Posts Tagged ‘measurement’

On the Height of a Field

January 1, 2013

This is a short story about belief and evidence, and it starts with the GPS watch I use when I go for a run. Here’s the plot of my elevation today:


It looks a little odd until I show you this map of the run:


Each bump on the elevation plot is one lap of the field. In the middle, I changed directions, giving the elevation chart an approximate mirror-image symmetry. (I don’t know what causes the aberrant spikes, but my friend reports seeing the same thing on his watch.)

According to the GPS data, the field is sloped, with a max height of 260 feet near the center field wall and 245 feet near home plate. It’s insistent on this point, reiterating these numbers each time I do the run (except once when the tracking data was clearly off, showing me running across parking lots and through nearby buildings.) I disagreed, though. The field looked flat, not sloped at 3 degrees. I was disappointed to have found a systematic bias in the GPS data.

But I occasionally thought of some minor consideration that impacted my belief. I remembered that when I went biking, I often found that roads that look flat are actually uphill, as can be verified by changing directions and feeling how much easier it becomes to go a given pace. I Googled for the accuracy of GPS elevation data, and found that it’s only good to about 10 meters. But I didn’t care about absolute elevation, only change across the field, and I couldn’t find any answers on the accuracy of that. (Quora failed me.) I checked Google Earth, and it corroborated the GPS, saying the ground was 241 ft behind home plate and 259 in deep center field. But then I read that the GPS calibrated its elevation reading by comparing latitude/longitude coordinates with a database, and so may have been drawing from the same source as Google Earth.

People wouldn’t make a sloped baseball field, would they? That would dramatically change the way it plays, since with a 15-foot gain, what was once a solid home run becomes a catch on the warning track. Googling some more, I found that baseball fields can be pretty sloped; the requirements are fairly lax, and in fact they are typically sloped to allow drainage.

I was starting to doubt my initial judgment, and with this in mind, when I looked at the field, it made more and more sense that it’s sloped. Along the right field fence, there’s a short, steep hill leading up to the street. It’s about five feet high and at least a 30-degree slope. It’s completely unnatural, as if it exists because the field as a whole used to be considerably more sloped, but was dug out and flattened. The high edge of the field was then below street level, so there’s that short, steep hill leading up. And if the field was dug out and flattened, maybe they didn’t flatten it all the way. The entire campus is certainly sloped the same general direction as the GPS claimed for the field. It drops about 70 feet from north to south, and it’s frequently noticeable as you walk or bike around. There’s another field I run on with essentially the same deal, and I found that when I knew what to look for, I could indeed see the slope there.

Eventually, the speculation built up enough to warrant a little effort to make a measurement. I asked a wise man what to do, and he suggested I find a protractor, hang a string down to detect gravity, and site from one side of the field to the other. I did so, expecting to feel the boldness of an impartial, truth-seeking scientific investigator as I strode across the grass. That wasn’t what I got at all.

First, I felt continuous fluctuations in my confidence. “I’m 60% confident I’ll find the field is sloped,” I told myself, then immediately changed it to 75, not wanting to be timid, then felt afraid of being wrong, and went back to 50. I’ve played The Calibration Game and learned what beliefs mean, and mostly what it’s done is give me the ability to not only be uncertain about things, but to be meta-uncertain as well – not sure just how uncertain I am, since I don’t want to be wrong about that!

Second, I felt conflicting desires. I couldn’t decide what I wanted the result to be. I wanted the field to be flat to validate my initial intuition, not the stupid GPS, but I also wanted the field to be sloped so I could prove to myself my ability to change my beliefs when the evidence comes in, even if it goes against my ego. (A strange side-effect of wanting to believe true things is that you find yourself wanting to do things not because they help you believe the truth, but because you perceive them to be the sort of things that truth-seekers would do.) I recalled a video I had seen years ago about Gravity Probe B, and the main thing I remembered from it was a scientist with long, gray hair and huge unblinking eyeballs explaining in perfect monotone that he didn’t have a desire for the experiment to confirm or refute general relativity; he only wanted it to show what reality was like.

