# The Universe of Discourse

Sun, 24 Apr 2016

[ Disclaimer: I know very little about basketball. I think there's a good chance this article contains at least one basketball-related howler, but I'm too ignorant to know where it is. ]

Randy Olson recently tweeted a link to a New York Times article about Steph Curry's new 3-point record. Here is Olson’s snapshot of a portion of the Times’ clever and attractive interactive chart:

(Skip this paragraph if you know anything about basketball. The object of the sport is to throw a ball through a “basket” suspended ten feet (3 meters) above the court. Normally a player's team is awarded two points for doing this. But if the player is sufficiently far from the basket—the distance varies but is around 23 feet (7 meters)—three points are awarded instead. Carry on!)

Stephen Curry

The chart demonstrates that Curry this year has shattered the single-season record for three-point field goals. The previous record, set last year, is 286, also by Curry; the new record is 406. A comment by the authors of the chart says

The record is an outlier that defies most comparisons, but here is one: It is the equivalent of hitting 103 home runs in a Major League Baseball season.

(The current single-season home run record is 73, and !!\frac{406}{286}·73 \approx 103!!.)

I found this remark striking, because I don't think the record is an outlier that defies most comparisons. In fact, it doesn't even defy the comparison they make, to the baseball single-season home run record.

Babe Ruth

In 1919, the record for home runs in a single season was 29, hit by Babe Ruth. The 1920 record, also by Ruth, was 54. To make the same comparison as the authors of the Times article, that is the equivalent of hitting !!\frac{54}{29}·73 \approx 136!! home runs in a Major League Baseball season.

No, far from being an outlier that defies most comparisons, I think what we're seeing here is something that has happened over and over in sport, a fundamental shift in the way they game is played; in short, a breakthrough. In baseball, Ruth's 1920 season was the end of what is now known as the dead-ball era. The end of the dead-ball era was the caused by the confluence of several trends (shrinking ballparks), rule changes (the spitball), and one-off events (Ray Chapman, the Black Sox). But an important cause was simply that Ruth realized that he could play the game in a better way by hitting a crapload of home runs.

The new record was the end of a sudden and sharp upward trend. Prior to Ruth's 29 home runs in 1919, the record had been 27, a weird fluke set way back in 1887 when the rules were drastically different. Typical single-season home run records in the intervening years were in the 11 to 16 range; the record exceeded 20 in only four of the intervening 25 years.

Ruth's innovation was promptly imitated. In 1920, the #2 hitter hit 19 home runs and the #10 hitter hit 11, typical numbers for the nineteen-teens. By 1929, the #10 hitter hit 31 home runs, which would have been record-setting in 1919. It was a different game.

Takeru Kobayashi

For another example of a breakthrough, let's consider competitive hot dog eating. Between 1980 and 1990, champion hot-dog eaters consumed between 9 and 16 hot dogs in 10 minutes. In 1991 the time was extended to 12 minutes and Frank Dellarosa set a new record, 21½ hot dogs, which was not too far out of line with previous records, and which was repeatedly approached in the following decade: through 1999 five different champions ate between 19 and 24½ hot dogs in 12 minutes, in every year except 1993.

But in 2000 Takeru Kobayashi (小林 尊) changed the sport forever, eating an unbelievably disgusting 50 hot dogs in 12 minutes. (50. Not a misprint. Fifty. Roman numeral Ⅼ.) To make the Times’ comparison again, that is the equivalent of hitting !!\frac{50}{24\frac12}·73 \approx 149!! home runs in a Major League Baseball season.

At that point it was a different game. Did the record represent a fundamental shift in hot dog gobbling technique? Yes. Kobayashi won all five of the next five contests, eating between 44½ and 53¾ each time. By 2005 the second- and third-place finishers were eating 35 or more hot dogs each; had they done this in 1995 they would have demolished the old records. A new generation of champions emerged, following Kobayashi's lead. The current record is 69 hot dogs in 10 minutes. The record-setters of the 1990s would not even be in contention in a modern hot dog eating contest.

Bob Beamon

It is instructive to compare these breakthroughs with a different sort of astonishing sports record, the bizarre fluke. In 1967, the world record distance for the long jump was 8.35 meters. In 1968, Bob Beamon shattered this record, jumping 8.90 meters. To put this in perspective, consider that in one jump, Beamon advanced the record by 55 cm, the same amount that it had advanced (in 13 stages) between 1925 and 1967.

