A Look Back at Alex Gordon's Mad Dash Home That Never Was

Introduction

Apparently, I like doing sports forensic analysis. I must, since I'm clearly not doing it for the money. So here's a third installment, after my look at Derek Fisher's 0.4 shot and Mookie Betts's encounter with Houston's right field fans. This makes it two for three on forensic analyses done many many years after the fact, haha.

Let's set the stage: It's October 29, 2014. The San Francisco Giants and the Kansas City Royals are locked in a tightly contested winner-take-all Game 7 of the World Series. Both starting pitchers have long since been knocked out, and the Giants are clinging to a 3–2 lead in the bottom of the ninth. Giants ace pitcher Madison Bumgarner entered the game in the bottom of the fifth inning and hasn't left. He gave up a sharply rapped single to right to Omar Infante, then subsequently mowed down twelve straight batters.

Here, in the bottom of the ninth, he strikes out Eric Hosmer for the first out and gets Billy Butler to foul out weakly to first base for the second out. Bumgarner has now retired fourteen straight batters and needs just one more to record a five-inning save and earn the Giants their third title in five seasons.

But Alex Gordon, after fouling off the first pitch, cues a tailing liner into left center that dies just in front of the hard-charging center fielder Gregor Blanco, then bounces by him toward the wall in left center. He pulls up as it appears clear that left fielder Juan Pérez is going to beat him to the ball.

Pérez boots the ball as he rushes to pick it up, though, and a tense couple of seconds pass before he succeeds and finally gets the ball back to shortstop Brandon Crawford in shallow left. By the time he does so, Gordon is pulling into third base on a single plus two-base error, having gotten the stop sign from third base coach Mike Jirschele. Crawford checks to make sure that Gordon doesn't keep running, then throws routinely to first baseman Brandon Belt near the pitcher's mound.

Salvador Pérez (hereafter "Salvy," his nickname, to avoid confusion with Juan Pérez) now comes to bat, and there follow six specimens of what writer Wade Kapszukiewicz calls "Golden Pitches": pitches whose end result could potentially win the World Series for either team. Such pitches can only occur in the bottom of the ninth or any later inning of Game 7. If Salvy hits a home run, the Royals win the World Series. If he makes an out without first driving in Gordon, the Giants win the World Series.

Bumgarner goes to a tactic that has been successful all game for him: climbing the ladder on his high fastball and daring the Royals to hit it. Salvy swings repeatedly at pitches that are nearly neck high, and on the sixth pitch of the at-bat (and the 68th pitch of Bumgarner's appearance), he finally fouls out meekly to third baseman Pablo Sandoval, Gordon is stranded at third, and the Giants win.

In the aftermath of the game, and indeed all throughout the offseason, Jirschele and Royals manager Ned Yost were repeatedly asked whether they could have or should have sent Gordon home on that second-to-last play. Both were adamant that they made the right decision to hold Gordon, but the fact that Salvy never seemed to be able to contend on an equal footing against Bumgarner in this final at-bat sustained the fervent wish that Gordon had gone home.

But should he have? The question is a tantalizing one and touches on many notions of tactics and strategy. In this post, I'll analyze the game footage and other media and create a framework for deciding whether the Royals would have been better off sending Gordon home.

Technology and Stuff

The first order of business is to establish the basic facts of the play. How long after the crack of the bat did Gordon take to reach first base, second base, third base? When did Crawford field the throw from Pérez in left field? How far was he from home plate at that time? And how long would it take him to deliver the ball to catcher Buster Posey at home, if Gordon were to run home?

The play's timing can be determined by counting frames of the MLB video of the play and its various live-speed replays. Here's the play on YouTube; this video is encoded at 30 frames per second, so by counting frames from the initial contact with the bat and dividing by 30 frames per second, we can construct a timeline of the play:

• 0.00 s: crack of bat
• 2.97 s: ball falls in front of Blanco
• 4.00 s: ball bounces a second time, then rolls to fence
• 4.73 s: Gordon touches first base
• 6.63 s: ball reaches the fence
• 7.80 s: Pérez boots ball along fence
• 8.43 s: Gordon touches second base
• 10.00 s: Gordon turns to look at the play in the outfield (approx)
  
• 10.17 s: Pérez throws ball to cutoff
• 10.67 s: Jirschele begins raising his hands (approx)
• 11.17 s: Jirschele's hands are now up to hold Gordon (approx)
• 11.77 s: Crawford fields ball (about 212 feet from home plate)
• 12.17 s: Gordon stops at third base
• 13.53 s: Crawford throws to Belt
• 14.93 s: Belt fields ball and time is called
  

For some of these events, I also used, as secondary event and time sources, this MLB Statcast video, this fan video from just above the Giants' dugout, and this other fan video from the left field stands, all encoded at 30 frames per second. None of these times are accurate to any better than 1/30 of a second, therefore, though they've been rounded to the nearest hundredth to simplify the arithmetic. I estimate these times have an error of about ±0.05 seconds.

Incidentally, ESPN also analyzed the MLB video, and somehow got 8.30 seconds for Gordon to touch second. I've measured it a few times, and I don't see how they get that. The rest of our times are within my ±0.05 second error bar, including notably the time it took for Gordon to reach third base, which makes that 0.13 second discrepancy even odder. I'll use my figure of 8.43 seconds, to keep the methodology consistent.

