even something as silly as this old folk song.
5. **Sonic Youth, “Jams Runs Free”.** A very typical Sonic Youth track off of their latest album. It’s a bit *smoother* than some of their older stuff, but it’s got the same sound to it – the semitones, dissonance, and other strangeness; it’s just more subtle.
6. **The Clogs, “Sticks and Nails”**. More post-rock. The Clogs are an excellent classical-leaning post-rock trio. This is a very dark, percussive, dissonant piece.
7. **Darol Anger’s Republic of Strings, “Father Adieu”**. A really wonderful track from of one of Darol Anger’s new projects.
8. **The National, “Friend of Mine”**. An alternate face for some members of the Clogs. The National is a very interesting band – you can definitely hear the connection to the Clogs, and yet it’s also very traditional country-rock style songwriting. Good stuff; not necessarily something that I’d want to listen to every day, but really great once in a while.
9. **Broadside Electric, “Seafood Invasion”**. An instrumental track from a really great Philadelphia area electric folk group. I learned to play the tin whistle from the whistle player in this group.
10. **Kaipa, “A Complex Work of Art”**. A *wonderful* track from the re-united Kaipa. Kaipa is the band where Roine Stolte of the Flower Kings got his start. It sounds like a cross between old Yes and the Flower Kings.

Pathological Programming: Ignorance is Bliss, or at least control.

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Todays dose of pathology is another masterpiece from the mangled mind of Chris Pressey. It’s called “[Version](http://catseye.mine.nu:8080/projects/version/)”, for no particularly good reason.
It’s a wonderfully simple language: there’s only *one* actual kind of statement: the labelled assignment. Every statement is of the form: “*label*: *var* = *value*”. But like last’s weeks monstrosity, Smith, Version is a language that doesn’t really have any flow control. But instead of copying instructions the Smith way, in Version the program is toroidal: it executes all statements in sequence, and when it reaches the end, it goes back to the beginning.
The way that you manage to control your program is that one of the variables you can assign values to is special: it’s called IGNORE, and it’s value is the *ignorance space*. The value of the ignorance space is an *irregular* expression (basically, a weak wild-card subset of regexps); if a statement’s label fits the ignorance space, then the statement is ignored rather than executed. The program will keep executing as long as there are any statements that are not part of the ignorance space.

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Navier-Stokes: Another Historic Problem Solved!?

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A few months ago, I wrote about the Poincare conjecture, and the fact that it appeared to finally have been solved by a reclusive russian mathematician named Grisha Perelman. Now there’s news that *another* classic problem may have been solved. This time, it’s the Navier-Stokes equation, apparently solved by [Professor Penny Smith](http://comet.lehman.cuny.edu/sormani/others/smith.html) of Lehigh University. She’s published the steps leading up to her solution in top peer-reviewed journals, and a [preprint of the final paper is now available via arxiv](http://arxiv.org/abs/math/0609740). There’s also a pretty good detailed description of the solution on [Christina Sormani’s website](http://comet.lehman.cuny.edu/sormani/others/SmithNavierStokes.html).
The Navier-Stokes equations form a classic problem that I actually know a bit more about, although I have to admit that the proof of the solution is beyond my ability to understand. Why should you care? Aside from the fact that it’s a famous problem with a million dollar reward posted by the Clay Institute for a solution, it’s *useful*. Unlike the Poincare conjecture, the reason why we care about solving the Navier-Stokes equations isn’t just theoretical. If Professor Smith’s solution and proof do turn out to be correct, it would be a really incredible accomplishment, with direct, immediate, practical implications.

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Meet the Manifolds

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So far, we’ve been talking about topologies in the most general sense: point-set topology. As we’ve seen, there are a lot of really fascinating things that you can do using just the bare structure of topologies as families of open sets.

But most of the things that are commonly associated with topology aren’t just abstract point-sets: they’re *shapes* and *surfaces* – in topological terms, they’re things called *manifolds*.

Informally, a manifold is a set of points forming a surface that *appears to be* euclidean if you look at small sections. Manifolds include euclidean surfaces – like the standard topology on a plane; but they also include many non-euclidean surfaces, like the surface of a sphere or a torus.

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More Bad Math from HIV Denialists

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As [Tara](http://scienceblogs.com/aetiology/2006/10/aids_and_viral_load.php), [Nick](http://aidsmyth.blogspot.com/2006/09/viral-load-paradigm-shift-not-really.html), and [Orac](http://scienceblogs.com/insolence/2006/10/more_distortion_of_peerreviewed_data_by.php) have already discussed, there’s been a burst of
activity lately from the HIV denialist crowd, surrounding [a new paper](http://jama.ama-assn.org/cgi/content/full/296/12/1498) studying the correlation between viral loads and onset and progression of symptoms in AIDS. For example, Darin Brown, allegedly a mathematician (and recently a troll in the comments here on GM/BM), has [written](http://barnesworld.blogs.com/barnes_world/2006/10/it_must_be_jell.html):
>Even if one is willing to endure the intellectual contortions necessary to
>reconcile these findings with the HIV/AIDS hypothesis, it is impossible to deny
>that they are incompatible with the justification for the treatment strategies
>advocated over the past 10 years.
>
>In case anyone was in a cave, for a decade, the treatment dogma has been:
>
>(1) CD4 counts and “viral load” are accurate predictors of progression to
>”AIDS” and death. In fact,
>
>(2) All three are correlated to each other. As viral loads go up, CD4 counts go
>down, and each indicates progression to “AIDS”. This is because HIV causes loss
>of CD4 cells. This is why they are called “surrogate markers”. This is why
>dozens and dozens of studies used viral load and CD4 counts as outcomes.
>Conversely, as viral load goes down, CD4 counts go up, and the patient is
>”healthier”.
>
>(3) If viral load goes up and CD4 counts go down sufficiently, you should go on
>ARVs immediately. Who knows how many healthy people have been put on these
>drugs on the basis of viral load and CD4 counts alone.
>
>The above 3 points have been drummed beyond belief over the past 10 years. For
>the AIDS establishment to deny now that this is what they have been saying all
>this time boggles the mind, but is not surprising.
When it comes to the science of it, I can’t contribute anything beyond what Tara and friends had to say. But the denialist argument around this is actually a classic example of one of my personal bugaboos concerning statistics. Details below the fold.