Progression of the world long jump record
The cliff at 1968 is Bob Beamon

Did Beamon's new record represent a fundamental shift in long jump technique? No: Beamon never again jumped more than 8.22m. Did other jumpers promptly imitate it? No, Beamon's record was approached only a few times in the following quarter-century, and surpassed only once. Beamon had the benefit of high altitude, a tail wind, and fabulous luck.

Joe DiMaggio

Another bizarre fluke is Joe DiMaggio's hitting streak: in the 1941 baseball season, DiMaggio achieved hits in 56 consecutive games. For extensive discussion of just how bizarre this is, see The Streak of Streaks by Stephen J. Gould. (“DiMaggio’s streak is the most extraordinary thing that ever happened in American sports.”) Did DiMaggio’s hitting streak represent a fundamental shift in the way the game of baseball was played, toward high-average hitting? Did other players promptly imitate it? No. DiMaggio's streak has never been seriously challenged, and has been approached only a few times. (The modern runner-up is Pete Rose, who hit in 44 consecutive games in 1978.) DiMaggio also had the benefit of fabulous luck.

Is Curry’s new record a fluke or a breakthrough?

I think what we're seeing in basketball is a breakthrough, a shift in the way the game is played analogous to the arrival of baseball’s home run era in the 1920s. Unless the league tinkers with the rules to prevent it, we might expect the next generation of players to regularly lead the league with 300 or 400 three-point shots in a season. Here's why I think so.

1. Curry's record wasn't unprecedented. He's been setting three-point records for years. (Compare Ruth’s 1920 home run record, foreshadowed in 1919.) He's continuing a trend that he began years ago.

2. Curry’s record, unlike DiMaggio’s streak, does not appear to depend on fabulous luck. His 402 field goals this year are on 886 attempts, a 45.4% success rate. This is in line with his success rate every year since 2009; last year he had a 44.3% success rate. Curry didn't get lucky this year; he had 40% more field goals because he made almost 40% more attempts. There seems to be no reason to think he couldn't make the same number of attempts next year with equal success, if he wants to.

3. Does he want to? Probably. Curry’s new three-point strategy seems to be extremely effective. In his previous three seasons he scored 1786, 1873, and 1900 points; this season, he scored 2375, an increase of 475, three-quarters of which is due to his three-point field goals. So we can suppose that he will continue to attempt a large number of three-point shots.

4. Is this something unique to Curry or is it something that other players might learn to emulate? Curry’s three-point field goal rate is high, but not exceptionally so. He's not the most accurate of all three-point shooters; he holds the 62nd–64th-highest season percentages for three-point success rate. There are at least a few other players in the league who must have seen what Curry did and thought “I could do that”. (Kyle Korver maybe? I'm on very shaky ground; I don't even know how old he is.) Some of those players are going to give it a try, as are some we haven’t seen yet, and there seems to be no reason why some shouldn't succeed.

A number of things could sabotage this analysis. For example, the league might take steps to reduce the number of three-point field goals, specifically in response to Curry’s new record, say by moving the three-point line farther from the basket. But if nothing like that happens, I think it's likely that we'll see basketball enter a new era of higher offense with more three-point shots, and that future sport historians will look back on this season as a watershed.

[ Addendum 20160425: As I feared, my Korver suggestion was ridiculous. Thanks to the folks who explained why. Reason #1: He is 35 years old. ]

Wed, 24 Jan 2007

Length of baseball games
In an earlier article, I asserted that the average length of a baseball game was very close to 9 innings. This is a good rule of thumb, but it is also something of a coincidence, and might not be true in every year.

The canonical game, of course, lasts 9 innings. However, if the score is tied at the end of 9 innings, the game can, and often does, run longer, because the game is extended to the end of the first complete inning in which one team is ahead. So some games run longer than 9 innings: games of 10 and 11 innings are quite common, and the major-league record is 25.

Counterbalancing this effect, however, are two factors. Most important is that when the home team is ahead after the first half of the ninth inning, the second half is not played, since it would be a waste of time. So nearly half of all games are only 8 1/2 innings long. This depresses the average considerably. Together with the games that are stopped early on account of rain or other environmental conditions, the contribution from the extra-inning tie games is almost exactly cancelled out, and the average ends up close to 9.