Next, how far was Crawford as he fielded the throw and prepared to throw home if necessary? For this analysis, as a reference frame for determining event locations, I used the special groundskeeping design for the World Series games at Kauffman Field in Kansas City, which is depicted here (click to enlarge):

Gorgeous setting, by the way. This photo is from before Game 1, but I don't think the pattern changed for Game 7. As you can see, left field (and right field, but we're focusing on left field) is criss-crossed with a lattice of intersecting light and dark bands, which will serve as a grid for us to identify the locations of players and events during the play. We'll need to fix this grid on a diagram of Kauffman Field, which we create from Google Maps's satellite view (click to enlarge):

North is up. This groundskeeping pattern is not the one from the 2014 World Series, so we can't simply use the satellite image as is. Rather, by comparing the two images, we create an overlay for the World Series groundskeeping pattern. I've rotated the image counter-clockwise by 2 degrees to line the field up horizontally and vertically, then added the relevant portion of the pattern as green outlines (click to enlarge):

Now that we have the grid laid out, we dispense with the satellite view, go through the video, and place the various events on our overlay (click to enlarge):

These events include positions of Blanco (B1 and B2), Pérez (P1, P2, and P3), and Crawford (C1 and C2) throughout the play, as well as the path of the hit ball through the outfield (H1, H2, H3, and H4). We now remove the grid as well, and connect the main events (click to enlarge):

The throw that Crawford would have had to make is the dotted orange arrow to home. By measuring against the 100-foot scale, we see that the throw is about 212 feet. I estimate this method to have an error of maybe ±10 feet, so it's somewhere between 202 feet and 222 feet, but the rest of my analysis will assume a distance of 212 feet.

That's well outside most casual estimates. ESPN's article gauged it at 180 feet, which is 15 percent low. My first off-the-cuff estimate was 140 feet, which is on the skinned infield and ridiculously low, though it was echoed by multiple commentators; then, I guessed 180 feet, in line with ESPN's estimate. Crawford himself thought he was 30 to 40 feet out onto the outfield grass, which would put him 180 to 190 feet from home plate. The longer throw makes the potential play closer than otherwise—but is it close enough to send Gordon?

So Much Crawford

The critical factor is how long Crawford reasonably needs to turn that throw around to home plate. Fortunately, we have plays to compare this one to. The closest play I could find is from September 9, 2016, with the Arizona Diamondbacks hosting the Giants. In the bottom of the seventh inning, with the Giants leading 5–4, Chris Owings hits a fly ball to deep center that bounces off the glove of Denard Span. Socrates Brito scores easily from second base to tie the game, and Owings tries to come all the way around to score also, but he's nipped at the plate by a strong throw from Crawford. (The game went into extra innings tied at 5, and the Giants eventually won 7–6 in 12 innings, so the play turned out to be critical.) Again, we can create a timeline:

• 0.00 s: crack of bat
• 11.47 s: Crawford fields ball (about 235 feet from home plate)
• 12.20 s: Crawford throws ball
• 14.33 s: Posey fields ball
• 14.80 s: Posey applies tag
• 14.93 s: Owings reaches home plate (already out)

This play is almost directly behind second base and there is a convenient sequence of 25 light and dark diamonds, again created by groundskeeping. Crawford is in the middle of the ninth diamond, counting from the edge of the skinned infield, at 155 feet—the skinned infield is a partial circle with a 95-foot radius centered on the pitching rubber—to the edge of the 16-foot warning track in deep center, at 391 feet. That gives us our final figure of 235 feet (again, ±10 feet).

On this play, Crawford took 0.73 seconds to throw the relay, which traveled 235 feet in 2.13 seconds, for an average speed of about 110 feet per second, almost exactly 75 mph. Posey then needed an additional 0.47 seconds to tag Owings. When Posey fielded the ball, Owings was about 13 feet from home plate (that's the radius of the circle surrounding home plate), and he applied the tag when Owings was about 3 feet from home plate.

It's worth noting that a thrown baseball loses a lot of velocity in the air, about 15 percent per second at typical speeds. Crawford probably threw the ball at around 90 mph, and it slowed down to around 60 mph by the time it reached Posey.

There is also a play from September 7, 2013, in the top of the eighth inning of a game in San Francisco between the Giants and Diamondbacks, where Crawford relays a throw from center fielder Ángel Pagán. It's difficult to determine Crawford's distance from home plate; I estimate that he's 195 feet away. The throw covers 1.77 seconds in the air, which is consistent with an average speed of 75 mph, but because of the uncertainty in the distance, it's not my primary comparison.

Additionally, in the second inning of this Game 7, Crawford threw a relay to home plate from right fielder Hunter Pence. Again, it's hard to tell just where Crawford is, but I estimate he's 50 feet past the skinned infield, or 205 feet from home plate, and the throw took 1.90 seconds to get there, an average speed of about 74 mph. The throw was not in time to catch Billy Butler scoring the Royals' first run, but Butler was already halfway from third base to home plate when Crawford made his throw.

Later, in the fourth inning, Crawford made a snap throw to first to complete a sparkling double play started by second baseman Joe Panik, who made a catch diving to his right and glove flipped the ball directly to Crawford. That throw was made under different circumstances, but Crawford's performance was similar: As this MLB Statcast video indicates, he needed 0.77 seconds to throw the relay, which he did at 72 mph, though he was forced to throw it flat-footed.

Reconstructing the Sequence

With all this in mind, let's run through the play again, annotated this time with the sequence and commentary (click to enlarge):

1. At 0.00 seconds, Gordon hits the ball to left center (blue dashed line). The ball is hit near the end of the bat, which causes it to tail away toward left field; see the path above. Blanco initially thinks he can catch the ball on the fly, and he charges forward. Pérez also thinks Blanco will catch the ball and starts jogging toward the infield; Posey similarly jogs to the mound, anticipating a celebration involving the notorious Posey Hug.
   
2. By the time the ball hits the ground at 2.97 seconds, Blanco has realized that he can't catch up to the ball, but it's too late for him to pull up to play it on the bounce. It squirts right by him toward the fence. This counters the notion that by running hard out of the gate, Gordon would risk being caught between first and second; he would never get to first base within 2.97 seconds. Pérez has to turn around and sprint toward the ball, and Blanco pulls up as he realizes he can't get there any quicker than Pérez.
   