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Square Root on the Abacus

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Doing square root on the abacus is a lot like doing it on paper. The big difference? It’s actually *easier* on the abacus. What I find pretty cool is that I’m a rank beginner at the abacus. I never actually tried to use one before I started writing these posts. But I can do that root *faster* on the abacus than I can on paper.
The one difficult step in the paper square root is guessing the approximate digits; as you get beyond the third or fourth digit, the numbers start getting a bit large, and it can be hard to guess the correct estimate. On the abacus, you can very rapidly do repeated subtraction, so you deliberately guess low, and then add on. You’ll see what I mean as we work through an example.
One thing about the square root is you need a bigger abacus. So far, we’ve used very small ones for the examples here. The more you want to do with an abacus, the bigger you want it to be. A small abacus typically has something like 9 columns; a medium abacus has 13 digits. But for more interesting calculations, the kind of thing that we westerners would have a slide rule or scientific calculator, the abacus equivalent is a *27* digit abacus with a couple of sliding markers for helping keep track of things. You want a nice big abacus for doing things like roots, because you’re going to partition it into multiple sections.

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Connected Topologies and Fixed Points

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If you’ve got a connected topology, there are some neat things you can show about it. One of the interesting ones involves *fixed points*. Today I’m going to show you a few of the relatively simple fixed point properties of basic connected topologies.
To give you a taste of what’s coming: imagine that you have two sheets of graph paper, with the edges numbered with a coordinate system. So you can easily identify any point on the sheet of paper. Take one sheet, and lay it flat on the table. Take the *second* sheet, and crumple it up into a little ball. No matter how you crumple the paper into a ball, no matter where you put it down on the uncrumpled sheet, there will be at least one point on the crumpled ball of paper which is directly above the point with the same coordinate on the flat sheet.
As a quick aside: today is Yom Kippur, which means that this post is scheduled, and I’m not anywhere near my computer. So I won’t be able to make any corrections or answer any comments until late this evening.

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Friday Random Ten, Sept 29

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1. **Steven Reich, “Explanations Come To An End Somewhere”**: one movement from one of Steven Reich’s recent works, the “You Are” variations. I think it’s some of the best stuff he’s ever written.
2. **Fiddlers Four, “Pickin’ the Devil’s Eye”**. Another Darol Anger project, and as usual, it’s very cool. This is basic old-time country fiddling, full of energy and fire.
3. **Seamus Egan, “To An Old Rose”**. Seamus Egan is a brilliant Irish musician; he’s one of the finest Irish flutists in the world, and he also plays tenor guitar, regular guitar, banjo, keyboards, lap steel, and who knows what else. This is a tenor guitar waltz, very mellow, very beautiful.
4. **Hamster Theatre, “La Sacre D’Merde”**. Strange but brilliant band. They’re a Rock-in-Opposition spinoff of Thinking Plague with very heavy early music influences. I also love their titles… “La Sacre D’Merde” is basically “The Rite of Shit”; the title of the album is “The Public Execution of Mr. Personality”.
5. **Rachel’s, “Old Road”**. “Rachel’s” is another of those post-rock ensembles that I’m so wild about. They’re a very classical one; the closest comparison would be to the Clogs, by Rachel’s is *better*. In fact, they’re by far the best of the PREs that I’ve found so far. There just aren’t words for this kind of music, it’s too good. And I discovered them entirely by accident! I was looking for something else, and just happened to notice this thing tagged “PRE”.
6. **Dirty Three, “Stellar”**. Yes, more post-rock. Dirty Three is wonderful.
7. **John Corigliano, “Elegy”, 2nd movement of the Clarinet Concerto .** Corigliano is one of the finest composers in the world today. As a clarinetist, I’m particularly enamoured of his clarinet concerto. This is played by Stanley Drucker. I used to think that Drucker wasn’t such a great clarinetist, based on his recordings of Mozart and Weber’s concertos. This recording completely changed my mind; Drucker is the very finest performer of modern compositions on the clarinet. It’s obvious that he’s just not as *interested* in the older stuff; the modern stuff is what he loves.
8. **The Andy Statman Klezmer Orchestra, “Galitzianer Chusud”**. There was also a piece from this group in last week’s FRT. I’m particularly fond of this one, because my father is a Galtizianer. (Galitzianers are Jews from the Galicia region of what was Russia when they lived there. My mother’s family are Litvak’s (Latvians). The Litvaks and the Galitzianers were traditionally rivals; the old joke among Litvaks is that if you’re not careful, your children will grow up to marry a Galitzianer.)
9. **Rachel’s, “And Keep Smiling”**. Two Rachel’s tracks in one FRT! Hurrah!
10. **Psychograss, “Stroll of the Mudbug”**. Psychograss is yet another Darol Anger project, also featuring my former banjo teacher, Tony Trischka, not to mention David Grier, Mike Marshall, and Todd Phillips. A group of musicians with more skill than this gang is damned hard to find, and they have a *great* chemistry as a group. It’s newgrass with a good bit of jazz.