By way of the astronomy picture of the day, I encountered a really fantastic site about the analemma.
The analemma is the apparent path that the sun takes in the sky during the year. If you record the
precise position of the sun at the same time every day, instead of being in exactly the same place
every day, it will traverse a figure eight, like in this image. This is an effect caused by a combination of the eccentricity of the earth's orbit, and the tilt of the earth's axis. It can be a bit hard to visualize just where the figure-eight shape comes from; the analemma site uses a combination of diagrams and animations to make it extremely clear, and works through the entire process of demonstrating where each component of the analemma comes from, and deriving the equations that describe it.
I've mostly been taking it easy this week, since readership is way down during the holidays, and I'm stuck at home with my kids, who don't generally give me a lot of time for sitting
and reading math books. But I think I've finally got time to get back to the stuff
I originally messed up about sheaves.
I'll start by talking about the intuition behind the idea of sheaves. The basic idea of
a sheave is to provide a way of taking some local property of a topological space, and
demonstrating that it holds everywhere. The classic example of this is manifolds, where the *local* property of being locally almost euclidean around a point is expanded to being almost euclidean around *all* points.
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Friday Not-So-Random Five
I decided in honor of the new year, I'd do something a bit different this week. Instead of
doing a random shuffle on my IPod, I separated out my favorites of the modern classical pieces that I discovered this year. Some of these are brand new recordings just released this year; others are older recordings that I just happened to discover this year.
1. **Igor Stravinsky, "Suite #1", from "Shadow Dances" performed by the Orpheus Chamber Orchestra.** Beautiful piece for a small orchestra. Very typically Stravinsky; some strange tonalities, but they're mostly very subtle. This is modern classical music that even people who don't generally like modern classical can appreciate.
2. **Tan Dun, "Water Passion after St. Matthew"**. A piece written by the Chinese composer Tan Dun in honor of the 250th anniversary of the death of JS Bach. This is *definitely* not a piece for people who don't like modern classical music. Mostly atonal, except for a few sections. It's got some fragments from Bach's St. Matthew's Passion, with a very strong influence from Chinese opera. It often sounds oddly Jewish. I haven't made up my mind about
this yet; it's going to take a few more listenings before I really get it. At times, I think it's brilliant, and at times, I think it's just strange. In any case, it's worth the
effort of listening to, to hear the voice of a very notable modern composer writing in
an utterly unique style.
3. **Steven Reich and Maya Beiser, "Cello Counterpoint"**. I've been a fan of Steven Reich for a long time. He's a modern composer from the minimalist school, whose music is strongly
influenced by the time he spent studying with African drummers. This piece is just dazzling; it's all played by Maya Beiser, but she's recorded 7 different tracks, and plays the 8th live over the mixed recordings. This is an amazing piece of music.
4. **John Corigliano, "Fantasy on a Bach Air"**. A piece by John Corigliano, also in honor of JS Bach, built around a melody from a Bach air. Corigliano is my favorite modern composer; he tends to write a lot of very atonal stuff, but unlike composers like Stockhausen, he manages
to do it in a way that's pleasant to listen to. He finds different kinds of musical structures for the music, which still appeal to your ear.
5. **Phillip Glass, "Overture from Les Enfants Terrible"**. "Les Enfants Terrible" is one of Phillip Glass's latest operas. It's distinctively Glass, but at the same time, it's very different from much of Glass's past work. It's much more willing to be openly dissonant, and
to use larger, longer structures and more complex rhythms than most of Glass's earlier work.
Like my friend and blogfather [Orac][orac], I'm a huge fan of Doctor Who, and I've been
greatly enjoying its renewed life in the new series on BBC. In fact, the current Doctor, played by David Tennant, has become my favorite of all of the Doctors - better than the
usual fan favorite of Tom Baker, better than my own former favorite, Jon Pertwee.
The reason why I'm such a fan of Tennant is that his Doctor combines many of the personality
traits of the past Doctors, while giving it his own unique spin. Tennant's Doctor has the
hands-on activity of Jon Pertwee, the exuberance of Peter Davison, the sense of *history* of
William Hartnell (the very first Dr), and the kind of goofy enthusiasm of Tom Baker.