3. At 4.00 seconds, the ball bounces a second time. Had Blanco played the ball safely, he would have caught it at about 3.90 seconds. He would have been about 210 feet from second base and would have gotten the ball back well in time to keep Gordon from advancing past first base. In reality, the ball continues bouncing toward the wall, Pérez in hot pursuit.
4. Meanwhile, Gordon has run toward first base, but not at top speed. At 4.73 seconds, he reaches first base, having seen the ball bounce once and then twice. At this point, Gordon knows he'll get to at least second and has a good chance at third. Posey returns to the plate for a potential play, and Bumgarner retreats toward the backstop to back him up. Crawford began the play at the edge of the skinned infield, but now runs out to short left field to act as the primary relay. Panik sets up about 40 feet behind him as the secondary cutoff.
   
5. At 6.63 seconds, the ball reaches the fence. Pérez gets there shortly thereafter, but at 7.80 seconds, he boots the ball about 10 feet leftward along the fence. However, even if he fields it cleanly, he is over 300 feet from third base. It would take a phenomenal throw to nail Gordon there, even with Crawford relaying. Blanco's misplay is almost solely responsible for Gordon advancing, and indeed the official scorer assigned an error only to Blanco, not Pérez.
   
6. At 8.43 seconds, Gordon reaches second base. He stumbles slightly as he rounds the bag, but regains his balance. He turns his head to the outfield to try to gauge the play, but it turns out he can't see it clearly because of glare from the outfield display.
   
7. At 10.17 seconds, Pérez has finally secured the ball and throws it (orange dashed line) to the cutoff man Crawford, who stands 212 feet from home plate. Panik is behind him, keeping an eye on the play in left field as well as Gordon's progress on the basepath. Meanwhile, Jirschele starts raising his hands, and at 11.17 seconds (give or take), the stop sign is up.
   
8. At 11.77 seconds, Crawford fields the throw, having had to "pick" it on the short hop. Normally, the cutoff man is supposed to avoid trying to catch a throw on the short hop; he should let it go to the secondary cutoff man to avoid the ball bouncing away and letting the run score uncontested. Crawford later said, "Nothing against Panik, who was the second cutoff man on the play, but I was going to catch the ball unless I couldn't catch it [that is, literally couldn't reach it]." Panik puts his hands up to forestall an immediate throw home.
   
9. At 12.17 seconds, Gordon pulls in at third base. Crawford has turned around, poised to throw home, but after seeing Panik's signal and checking that Gordon isn't going, he tosses a more routine 140-foot throw at 13.53 seconds to Belt, who catches it at 14.93 seconds. The umpires call time.
   

So much for what actually happened. It's time to speculate! Suppose that Crawford again takes 0.73 seconds to turn around his relay throw, which we'll suppose averages 75 mph. (At 212 feet, this throw is somewhat shorter than our comparison, but it's close enough that the difference is probably minor. If anything, however, this approximation overestimates the time elapsed by the throw.) He would then make that throw to the plate at 12.50 seconds, and it would be fielded by Posey 1.92 seconds later, at 14.42 seconds.

Would that be in time to catch Gordon? He actually got into third base at 12.17 seconds. Let's suppose he could have gotten home in another 3.50 seconds, landing him there at 15.67 seconds. He would be more than 30 feet from home plate as the ball reaches Posey's glove. That gives Posey more than a second to apply the tag; in the play on Owings, Posey needed less than half a second to apply the tag.

The What-If Scenario

But suppose that Jirschele hadn't put on the stop sign, and encouraged Gordon to run all the way home. It remains to be seen whether that would be a good idea, but suppose he did that. Let's also assume that Gordon ran flat out all the way and didn't stumble going around second. Gordon would then have reached third earlier, but how much earlier?

On a triple the previous season, on April 5, 2013, in which Gordon seems to have run hard the whole way, he slid into third base 11.90 seconds after the crack of the bat. (AZ Central ran an article on this play, and somehow measured the run at 11.03 seconds. Again, I'll stick with 11.90 seconds to keep the methodology consistent.) If he ran the same way in Game 7, his time to home plate would be longer, by about the time on one of his intermediate legs (first to second, or second to third). As far as I can tell, Gordon is rarely better than 3.60 seconds on any of these—his intermediate first-to-second leg in Game 7 was 3.70 seconds—but again, let's say it adds 3.50 seconds. That gets him to home plate at 15.40 seconds, and still gives Posey nearly a full second to apply the tag. Gordon would be about 25 feet from home plate when Posey fielded Crawford's throw.

All in all, it seems as though Posey would tag Gordon comfortably out in almost any circumstance—barring an error. So how often does Crawford uncork a wild throw? In 2014, Crawford committed 21 errors, on 634 opportunities, which included 185 putouts and 428 assists (throws that lead to a putout). It's unlikely that all 21 errors were throwing errors, and also unlikely that he only threw 428 times, but let's assume both of those are true to put an upper bound on his error rate. In that case, he would have 21 errors on 449 throws, for an error rate of 0.047, a bit under 5 percent. You'd never send a runner if you thought his chances of making it were under 5 percent.

Of course, most of those throws were from shortstop to first base, a throw that averages about 120 feet. The throw in this case was 80 percent further, certainly well within Crawford's range, but probably it increases his error rate. Let's say it doubles it, to 10 percent. Is that high enough to send Gordon?

Probably not. Statistically, for the 2010–2015 era, with a runner on third base and two outs, that runner scores about 26 percent of the time. That itself should be enough to settle the matter, but there's more. In that situation, the team scores an additional run (or more) about 7 percent of the time; otherwise, in this case, the game goes to extra innings and it's a coin flip as to who wins. Maybe the home team has an edge, but it's small. Rob Mains's study in Baseball Prospectus suggested it was about 52–48 to the home team (less than it is in regulation, interestingly).