On top of all this, there was the sense of irony at so much mental gymnastics over a triviality like the slope of a baseball field, and the self-consciousness at the absurdity of standing around in the cold pointing jerry-rigged protractors at things. So at last I crossed the field and lined up my protractor for the moment of truth

It didn’t work. I had placed my shoes down on the grass as a target to site, but from center field they were hidden behind the pitcher’s mound. I recrossed the field and adjusted them, and went back. I still couldn’t see the shoes; they were too small and hidden in the grass. I could see my backpack, though, so I sited off that. But it still didn’t really work. I didn’t have a protractor on hand, so I had printed out the image of one from Wikipedia and stapled it to a piece of cardboard, but the cardboard wasn’t very flat, making siting along it to good accuracy essentially impossible.

I scrapped that, and after a few days went to Walgreens and found a cheap plastic protractor and some twine that I used to tie in my water bottle as a plumb bob. Returning to the field, I finally found the device to be, well, marginal. Holding it up to my eye, it was impossible to focus along the entire top of the protractor at once, and difficult to establish unambiguous criteria for when the protractor was accurately aimed. I was also holding the entire thing up with my hands, and trying to keep the string in place between siting along the protractor and moving my head around to get the reading.

Nonetheless, my reading came to 87 degrees from center field to home plate and 90 degrees from home plate back to center field. This three-degree difference seemed pretty good confirmation of the GPS data. In a final attempt to confirm my readings, I repeated the experiment in a hallway outside my office, which I hope is essentially flat. It’s 90 strides long, (and I’m about two strides tall) and I found 88 degrees from each side, roughly confirming that the protractor readings matched my expectations. (I’d have used the swimming pool, which I know is flat, but it’s closed at the moment.)

I’m now strongly confident that the baseball field is sloped – something around 95% after considering all the points in this post. That’s enough that I don’t care to keep investigating further with better devices, unless maybe someone I know turns out to have one sitting around.

Still, there is some doubt. Couldn’t I have subconsciously adjusted my protractor to find what I expected? There were plenty of ways to mess it up. What if I had found no slope with the protractor? Would I have accepted it as settling the issue, or would I have been more likely to doubt my readings? It’s perfectly rational to doubt an instrument more when it gives results you don’t expect – you certainly shouldn’t trust a thermometer that says your temperature is 130 degrees – but it still feels intuitively a bit wrong to say the protractor is more likely to be a good tool when it confirms what I already suspected.

The story of how belief is supposed to work is that for each bit of evidence, you consider its likelihood under all the various hypotheses, then multiplying these likelihoods, you find your final result, and it tells you exactly how confident you should be. If I can estimate how likely it is for Google Maps and my GPS to corroborate each other given that they are wrong, and how likely it is given that they are right, and then answer the same question for every other bit of evidence available to me, I don’t need to estimate my final beliefs – I calculate them. But even in this simple testbed of the matter of a sloped baseball field, I could feel my biases coming to bear on what evidence I considered, and how strong and relevant that evidence seemed to me.  The more I believed the baseball field was sloped, the more relevant (higher likelihood ratio) it seemed that there was that short steep hill on the side, and the less relevant that my intuition claimed the field was flat. The field even began looking more sloped to me as time went on, and I sometimes thought I could feel the slope as I ran, even though I never had before.

That’s what I was interested in here. I wanted to know more about the way my feelings and beliefs interacted with the evidence and with my methods of collecting it. It is common knowledge that people are likely to find what they’re looking for whatever the facts, but what does it feel like when you’re in the middle of doing this, and can recognizing that feeling lead you to stop?

Reflections on the Moon

September 26, 2009

Q: Why don’t elephants play tennis?
A: They prefer squash.

Two players compete in a fast-paced game at the gym. I exercise across from them, watching as they smash a blue rubber ball in turns. The game is in a small indoor court, and the ball moves very quickly. Its motion, followed from a distance, appears nearly rectilinear. All of the walls, the floor, and the ceiling are in play, and the ball’s wild meanderings, jots of wild color, mesmerize me as I stretch.

The game is squash (Wikipedia), which I had heard of but not seen played before. Watching the squash players, I was at first surprised by their ability to predict the seemingly-chaotic motion of the ball. However, a geometric property of the court aids them.

The squash ball moves very quickly (faster than a major-leaguer’s fastball), so over the short distances of the court, we can approximate its motion as roughly a straight line. The court is a rectangular prism, and this shape has the property that if a player smashes the ball at a corner, the ball will pop back out right at them, parallel to the way it came in. In two dimensions this is shown in the following diagram:

The squash ball follows the dark blue path, bouncing off the corners at points A and B.  The light blue line shows the continuation of the paths, with a hypothetical meeting point D if the incoming and outgoing lines are not parallel.