Thu, 16 Nov 2006

Etch-a-Sketch blue-skying, corrected
In my last article I discussed a scheme for improving the Etch-a-Sketch which contained a serious mechanical error. I was discussing attaching gears to the two knobs of the Etch-a-Sketch to force them to turn at the exact same rate. Supposing that the distance between the knobs is 1 unit, I said, then we can gear the two knobs together by attaching a gear of radius 1/2 to each knob; the two gears will mesh, and the knobs will then turn at the same rate, in opposite directions. This was fine.

Then I went astray, and suggested adding an axle peg midway between the two knobs, and putting gears of radius 1/3 on the peg and on the two knobs. This won't work.

The one person who wrote to me to ask about the problem is a very bright person, but been seriously confused about how I was planning to set up the gears, so I evidently I didn't explain it very well. It needed a picture. So this time I'm going to try to get it right, with pictures. Here is an Etch-a-Sketch:

Recall that the distance between the centers of the two knobs (shown here in gray) is defined to be 1.

Here are some gears, which happen to have radii 1/3, 1/4, and 1/6:

The dark spots are the axle pins, which are at the centers of the gears.

Here's a picture of an Etch-a-Sketch with a radius-1/2 gear mounted on each knob:

Since the two gears are of equal size, the knobs are constrained to turn at the same rate, in opposite directions. This forces the Etch-a-Sketch to draw a line with slope -1, from northwest to southeast or vice-versa.

Here the knobs have been fitted with different-sized gears, one with radius 1/3 and the other with radius 2/3:

The left-hand knob is forced to turn exactly twice as fast as the right-hand knob, producing a line with slope -1/2. To get a line with slope -2, just reverse the positions of the two gears. I got this much right in the original article. (Although it didn't occur to me, before I saw the pictures, just how much of the screen would be occluded by the gears. Better make them transparent.)

Then I suggested that you could drill a little hole in between the two knobs, and use it to mount a third axle and a third gear. If all three gears are the same size, the two knobs are forced to turn at the same rate, this time in the same direction, and you get a line with slope 1, from southeast to northwest:

All fine, except that I said that since all three gears are the same size, they must each have radius 1/3. Wrong. They must have radius 1/4, as the diagram above should make clear: from the center of the left knob to the rightmost edge of its gear is one radius; the width of the middle gear is two radii, not one, and from the edge of the right gear to the center of its knob is the fourth radius. Since the total of four radii is 1, each radius must be 1/4. Oops.

This wrecks the rest of the details of my other article. Since we were already including gears of size 1/2 and 1/3, I reasoned, we can throw in a gear of size 1/6 and get some new behaviors from the 1/2 + 1/3 + 1/6 combination. The corresponding combination for 1/2 and 1/4 is 1/8:

And this gets us lines of slope 2, or, reversing the order of the gears, 1/2.

So what next? The calculations are a bit less obvious than they were back in the happy days when I thought that installing two gears of size p and q left space for one of size 1-(p+q). It's tempting to consider a radius-1/3 gear next, since it's the simplest size I haven't yet installed. But to mount it on the knobs along with a size-1/2 gear, we need to include a size-1/12 gear to go in between:

This produces a 2/3 or 3/2 gear ratio.

Once we have the size-1/12 gear, we can mount it with the size-1/4 and size-1/3 that we already had:

Well, you get the idea. I probably would not have gone on so long, but I was enjoying drawing the diagrams. I used linogram, which rocks. it's almost exactly the diagram-drawing program I've always wanted; expect an article about this next week sometime.

Sun, 12 Nov 2006

Etch-a-Sketch
I've always felt that the Etch-a-Sketch is a superb example of a toy that doesn't do as much as it could.

An Etch-a-Sketch is a drawing toy invented in 1959 by Arthur Granjean and marketed by the Ohio Art company since shortly afterward. It looks superficially like a flat-screen television with two knobs.

The underside of the screen (that is, the inside surface) is coated with a fine aluminum powder. Also under the screen is a hidden stylus. One knob moves the stylus horizontally, the other vertically. As the stylus moves, it scrapes the aluminum powder off the screen, leaving behind a black line. If you hold the Etch-a-Sketch upside-down and shake it, the powder again coats the screen, erasing the lines.