That means that with Gordon stopping at third, the Royals have about a 17 percent chance of eventually winning the game (a win in nine innings with 7 percent probability, and a win in extras with 0.52 times 0.19, or 10 percent). If he goes home and makes it, the Royals have about a 55 percent chance of eventually winning the game (a win in nine innings with 7 percent probability, and a win in extras with 0.52 times 0.93, or 48 percent). If he goes home and is tagged out, of course, the Royals simply lose.

So in order for it to be worth it to send Gordon, he has to have a success rate of at least 17/55 or 31 percent. Incidentally, Nate Silver did only this part of the analysis, arriving at a figure of 30 percent using slightly older scoring statistics. (He then simply assumed that Gordon would score more often than that and therefore advocated sending him. Very lazy, Nate!) David Freed, writing for the Harvard Sports Analytics Collective, determined a threshold of 29.6 percent based on more specific statistics (though they use the dramatic underestimate that Crawford stands 140 feet from home plate for the rest of their analysis). So there's general agreement on that roughly 30 percent figure.

I don't see Gordon scoring with anything like that probability. Maybe the long throw increases Crawford's error rate a bit more than double, maybe that 30 percent can be edged a little downward because Salvy was hit by a pitch earlier in the game, but I just don't think those two lines cross. Crawford was no playoff newbie in 2014, and sending Gordon just to force him to make a play isn't the percentage call. I'm sympathetic to those who wanted Gordon to be sent home for the excitement value, but it should be recognized for what it is: a gut reaction call that goes against both traditional baseball judgment and post-mortem analysis.

Thursday Morning Third-Base Coaching

Afterwards, there were a lot of fans who insisted that not only should Gordon have been sent home, but that people who agreed with holding Gordon were flat out wrong. Frankly, I think that's a little crazy. It comes from thinking that because Salvy did in fact pop out, he was destined to pop out. Even having been hit by a pitch back in the second inning, Salvy had a chance of walking it off against Bumgarner. He had hit a homer back in Game 1, accounting for the Royals' only run against Bumgarner. And there's the history of Kirk Gibson hitting a home run off the great Dennis Eckersley in Game 1 of the 1988 World Series. Does anyone watching that video think that either of Gibson's legs was in better shape than Salvy's? Salvy would go on to earn the 2015 World Series MVP when the Royals came right back to win the title.

Other fans thought that Bumgarner's admittedly dominant performance argued for a more aggressive stance on sending Gordon home. But again, this smacks of after-the-fact destiny. Bumgarner had already thrown 62 pitches (before facing Salvy), after throwing 117 pitches in Game 5 just three nights earlier. It was by no means a foregone conclusion that he was unhittable. Certainly Yost thought they would get to Bumgarner.

Tim Kurkjian wrote the ESPN article that analyzed the video for timings. That same article also collected quotes and observations from many of the principals involved. To a man, they all agreed that the right call was made. Most of those interviewed thought it wasn't close. (Yost thought Gordon would have been out by 40 feet, which I think is a bit of an overestimate.) The only player who was even halfway wondering what would have happened was Gordon himself, and by his own admission, he couldn't clearly see what was going on in the play at the time, because the bright display in center field cast a glare that obscured Blanco's and Perez's hijinks.

Some other observations out of that article: Jirschele claimed that he was waiting for Crawford to field the throw from Pérez cleanly before holding Gordon up. But Jirschele began holding his hands up over a second before the ball had gotten to Crawford. I suspect he felt Crawford's chances of fielding the throw cleanly were too high not to put the stop sign up before it was too late; if so, his intuition was vindicated.

Gordon recalls running hard out of the box. It didn't seem that way to most observers, including Jirschele, and in fact some fans thought he was just jogging to first until the ball dropped. The 4.73 second time to first base suggests that he was moving faster than that, but not running all out. The explanation is pretty straightforward—Gordon clearly thought that he could expect no more than a single and ran accordingly—but the charge that under the circumstances he should have been running harder than he did is a reasonable one. As we've seen, though, even his fastest run would have had a hard time beating a halfway accurate throw home.

During the following offseason, a local college baseball team reenacted the play and nailed the runner five times out of six, failing only the first attempt on an overthrow. Some fans pointed to that one failure as an additional point in favor of sending Gordon, but that first reenactment got the timing wrong; the shortstop didn't throw until the runner was nearly a full second past third base. (This video is encoded at 24 frames per second.) Seeing the runner that far ahead may have caused the shortstop to rush the throw; also, with that extra time, the catcher could plausibly have retreated to catch the ball properly and race back to tag the runner. This experiment differed too much from the original game conditions to be of much probative value, though.

Finally, five years after the game, Jirschele revisited the call, affirming that he made the right call, and capping it all with an amusing anecdote. But the plain fact of the matter is that if he had sent Gordon home, there would very likely be no debate, and instead Jirschele would be held up as the Royals' third base coach who made the call that ended his team's season.

Bias, Unseen But Not Unfelt

 [This post is adapted lightly from a Facebook post I just made.]

<tl;dr> The Atlanta crime doesn't need to have been racially or sexually motivated, per se, for race or sex to have been a factor. </tl;dr>

 

So the other day, about a week and a half ago, I was accosted at the drug store where I was picking up some medications for the family. Harassed, really. Some fellow had come in line behind me, rather closer than six feet away. So (as is my wont) I moved forward a bit so I was halfway between him and the person in front of me. Something about that set him off and there followed a fairly dreary 30 to 45 seconds of him pointing in my face and accusing me of racism. He was Black, you see.

 

I'm not in need of any kind of support over this; it didn't last long, and I'm afraid I have too high an opinion of myself to be too upset by it. Mostly, I was anxious that he was breathing at me pretty heavily from close up (12 to 18 inches?). Only now, as the usual incubation period has more or less passed without any symptoms, have I gradually relaxed—about that, anyway. (And yeah, I'm aware the chances of my contracting COVID from him were pretty low—I figure about one in a thousand, something on that order. We were both masked. But I'm the sort to obsess about it a little.)