The squash ball follows the dark blue path, bouncing off the corners at points A and B. The light blue line shows the continuation of the paths, with a hypothetical meeting point D if the incoming and outgoing lines are not parallel.

The ball comes into the corner bouncing at point A, making an angle \theta with the wall. By hypothesis, it bounces off with the same angle, then comes into the next wall with an angle \phi at point B. It bounces off with this angle as well and returns to the court. We’re trying to show that the incoming and outgoing paths are parallel.

To do this, I’ve drawn in light blue the continuation of the incoming and outgoing paths. If they’re parallel, they never meet, and the angle drawn as \delta should be zero. Notice that \theta and \phi are the small angles of a right triangle ABC, so they add to a right angle. Angle CBD is opposite \phi, and so equal to it. That means angle ABD is 2\phi and similarly angle BAD is 2\theta. Those two angles, along with \delta form the triangle ABD. However, they add to a straight angle by themselves, and so we must have \delta = 0, showing that the ball pops out parallel to the way it came in, allowing the players to predict its motion easily.

In three dimensions, this is just the same, except that you have to work through the argument over again. A student of mine pointed out a different argument to come to the same result. Set up and x-y-z coordinate system at the corner along the intersections of the planes. Then one wall works by flipping the x-coordinate of the incoming ball’s velocity vector, and the other two flip the ball’s y and z-coordinates, so that after bouncing off all three, the velocity vector is reversed.

In squash, this result is far from perfect because gravity affects the ball’s motion, and its rotation, along with friction from the walls, may affect its angle of reflection. Energy is lost in each reflection as well, and the ball will slide some against the wall, so all in all it’s a rough approximation.

For light the approximation is much better as long as the wavelength of light is much smaller than the size of the mirrors. The setup with three orthogonal plane mirrors like this is a called a retroreflector because it reflects light back the way it came. If you look into one from any angle, you will see your own pupil at the corner, because at the corner the incoming and outgoing rays are not only parallel, but on top of one another, so light must start and end at the same place after reflecting there. All the light you see ends at your pupil, so that’s what you see in the corner. If you have three hand mirrors, it’s an easy experiment to try.

One interesting application of this idea is shown here:

A retroreflector on the moons surface.  The Apollo missions left this array of hundreds of reflectors intended for extremely accurate measurements of the Eart-Moon distance.

A retroreflector on the moon's surface. The Apollo missions left this array of hundreds of reflectors intended for extremely accurate measurements of the Eart-Moon distance.

This is a retroreflector array on the moon. When an Earth-based laser sends a pulse of light at the moon, the retroreflectors send the pulse back to Earth. If you could measure the trip time very precisely, you can multiply by the speed of light to find the distance to the moon. The APOLLO project (not the lunar orbiters, but the ground-based Apache Point Observatory Lunar Laser-ranging Operation) is trying to do this to an accuracy of one millimeter.

I’ve sometimes heard silly things like “the Campbell’s soup cans thrown out by Americans in a single month could stretch to the Moon and back three times.” I say this is silly because

  • Why would you want to do that?
  • I totally just made the statistic up because it is meaningless and nonmemorable. Things like per-capita consumption, percentages of usable land being turned into dumps, and statistics about ecological impact actually mean something.
  • No they can’t. They would fall down if they tried.

Regardless, if you know the distance to the moon with one millimeter accuracy, then your estimate of how many Campbell’s soup cans away it is is limited by how accurately you know the size of a Campbell’s soup can until you measure the can to an accuracy of single atom (and Campbell’s soup cans vary from one to another by a lot more than that, and don’t stack perfectly regularly, and shift around, and get hotter and colder, etc).

There are a few good reasons you’d want to know the position to the moon so precisely. Perhaps the most striking is as an extremely tight test of the equivalence principle of general relativity. The Earth and Moon have different densities, and so might conceivably fall towards the sun at different accelerations, even when they’re the same distance away. Modern cosmology and theoretical physics frequently explore theories of modified gravity in attempts to explain the acceleration of the universe’s expansion or create quantum theories of gravity. If the equivalence principle doesn’t hold, watching the acceleration of the moon very closely could be the first place we’d get a hint of it.