It is very easy to draw horizontal and vertical lines, but very difficult to draw diagonal lines. (Wikipedia says "Creating a straight diagonal line or smoothly curved line with an Etch A Sketch is notoriously difficult and a true test of coordination.") So although extremely complex drawings can be made with an Etch-a-Sketch:

(Etch-a-Sketch drawing of Albert Einstein by Nicole Falzone)

Most people can only produce something like this:

As a child, I had an Etch-a-Sketch. I did get some entertainment out of it by figuring out how it worked: if you manipulate the knobs to remove the aluminum powder from a large part of the screen, you can see the stylus inside. But for the most part I found it frustrating and disappointing. As an adult, I still find it so.

But it needn't be so. The most frustrating thing about the Etch-a-Sketch, I think, is that its potential has not yet begun to be unlocked.

Consider a forty-five degree line. To draw such a line, one must turn both the horizontal and the vertical knobs at the same time, at exactly the same rate. Suppose, for concreteness, that we're drawing a line from the upper left to the lower right. If you turn the horizontal knob a little too quickly, the diagonal line will bend rightward; if you turn it a little too slowly the diagonal line will bend downward. So in contrast to the mathematically exact vertical and horizontal lines that are easy to draw, it's next to impossible to draw a diagonal line that doesn't wiggle. And when you screw up, you can't fix the mistake without erasing the whole thing and starting over.

But the solution is obvious: If you can link the two knobs somehow, so that they can only turn simultaneously, you can easily draw a diagonal line. As a child, I experimented with rubber bands, trying to get one knob to drive the other. This wasn't successful. Clearly, a better solution is to use gears.

There are plenty of examples of toys that have good-quality cast-plastic gears. (Spirograph is one such.) The knobs on the Etch-a-Sketch could be geared together. If the gears are the same size, the knobs will rotate at the same rate, and the result will be a perfect 45° line.

If you gear the two knobs together directly, they will rotate in opposite directions, so that you can only draw lines with slope -1 (northwest to southeast), not with slope 1 (northeast to southwest). To fix this problem, we need to introduce more gears. There can be an axle peg sticking up from the case of the Etch-a-Sketch, in between the two knobs. Mounting three equal-sized gears on the two knobs and the axle peg gears will force the knobs to rotate in the same direction, at the same rate.

[ The remainder of this article contains a number of very dumb arithmetic errors. For example, you cannot fit three gears of size 1/3 on the knobs and pegs; you need to use three gears of size 1/4 instead. I will correct this on Monday, and provide an illustration to make it clearer what I mean. —MJD ]

[ Addendum 20061116: I have posted the correction, with illustrations. ]

Let's say that the distance between the centers of the two knobs is 1. We can get a line of slope -1 by mounting two gears, each with radius 1/2, on the two knobs; we can get a line of slope +1 by mounting three gears, each with radius 1/3, on the two knobs and on the axle peg. If we want to do both, we had better make the axle peg removable, or else it will interfere with the size-1/2 gears. This is no problem. It can mount into a socket on the front of the Etch-a-Sketch, and be pulled out when not needed.

But why have only one socket? We're including five gears already (two of size 1/2 and three of size 1/3) so we may as well put them to some more use. Throw in a size 1/6 gear, and add another socket for the axle peg, this time 1/3 of the way between the two knobs. Now you can mount a size 1/2 gear on the left knob, a size 1/6 gear on the axle peg, and a size 1/3 gear on the right knob. If the left knob turns at rate r, the middle gear turns at rate -3r and the right knob turns at rate 3r/2. This produces a line with slope 3/2, which is about a 56-degree angle.

Or put in another socket for the axle peg, 1/6 of the way between the knobs, and then mount size 1/2, size 1/3, and size 1/6 gears, in that order. The knobs are now producing a line with slope 3, a 72-degree angle. If you want a line with slope 1/3 (18°) instead, just reverse the order of the gears. (That is, exchange the large and the small ones.)

At this point adding a few more gears expands the repertoire significantly. Add a radius-2/3 gear and another radius-1/6 gear and you can mount [2/3, 1/3] to get lines with slope -1/2, [1/3, 2/3] to get slope -2, [2/3, 1/6, 1/6] to get slope 4, [1/6, 1/6, /23] to get slope 1/4.

Clearly, you can carry this onwards, limited only by the space for the axle holes and the expense of adding in more gears. Spirograph used to deliver fifteen or twenty plastic gears for a reasonable price, so it's clearly not implausible that Ohio Art could have done something like this.

Sometimes I even dare to think that they might have provided cams or elliptical gears. Properly designed cams could gear together the knobs to produce mathematically exact curved lines, squiggles, maybe even circles.

But no, as far as I can tell, it's never been done. Why not?