This is an archive page. What you are looking at was posted sometime between 2000 and 2014. For more recent material, see the main blog at http://laboratorium.net

Take, for example, the problem of levelling. One of the most fundamental techniques of surveying involves looking through a telescope which is "absolutely" level. Once you have a telescope known to be level, you can look through it at two different points where your assistants are holding up vertical rods with heights marked along them. The difference in the measurements tells you how far the one point is above the other. Of course, your computed elevations will only be as correct as your telescope is actually level. It's thus of paramount importance to have a way of making your telescope very close to horizontal.

The most common technique of doing this is to fit the telescope with a "level tube:" a fluid-filled tube, parallel to the telescope, with an air bubble inside. When the bubble is exactly centered in the tube, the tube is horizontal. And hence the telescope is horizontal.

Well, more or less horizontal. What if the tube is mounted out of alignment, so that there's a slight angle between tube and telescope? Well, then, and this is the brilliant part, you use a technique called reversion to align the telescope and tube.

So first, you level the tube telescope as normal. When the bubble is nice and centered, you look through the telescope, which will be some place close to -- but not exactly -- horizontal. You have your assistant pick up a measuring rod, walk a ways away, and hold it vertically in the line of sight of the telescope, with the bottom end touching the ground. You take note of where along the rod the telescope is aimed. Say, for sake of example, it's at 1.137 meters off the ground.

Now, you take the telescope and flip it upside down. In your standard surveying device (called a "transit"), the telescope is mounted on a pair of wheels, one vertical and one horizontal, so it can point in any direction and do complete 360-degree turns. Upside-down is no problem; you just give the vertical wheel a half turn. Since the telescope is now pointing in the exact opposite of the way it used to be pointing, you need to give the horizontal wheel a half turn, too. Now you take the scope and look through it and make small adjustments until you're looking at the rod your assistant is holding, and then you make tiny adjustments until you're staring right at the 1.137-meter mark again.

See what that's gotten you? The telescope is aligned exactly the same way it used to be, because you're still sighting along the same straight line. But the level tube has been flipped upside-down. Instead of hanging down from the telescope, it's now projecting up. If there was an angle a between the tube and telescope, the tube is now 2a away from being horizontal.

Think of it this way. You levelled the tube. It was horizontal to start with. The telescope was a away from being level (let's think of it as being slightly too high at the front end). When it was that far away from being level, you were looking at the 1.137-meter mark on a fixed rod. Then you fiddled with the wheels and did some mojo that ended with the telescope looking at the exact same spot. So it's still a above the horizontal The tube, however, has been mirror-reflected, as it were, through the telescope. If the front end were some smidgen further away from the scope than the back end, it's still that smidgen away, just on the other side. So the tube is at an angle of 2a to its original position.

From here, the game is easy. You adjust the screws holding the tube to the scope. You make sufficient adjustment to bring the the bubble halfway from where it is towards the center. (That's because a, the actual bad angle between scope and tube, is only half of 2a, which is how far the tube is currently away from level.) Now, you go back to the start. Put the telescope back in its normal position and level the tube. Take a sight on the rod; flip the scope over and align it with that sight again. If you've done things well, the bubble in the tube will still be precisely centered.

This is a great trick. It works anywhere you have two different systems of measurement, each of which can be checked for internal consistency, and whose relative divergence, while not itself directly measureable, can be reversed. By means of the reversal, you turn the error term into something you can add to itself and then measure on one of your two scales.

Its partner-in-crime involves the subtraction of systematic errors: you rig things so that the same error shows up on both sides of an equation and cancels with itself. For example, rather than measure difference in heights by starting at one point and looking at the other; you measure both points by looking at them from some arbitrary third point. That way, you don't need to worry about how far the telescope is off the ground; you just establish that height, whatever it happens to be, as the zero of your coordinate system.

Provided this neutral third point is roughly equidistant from the two endpoints, you'll also get rid of a couple of other systematic errors this way. You'll cancel out any residual levelling error too small to detect with the tube. It doesn't matter if you're dropping 2cm in every 100m; if both points are 200m away from your scope, you'll be 4cm too low on both measurements, but that 4cm will be subtracted from itself when you take the difference of your measurements. On long-haul high-precision surveys, you'll even cancel out the effects of the earth's curviture and of atmospheric refraction.