 

I've been thinking about that episode the last few days, though, in light of recent events in Atlanta. And it's not just about the fact that it happened, or that a police captain characterized the shooter as "having a bad day." These things are bad enough, but I think it's clear that they're bad. Few people are having trouble understanding that.

 

What got to me was a side line after the captain was taken out of his spokesperson role. Officials were quoted as saying that although the shooter denied a racial motive, they weren't ruling one out. And though that's not a bad thing as far as it goes, I'm concerned that it focuses on what is really a small percentage of a very large problem.

 

See, as I say, most people understand that racism is a bad thing, or at the very least, they understand that it's generally viewed as a bad thing. So as far as overt expressions of racism go, they know not to do it, or if they do do it, they keep it among like-minded people. But that's just the tip of the racist (or sexist, or any otherist in general) iceberg. Underneath all of that is a much larger mass of subliminal prejudicial behavior that mostly goes unnoticed.

 

Maybe that fellow would have harassed me anyway, but I think he was just that little bit more likely to do it because I was Asian. Or maybe that shooter would have been up for shooting someone who wasn't Asian, or wasn't a woman, but I think he was that much more likely to do it because they were. And a million other things that happen every day, of lesser consequence, but are just a bit more likely to have happened to the people they did in fact happen to.

 

And what makes them so insidious is the spectre of plausible deniability, that in any individual situation, one can defend oneself sincerely and successfully against charges of bias. Only in the large, statistically, can these biases be seen.

 

Most of these are not racist or sexist motives per se. Most of the time, the person is not actively (consciously or otherwise) seeking out someone who fits a particular profile. But by the same token, when the situation hits them, the voice inside them that says, hey, maybe let's not escalate this—that voice is just a little bit softer when it's someone they don't sympathize with for those reasons.

 

That voice is inside us all (mostly). But I don't believe that this voice speaks equally in response to all people of all creeds, colors, and sexes. I certainly don't believe mine does. Oh, I don't think I'm exceptionally biased or anything; I'm quite ordinary.

 

But part of the reason I wasn't more upset about being called racist, I think, is that I deeply believe bias exists in us all, and it's not possible to eliminate it. We can reduce it, but there's a part of being human that makes kneejerk classification a bit too automatic. The only real way to address that irreducible core of bias, I feel, is to explicitly bend over backward to counteract it; it's just too easy, otherwise—too human—to believe, honestly, that one is free of bias. And maybe I gave this man a pass for that reason.

 

Or maybe I just don't like to think of myself as being upset by it. Who knows?

High-Dimensional Weirdness

At work, I run a mathematics colloquium that meets every other Thursday.  I don't always present—I probably present about 20 to 25 percent of the time—but I did a recent one on the behavior of high-dimensional spaces.  I then came upon an oddity that I thought was worth sharing, for those three or four of you who might like that kind of thing.

In this presentation, I made reference to some dimensional weirdnesses.  While making the point that additional dimensions make room for more stuff (as I put it), I pointed out that if you put four unit circles in the corners of a square of side 4, you have room for a central circle of radius r = 0.414.  (Approximately.  It's actually one less than the square root of 2.)

 

Correspondingly, if you put eight unit spheres in the corners of a cube of side 4, you have enough space for a central sphere of radius r = 0.732 (one less than the square root of 3), because the third dimension makes extra room for the central sphere.

If you were to put a sphere exactly in the middle of the front four spheres, or in the middle of the back four spheres, it would have a radius of r = 0.414, just as in two dimensions, but by pushing it in between those two layers of spheres, we make room for a larger sphere.

Finally (and rather more awkwardly, visually speaking), applying the same principle in four dimensions makes room for a central hypersphere of radius r = 1 (one less than the square root of 4).

The situation for general dimension d (which you've probably guessed by now) can be worked out as follows.  Consider any pair of diametrically opposed unit hyperspheres within the hypercube (drawn in orange below).  Those two hyperspheres are both tangent to the central green hypersphere, and they are also tangent to the sides of the blue hypercube.

We can figure out the distances from the center of any unit hypersphere to its corner of the hypercube, as well as to the central hypersphere.  Since we also know the distance between opposite corners of the hypercube, we can obtain the radius of the central hypersphere:

One interesting consequence is that at dimension d = 4, the central green hypersphere is now as large as any of the orange unit hyperspheres, and above dimension d = 9, the central hypersphere is actually large enough to poke out of the faces of the hypercube.  Keep that in mind for what follows.



One other oddity had to do with the absolute hypervolume, or measure, of unit hyperspheres in dimension d.  A one-dimensional "hypersphere" of radius 1 is just a line segment with length 2.  In two dimensions, a circle of radius 1 has area π = 3.14159; in three dimensions, the unit sphere has volume 4π/3 = 4.18879....  The measure of a unit hypersphere in dimension d is given by

For odd dimensions, this requires us to take a fractional factorial, which we can do by making use of the gamma function, and knowing that

With that in mind (and also knowing that n! = n (n – 1)! for all n), we can complete the following table for hyperspace measures:

That last entry may come as a bit of a surprise, but it is simply a consequence of the fact that as a number n grows without bound, πⁿ grows at a constant pace (logarithmically speaking), while n! grows at an ever increasing rate.  As a result, the denominator of V_d totally outstrips its numerator, and its value goes to zero.



But what if we combine the two, and ask how the measure of the central green hypersphere, expressed as a proportion of the measure of the blue hypercube, evolves as the number of dimensions goes up?  On the one hand, we've seen that the measure of a unit hypersphere goes to 0 as the number of dimensions increases, but on the other hand, the central green hypersphere isn't a unit hypersphere; rather, its radius goes up roughly as the square root of the number of dimensions.  How do these two trends interact with increasing dimensionality?  In case it helps your intuition, here's a table for the ratios for small values of d.