But he's more than just a mish-mash of past Doctors - he's got those traits that connect his
character to his past incarnations, but he's definitely his own person. His hyperactive
exuberance is quite different from any Doctor we've seen before. Now that's partly just the
style of writing in the new series: the Christopher Eccleston Doctor had a similar
hyperactivity. But Eccleston's hyperness was much more self-focused: his Doctor was almost as
egocentric as the Colin Baker incarnation. Tennant's Doctor is just enthusiastic about the
universe, about humanity, about the excitement of living. But that's also played in conflict
with the other major trait of this Doctor: he's got a darkness, an edge of bitterness and
alienation just below the surface.
One of my favorite examples of this comes from last season's episode "School Reunion" which
features the return of Sarah Jane Smith, the old companion of Tom Baker's Doctor. Overall, Tennant's Dr. in the episode is the bouncy happy silly doctor - particularly when he finally gets to admit to Sarah Jane that he is the Doctor, and when SJ shows him K9. But there's
a scene, where he's alone with the leader of the Krillitane, the villains of the episode. The
Krillitane tries to entice the Doctor to join forces with him, and Tennant sneers and replies
that in his youth he was so very patient, so very merciful, but *not any more*. Now he only
gives one warning, and this is it.
The reason I'm writing all this today is that I just watched a bit-torrented download of this
years special Christmas episode. Overall, it's a rather goofy (if fun) affair - a rather
over-the-top monster story. But it's got another really wonderful scene that is so typical of
what I like about this Doctor. He's coming in to confront the monster, disguised as one of her
robots. After revealing himself, he shuts down the rest of her robots with great humorous
flair using the (extremely large) remote control hidden in his pocket (it's larger on the
inside!); and he offers the monster one chance to let him move her and her children to some
other planet, where they won't harm anyone. She refuses, and he replies "then this is *your*
doing", reveals that he is from Gallifrey, and destroys all of her children. It's a great
scene, and one that does a remarkable job of demonstrating just *who* this Doctor is.
At the end, the woman who appears to be the new companion doesn't join the Doctor, but
tells him to find someone. He naturally replies, "I don't need anyone". She disagrees,
and tells him that she believes that he sometimes needs someone "to stop him" - and
in one of those perfectly played bits of alienation and loneliness at the heart of this
doctor, he agrees. It's a subtly played scene, but it's perfect.
Anyway, overall, this years Christmas special is good, but not great. As I said above,
it's really a bit over-the-top and goofy. Definitely *not* the best writing of the
new series. But it's got a few really tremendous scenes, and overall, it leaves me
with confidence that Russell Davies and crew are going to continue to do a great job
of keeping Doctor Who alive and kicking.
Being a Nice Jewish BoyTM, Christmas is one of the most boring days of the
entire year. So yesterday, I was sitting with my laptop, looking for something interesting to read. I try to regularly read the [Panda's Thumb][pt], but sometimes when I don't have time, I just drop a bookmark in my "to read" folder; so on a boring Christmas afternoon, my PT backlog seemed like exactly what I needed.
[One of the articles in my backlog caught my interest.][pt-sc] (I turned out to be short enough that I should have just read it instead of dropping it into the backlog, but hey, that's how things go sometimes!) The article was criticizing that genius of intelligent design, Sal Cordova, and [his article about Zebrafish and the genetics of regeneration
in some zebrafish species.][sc] I actually already addressed Sal's argument [here][bm-sc].
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1. **Lunasa, "Feabhra"**: My favorite traditional Irish band. These guys are *really* traditional
instrumental Irish - Uillean pipes, flute, guitar, bodhran, and bass. The pipe player is
without doubt one of the best, if not *the* best in the world. I thought that I hated all kinds
of bagpipes until I saw Cillian Vallely performing live (before he joined Lunasa).
2. **Darol Angers Republic of Strings, "Bluebird"**: A track from Darol Angers latest project. Pretty much anything Darol does is gold; this isn't one of my favorite tracks, because I don't
like the singer, but it's got red-hot fiddling holding it all together, which makes up for it.
3. **Tortoise, "Unknown"**: Tortoise is another post-rock ensemble; one of the earliest ones. They're good, but not great.