Or, even more cleverly, you can use this trick to get rid of horizontal deviations in your telescope when you're constructing a straight line. Typical surveying straight lines are set up by starting at A, marking a spot B along the desired line, then taking the transit out to B, flipping the scope over vertically to line up with A, then flipping it back to point in the correct direction, where you set up a new point C and repeat. But this technique is vulnerable to unevenness in the vertical wheel; if you skew a little left when you flip, your line will skew progressively more to the left as you go along.

Let's think about the geometry for a moment. Say that when I spin the wheel vertically, I'm off by one degree to the right. If I now turn around, face the way I was coming from, and spin the wheel back to point the way I came from, it'll be back where it started. This means that the reverse 180-degree vertical flip causes a deviation of one degree to the left. Forward gives me a one-degree deviation right; backwards gives me a one-degree deviation left. Left, right. Those sound like opposites to me. I smell a reversion, don't you?

When I get to B, then, I start off as above. I take a "backsight" (a sighting with the scope upside-down) to A, then flip the scope over vertically and have my assistant mark out the appropriate point C. (That is, along the line I'm sighting, we lay out a known length (50 meters might be typical) using the tape, the other standard surveying tool.) Now, I know that C is likely to be slightly out of alignment. So I turn my scope around horizontally and take a regular sighting of A again. Now I flip it vertically and take a backsight, using that line to mark out some other point C' (at the same known distance along the line I'm now sighting).

If there's, say a leftwards angular error a due to skewing when I use the vertical wheel to go from backwards to forwards, then C is 50 cos a meters too far to the left. On the other hand, C' is 50 cos a meters too far to the right. The true point I'm looking for should be exactly halfway between C and C'. But that point is amazingly easy to find; we just tromp on over to C and C', lay down the tape between them, and find the halfway point. This new point, C*, becomes the point we use when we extend the line forwards from B, and so on. Any error introduced by eccentricity of the vertical wheel's mounting point in the transit has been cancelled out.

It is left as an exercise for the reader to work out other, more metaphorical, applications of the principle of reversion.

Previous such incidents left me with such treasures as my annotated copy of the fire code. This time around it was an encyclopedia of the rules to 150 major world sports (including jai alai, canoe sprint racing, bandy, and flat green bowls) and a 1966 surveying textbook. I'm especially happy about the latter; I've been looking for a good technical introduction to surveying for a while. Most modern books on the subject seem to omit much of the underlying math and physical principles, but this one predates all our modern fancy-schmancy electronical devices. GPS is for amateurs, and electronic sights are for cheaters; I want to know how Mason and Dixon did their job.

Other than these issues, it was a great show. Somewhere while I wasn't paying attention, they added a sixth member; their sound is richer without being thicker. They played a few new songs, some of which are gorgeous. And, most wonderfully of all, they had a not-available-in-stores tour CD with them, on which I quite literally spent every dollar I had on me. If they were a band that released singles, it would have been a B-sides collection: some freaky oddities and remixes, but also some beautiful songs that inexplicably missed the album cut.

I cannot do proper justice to how profoundly at peace their music puts me, though I have tried several times. This time around, I'll just say that the world is a much better place for their existence, and leave it at that.

That is, Napster begat Aimster (later Madster) and Grokster and Blubster and now Friendster, which isn't even a file-trading program. Twenty years from now, kids will wonder what's up with words like "monster," "shyster," and "lobster."

And all of this because Shawn Fanning was made fun of for his nappy hair.

Sakiyama v. AMF Bowling Centers, 17 Cal. Daily Opinion Serv. (The Recorder) 6099 (Cal. App. July 14, 2003).

A couple of days ago, I managed to combine these two interests: I've come up with a metaphor so distasteful that I think it all but precludes rational discussion.

One of the common arguments against affirmative action is stereotype regeneration: if some large fraction of minorities at a particular institution are there because of affirmative action, a modern generation of racists will develop that assumes that all minorities are there because of affirmative action, and that none are there because they would be "qualified" were they not minorities. This is the assumption that drives Kwesi Mfume and Clarence Thomas absolutely bugfuck; Mfume's response is to blame the new racists, while Thomas's response is to blame affirmative action itself.