Those of you who want to work it out for yourself may wish to stop reading here for the moment.  Steven Landsburg, who is a professor of economics at the University of Rochester but earned his Ph.D. in mathematics at the University of Chicago, told a story of attending a K-theory conference in the early 1980s, in which attendees were asked this very question.  Actually, they were specifically asked not to calculate the limiting ratio, but rather to guess what it might be, from the following choices:

• –1
• 0
• 1/2
• 1
• 10
• infinity

Attendees were invited to choose three of the six answers, and place a bet on whether the correct answer was among those three.  Apparently, most of the K-theorists reasoned as follows: Obviously, the measure can't be negative, so –1 can safely be eliminated.  Then, too, the central green hypersphere "obviously" fits within the blue hypercube, so its volume can't be greater than that of the hypercube, so the ratio of the two can't be greater than 1, so 10 and infinity can likewise safely be eliminated.

Well, "obviously," you know that the hypersphere can in fact go outside the hypercube, so 10 and infty can't actually be eliminated.  So what is the right answer?

At the risk of giving the game away so soon after offering it, I'll mention that the answer hinges on, of all things, whether the product of π and e is greater or less than 8.  Here's how that comes about: We know that the measure of a unit hypersphere in dimension d is given by

But that's just the unit hypersphere.  If we take into account the fact that the radius of the central green hypersphere is

then the question becomes one of the evolution of the measure G_d of the central green hypersphere:

To figure out how this behaves as d goes to infinity, we first rewrite it as

Next, we make use of Stirling's approximation to the factorial function:

Applying this to n = d/2 gives us

and when expressing it as a proportion of the measure of the hypercube of side 4, we get

Finally, we observe that we can write (by taking into account one extra higher-order term in the usual limit for 1/e)

and we see that

The right-hand side is eventually dominated by the factor involving πe/8 = 1.06746..., which drives the ratio G_d/4^d to infinity as d increases without bound—but it takes a long time.  A more precise calculation shows that the fraction first exceeds 1 at dimension d = 1206.  A plot of the ratio as a function of dimension looks like this:

Notice that the ratio reaches a minimum of very nearly 0.00001 at 264 dimensions; the exact value is something like 0.00001000428.  As far as I know, that's just a coincidence.

Misunderstood Rules in Sports, Part One of a Trillion

Because I apparently don't have enough random crap on my plate, I occasionally participate on Quora.  I'm there as Brian Tung; I'm not hard to find, other than you actually have to want to find me, and so far, that's not a very common thing.

Anyway, I often find myself embroiled in various debates (generally well-mannered, if not always good-natured) about various sports rules.  Most recently, the question was about passes or shots that go over the backboard.  For example, should this shot from 2009 by Kobe Bryant count?

Or how about this one from Jamal Murray, in 2019?

The common feeling is that these should not count, because the ball goes over the backboard, and everyone knows that a ball that goes over the backboard is out of bounds, right?

Right?

Well, it's complicated.  Complicated enough that I'm just going to drop this here for the next time this comes up.  Here's Rule 8, Sections II.a and II.b from the official NBA site:

a. The ball is out-of-bounds when it touches a player who is out-of-bounds or any other person, the floor, or any object on, above or outside of a boundary or the supports or back of the backboard.

This part of the rule is about what the ball touches, not where it goes.  There's a bit of excitement in that it uses the word "above," but in context, I think it's pretty clear that it refers to the ball touching something or someone above the boundary (the out-of-bounds line).

b. Any ball that rebounds or passes directly behind the backboard, in any direction, or enters the cylinder from below is considered out-of-bounds.

This is the relevant part.  Note that it uses the wording "directly behind the backboard."  To me, that means you take the backboard, and project it back away from the court; anytime the ball passes through that imaginary three-dimensional box, it's out of bounds.  It says nothing about the ball passing over the backboard.  If it meant that, I think it would have said that.

In both cases, the ball clearly goes over the backboard, but it never goes directly behind the backboard.  In the case of Kobe's shot, the best angle in this video (pretty poor resolution, but it was the best I could find) is found at about 0:48.  As for Murray's shot, well, read on.

I think the phrase "directly behind" is crucial.  It isn't enough that the ball go behind the plane of the backboard (which is four feet inside the baseline, so that would happen all the time).  It has to go somewhere where, if you were to look from the opposite baseline, you would see the ball through the backboard, not around it.

If you go online, you will see a majority of the web sites that discuss this question insist, quite authoritatively, that such shots are not to be counted.  As irritating as I sometimes find this, it's sort of understandable, because the wording of the rule is a bit terse, and also because the rules vary from governing body to governing body, as well as era to era.  For instance, these shots would be illegal in the NCAA:

Rule 7-1-3.  The ball shall be out of bounds when any part of the ball passes over the backboard from any direction.

This rule is stated again, almost verbatim, as Rule 9-2-2.

On the other hand, they're legal in FIBA:

Rule 23.1.2.  The ball is out-of-bounds when it touches:

• A player or any other person who is out-of-bounds.
• The floor or any object above, on or outside the boundary line.
• The backboard supports, the back of the backboards or any object above the playing court.

So there's some excuse for getting this wrong (plus they eschew the Oxford comma, but that's another blog post for another time).  If that's not enough, the rule in the NBA has changed—see the postscript below.

Fortunately, we have an approved ruling, from none other than Joe Borgia, NBA Senior Vice President of Replay and Referee Operations (I'll bet you already knew that):

Joe Borgia, NBA Senior Vice President of Replay & Referee Operations, joined @NBATV to discuss three plays from Sunday's NBA Playoff action:
- Butler charge in Q1 of #TORatPHI
- Gasol offensive foul in Q4 of #TORatPHI
- Murray shot over backboard in Q1 of #DENatPOR pic.twitter.com/5Lto9JNxOr

— NBA Official (@NBAOfficial) May 6, 2019

 
Jamal Murray's shot is discussed as the third case, at about 1:38 of the video.