4. **Bach, "Mer Sprach: Gehet Hin" from St. Matthews Passion**: Bach is the greatest composer
who ever lived; and I think that St. Matthews Passion is his finest work. A little slice
5. **Harry Bradley, "Dave Maguires/Gary Hastings Reels"**: very minimal traditional Irish fluting by a master. Harry is a brilliant flautist - he knows how to play with *enough* ornamentation to really punch the rhythm, but he never plays a single note more than he needs to. These reels feature him playing with nothing but a bouzouki and a trace of bohdran backing him. And it doesn't need any more. It's got amazing bounce and spirit to it, played by Harry at his reedy-sounding best.
6. **Rachel's, "4 or 5 Trees"**. One of my favorite post-rock ensembles. Rachel's is a very
classical-leaning PRE, and everything they do is brilliant.
7. **ProjeKct Two, "Escape from Sagittarius A"**: free improv by one of the trios that made up the last incarnation of King Crimson: Trey Gunn on bass/stick, Fripp on Guitar, and Adrian Belew playing a drum synth. Wierd, but good. It's great to hear Fripp when he's getting way
out there; he's often so disciplined that he holds back, so it's amazing to hear him really
kick loose. Sure, some of it isn't great - but some of it has a brilliance that can only come
8. **Mogwai, "I Chose Horses"**: yet another post-rock ensemble, this one from the more rock-oriented end of the genre. This is a mellowish track from them, with a very distinctively Mogwai sound to it.
9. **Miles Davis, "How Deep is the Ocean"**. Miles Davis. What more need be said?
10. **Godspeed You! Black Emperor, "Sleep: They Don't Sleep Anymore On The Beach/Monheim/Broken Windows, Locks Of Love Part III"**. Godspeed is the absolute unquestionably greatest of the rock-leaning post-rock ensembles. This track is very typical.
Today, I'm going to show you a very simple, very goofy little language called "SCEQL", which standards for "slow and clean esoteric queue language". It's based on nothing but a circular queue
of numbers with nothing but 8 commands. It's not one of the more exciting languages, but it can
be a lot of fun to figure out how to make the circular queue do what you want it to.
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The topology posts have been extremely abstract lately, and from some of the questions
I've received, I think it's a good idea to take a moment and step back, to recall just
what we're talking about. In particular, I keep saying "a topological space is just a set
with some structure" in one form or another, but I don't think I've adequately maintained
the *intuition* of what that means. The goal of today's post is to try to bring back
at least some of the intuition.
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In Haskell, there are no looping constructs. Instead, there are two alternatives: there are list iteration constructs (like
foldl which we've seen before), and tail recursion. Let me say, up front, that in Haskell if you find yourself writing any iteration code on a list or tree-like structure, you should always look in the libraries; odds are, there's some generic function in there that can be adapted for your use. But there are always cases where you need to write something like a loop for yourself, and tail recursion is the way to do it in Haskell.
Tail recursion is a kind of recursion where the recursive call is the very last
thing in the computation of the function. The value of tail recursion is that in a tail
recursive call, the caller does nothing except pass up the value that's returned
by the the callee; and that, in turn, means that you don't need to return to the caller
at all! If you think of it in terms of primitive machine-level code, in a tail-recursive call, you can use a direct branch instruction instead of a branch-to-subroutine; the tail-recursive call does *not* need to create a new stack frame. It can just reuse the callers frame.
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Over the weekend, in an attempt to cheer me up, a kind and generous reader sent me a link
[to a *really* wonderful site of crackpot science][adams]. It's a crackpot theory about how physics has it all wrong. You see, there is no such thing as gravity - it's all just pressure. And the earth (and all other planets) is actually a matter factory - matter is constantly created in the *hollow* center of the earth, and the pressure of all the new matter forces the earth to constantly expand. And the pressure of expansion creates the illusion of gravity. And according to the crackpot behind it all, the best part is that [*the math works!*][mathworks]
The site is the masterwork of graphic artist Neal Adams. Mr. Adams is a computer animation
guy; he's responsible for the obnoxious bumblebee "nasonex" ad. Mr. Adams believes that in
addition to drawing comic books and animated TV commercials, he's also a genius who's going to
totally reinvent all of physics, and show how all of those bigshot physicists and geologists are all wrong about everything.
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