In any case, this new form of stereotyping is mostly an exercise in botched conditional probabilities. Faced with a population whose variation on some metric is partially explained by a hidden variable, people just give up on the variable entirely and assume away the variation attributable to it. People don't deal well with r values other than 1, 0, and -1.

Having put the fallacy in this absurdly abstract setting, I then remembered where I'd seen something like it before. In some firing squads (possibly including Utah's), one of the shooters is (unknowingly) given a rifle loaded with blanks. This way, the executioners are allowed to believe that they are innocent of the killing.

Now, the parallelism isn't exact. The firing-squad issue has more to do with the difficulty of factoring causation and morality through probabilities. The squad is collectively responsible for a death; no hidden random fiddling with blanks and bullets changes this basic fact. Trying to parcel out portions of that responsibility is an open question in moral philosophy, but presumably one that shouldn't depend on the results of a die roll. And as for the legal treatment of statistical causation, it's not even coherent.

Anyway, these two scenarios do share a certain common structure. There's an identifiable group, some of whom are different from the others, in a way that's not easily observable, probably not even to them. And so the temptation arises, to attribute to the entire group the attributes of that hidden subgroup . . .

What makes this metaphor particularly egregious is that the offensiveness of the implied comparison cuts in the opposite direction from the point of the metaphor. It's pretty bad to compare minorities to executioners, but the thrust of the comparison is that the stereotyping created by affirmative action is stupid as a matter of profound irrationality.

After all, as a matter of principle, it's not so clear how much we should let important societal decisions be dictated by the known irrationality of particular groups of people. But making this argument isn't easy. I'm always on the lookout for catchy comparisons to help. If you come across any, let me know.

(Something similar sooner or later happens to all euphemisms, I think. The process is almost healthy; it means we still have at least a little bit of linguistic conscience left. "Ethnic cleansing" is the prize example; it made the descent from evasion to epithet quite quickly.)

In any case, now that Pentagon briefings are part of the daily news cycle, I'm guessing that it's only a matter of time before people start referring to hats as "head-mounted clothing systems."

It's been a busy week, and my dialup access has been even worse than usual, so I haven't had the chance to investigate in much detail, but I'm looking into this one. I think there may be an interesting back-story involved.

"about:blank"

It's not a particularly pretty place, nor are the chairs comfortable. It's just a couple of floors, extremely quiet and filled with the collected legal knowledge of our country. Damn, it's inspiring.

See, we've built ourselves quite a marvelous edifice in the law. It takes something for a society to be able to sustain such a sophisticated legal system, the same way it takes something to support Major League Baseball or the Space Needle. We made this. Maybe I'm just looking up some obscure points of federal standing law, but they fit into the rest of it. If I wanted to, I could go through the whole library, book by book, and see how it all fit together.

Sure, it creaks along here and there, and some parts of it are downright awful, but as a whole, it's quite an accomplishment. The trains run most of the time, people eat most of the time, we haven't had a serious war against ourselves in well over a century. And when problems crop up, we turn to the legal system to resolve them. My plucky little summer employer is dedicated to the proposition that there are many things wrong with this country, but that an awful lot of those things, maybe even all of them, can be addressed by working within the system we've got. And that's not to be sneezed at.

And--and this is the part that gets me--I can just walk in off the street at Hastings and start in on working within the system. It's not just that we have laws and books about laws, or that what's in the books really works when you take it to a courtroom: it's that we have public libraries full of those books, and that anyone can read those books and take their contents to the courtroom. We don't make it easy, but we don't make it impossible, either.

When I go up to Hastings, I'm binding myself to a democratic, egalitarian tradition. We made these laws, and we are all equal before them. From Abe Lincoln and Thomas Jefferson to inmates teaching themselves law in a prison library, this is the sweep of learning and living law in this nation, and this is the tradition in which I am taking my place. It makes me proud to be an American, and proud to be doing what I can to make America better.

On the other hand, I feel no great pride over the folks setting off firecrackers in the street, close enough for me to smell the smoke.