"...When you look at this angle, our rule is the ball cannot pass directly behind the backboard.  So when you saw that replay, you saw the ball went up, and it went over, but it never went directly behind it.  Otherwise, we would have seen it through the glass; that would have been illegal.  But up and over is fine, so that is a good basket."

I think that should settle the matter fairly nicely.

---

Here's more from Borgia:

"The old rule stated it was illegal when the ball went over the backboard (either direction). So imagine the backboard extending up to the roof—if the ball bounced off the rim and hit any part of the imaginary backboard a violation was assessed. We had too many game stoppages when the ball bounced over the edge so we changed the rule to say the ball cannot go directly behind the backboard. That is why I said the backboard is now an imaginary ‘tunnel’ that goes back, not up to the roof like in the old rule."

Postmodernism and a Classic of Chinese Literature

Bottom line up front: This is probably going to end up long, longer than it is now.  That might be true no matter when you're reading this. (Update 2022-01-26: I have indeed added more to it, mostly in the last section.)

A couple of years ago, I detailed on this blog a series of Chinese novel reading projects: 西遊記 Journey to the West by 吳承恩 Wú Chéng'ēn, 生死疲勞 Life and Death Are Wearing Me Out by 莫言 Mò Yán, 邊城 Border Town by 沈從文 Shěn Cóngwén, and 圍城 Fortress Besieged by 錢鐘書 Qián Zhōngshū.

After that, I took a bit of a break.  I had intended to continue on to 紅樓夢 A Dream of Red Mansions by 曹雪芹 Cáo Xuěqín, and had even read a couple of pages, but my father warned me against that one, suggesting instead 三體 The Three-Body Problem by 劉慈欣 Liú Cíxīn.  Well, I read a couple of pages of that too, but put it aside, probably because I read the Wikipedia plot summary and I decided I didn't like the conspiracy-theory angle.

Then sometime in the spring of 2018, I restarted Red Mansions once again, this time in (relative) earnest.  I had bought David Hawkes's English translation around the time of my first abortive attempt, and I now followed along in both languages, more or less as I had with my previous projects. It took a year and change, but I did finally finish it. And far from a chore, I enjoyed most every step of the way. (Though I did occasionally lose patience with some of the characters...)

Red Mansions (more commonly translated as The Dream of the Red Chamber, but Hawkes suggests this is misleading, and I tend to agree) is unusual—perhaps even unique—in Chinese literature for persistently and insistently asserting its own fictionality.  Other Chinese novels exhibit an array of the magical and the mystical, more so than Red Mansions, but even with that wink and nod to the reader, the novels themselves typically present the events as though they really happened, usually tying the events to a specific epoch in Chinese history (for example, such-and-such a year in so-and-so's reign).  Historicity is a big deal in Chinese fiction, ironically enough.

Not so Red Mansions.  After Cao motivates his novel with the desire to commemorate the young girls he knew as a well-to-do boy, the rest of the novel is said to be a story engraved on a consciousness-endowed, polymorphic jade stone, whose own story frames the central story, and who is brought down to earth to experience life by a Daoist priest and a Buddhist monk.  Echoes of all three (or perhaps it is they themselves) reverberate throughout the book, pushing the plot—engraved on the stone, remember!—this way and that.  Such adumbrations seem familiar to those of us looking back at the evolution of 20th-century Western literature; see James Joyce's Finnegans Wake for a notable, if rather denser, English analogue.  But for a novel written in 18th-century China (manuscripts were circulating at the time of Cao's death in 1763 or 1764, and the first printed edition arrived in 1791), it was positively revolutionary.

Perhaps because of that, perhaps because of the iconic love triangle in the central story, or perhaps it is supposed to be revered in the annals of Chinese literature, Red Mansions occupies a central position in the Chinese collective literary consciousness.  (My mother started reading it when she was younger, and never finished it.  She found it fairly ordinary, but in addition, she has a tendency to mistrust any hyperbolic criticism, positive or negative, and the mountains of praise heaped on the story, amounting almost to hysteria, turned her off to reading it.)  When I went to Taiwan earlier this year, I stopped in a bookstore, and there were no fewer than a dozen different editions of Red Mansions, along with at least as many critical studies and examinations. 

And Red Mansions is enormous.  I read a version I had found online, cobbling it together and having to fix occasional typos, and in one case, replacing three pages that had strangely gone missing.  At a normal font size, it occupied nearly 1400 pages; this is typical of printed editions too.  The English translation by Hawkes and John Minford (Hawkes's student) runs about 2500 pages, in five volumes.  (This kind of expansion is typical of translations from Chinese to English, and there's plenty of speculation as to why that is.)  This is something you have to commit to.

Speaking of the translation, Hawkes and Minford are meticulous, translating every detail of Cao's versatile prose and poetry.  As is typical, the author makes assumptions of his readership, assumptions that are still reasonable-ish for well-read modern Chinese, but which native English readers have no hope of meeting.  Hawkes and Minford usually meet the reader halfway, finding the corresponding English connotations whenever possible, and also choose the expedient of weaving historical context into the main text, resorting to footnotes and appendices only when absolutely necessary to avoid an abrupt dump of background.  Some appendices also explain some editorial choices in the translation.

Some of the word choices are oddly obscure, opting for 75-cent words (accounting for inflation) when a nickel will do without interrupting the tone.  And when I say 75-cent words, I mean words that I had never heard of in my entire life until now.  I'll try to collect a selected list of them so you know what I mean.  But by and large, the text fits what I read in the original Chinese.  There is another complete English translation, by the husband-and-wife team of 楊憲益 Yáng Xiànyì and 戴乃迭 Gladys Tayler Yang, that is also supposed to be good, and a bit more literally faithful, at the cost of being occasionally more opaque to Western readers.

The Story

At the center of the story that occupies the vast majority of Red Mansions' 120 chapters is the 賈 Jiǎ family.  Attached to the emperor by virtue of the service of past family members, long since dead, they are wealthy and extravagant.  People dress up to have tea, to move from one house to another in the compound, to go to bed.  They live a life of leisure, eating rare delicacies and drinking fine wine.  Even when they fall ill, their medicines (Chinese traditional, naturally) are the most exquisite available.  Their ginseng has to be picked at just the right time, with just the right shape to it.

The young scion of the family is 賈寶玉 Jiǎ Bǎoyù, a precocious and willful boy of about 13 at the start of the novel, who is pressured by his father to study the Confucian classics, but who mostly only has eyes for the girls of the family.  His name means "treasured jade," because he was born with a jade stone in his mouth—the magical stone from the frame story.  (An alternate title for the novel in both Chinese and English is 石頭記 Shítoujì The Story of the Stone.)  The two principal girls in the story are 薛寶釵 Xuē Bǎochāi, the only daughter of Baoyu's mother's sister, and 林黛玉 Lín Dàiyù, the only daughter of his father's sister.

Daiyu and Baochai are complementary yin and yang.  Daiyu is artistic, mercurial, and consumptive; Baochai is sensitive, compassionate, and robust.  A combination of dream sequences and wordplay implies that Baoyu's ideal woman would be a combination of the two: Both Daiyu and Baochai share one character of their given name with Baoyu.

But most of the family's younger generation is girls—a circumstance that exerts multiple forces on the main characters.  Baoyu is the only proper male member of the Jia family in his generation; he has only a half-brother Huan who is miserably jealous of Baoyu and who spends most of the novel plotting against him and otherwise acting like a dog who has been kicked to the curb rather too often.  As a result, tremendous pressure is brought to bear on Baoyu to continue the line and to sustain the emperor's favor.  As the family holdings slowly dwindle as the combined result of extravagance, bad luck, and traitorous servants, the family feels with greater urgency every ebb and flow in the affairs of Baoyu.

It is not only Baoyu who feels the effect of the gender imbalance in the household.  Daiyu comes to the family grounds when her mother dies and her father, who cannot bring her up, sends her to his in-laws.  From the beginning, she feels like an outsider with almost all of her relatives, despite their best efforts—all, that is, except Baoyu, to whom she feels an almost instant connection and affinity (and vice versa).  Otherwise, she is in constant fear of being left out on her own in the cold.

It is their romance, suppressed and sublimated by the strictures of Chinese tradition (in which marriage is a matter of parental prerogative), that forms the backbone of the novel, and which plays against the backdrop of the slowly declining Jia family fortunes.  Daiyu yearns with all of her heart to marry Baoyu, both for survival and because she loves him, but it is not up to her.  And because there are no other eligible Jia boys, any other girl—meaning Baochai, first and foremost—represents potential competition for a prize that only one of them can win.  In the end, the resolution of this emotional struggle also serves to drive the resolution both of Baoyu's psychological development and, at a larger scale, of the Jia family's fate.

The Authorship Question

It almost wouldn't be a classic Chinese novel if there weren't some question about its provenance.  Journey to the West, for instance, is merely attributed to 吳承恩 Wú Chéng'ēn; it is not actually known with certainty that he wrote it.  He is known to have written something by that name, but because there are in fact many writings of various lengths and degrees of historical accuracy by that name (it is rather generic, after all), and it was not found in his possession after his death, the attribution is only probable.

In the case of Red Mansions, there is no such question regarding Cao and the first two-thirds of the novel.  Though there are a dozen or so different manuscripts, the differences are generally minor and betoken no substantial variance on plot or characterization.  Nor is there nowadays any question that Cao is responsible for them.

The problem arises with the remaining 40 chapters.  There seem to be no fair copies that date back to Cao's day that contain anything past Chapter 80, at all.  And the plot moves along with sufficient leisure—the leisure that eventually dissuaded my mother from finishing the book—that by Chapter 80, things only then seem to begin to climb toward a climax.

Nevertheless, in 1791 (when Cao had been dead for nearly three decades), for the first printed edition, 高鶚 Gāo È, along with his friend 程偉元 Chéng Wěiyuán, cobbled together a collection of manuscript drafts that together appeared collectively to comprise the 40-chapter conclusion of the novel.  By this time, the authorship of the novel had been forgotten and would have to await future literary investigation to rediscover.

But there would be other, thornier questions to resolve almost immediately.  The general public had been clamoring for the end of Red Mansions, and Gao's completion served to satisfy their needs. The more dedicated aficionados of the book were another matter. At issue are an array of intimations and premonitions in the first part of the book, notably a series of poems in Chapter 5, which seem to impose quite clear restrictions on the eventual fate of many of the main characters (including the "big three").  These are further reinforced by a series of well-known annotations by anonymous commenters who are nevertheless clearly intimate friends or relations of Cao. But Chapters 81 through 120 in Gao's edition seem to contravene much of this material, some of it quite severely.

For example, in Chapter 5, Baoyu dreams that he sees a book that depicts, in pictorial and textual riddle form, the fates of the girls in the family.  One of them is 香菱 Xiānglíng, which Hawkes renders as Caltrop.  The picture associated with Caltrop makes it clear that she will die at the hands of the jealous stepwife of her master.  But in Gao's ending, it is the stepwife who dies, accidentally poisoned by her own hand when she tries to murder Caltrop.  What's more, it seems likely, in the light of various suggestive passages, that Cao originally had planned a much more harrowing ending for the Jia family than what was eventually presented in Gao's ending.

There are lesser inconsistencies, different manners of death from what seems preordained.  Together, they seemed to indicate to the increasing number of close students of the novel that the completion that Gao edited was not Cao's.  Either Gao edited material that was written by someone else, or (it was suggested increasingly often as decades passed) Gao wrote it himself.  This is still the orthodox position.  In recent years, statistical stylometry has even been employed to show that there is a substantial discontinuity in style between the first 80 chapters and the last 40.

On the other side of the ledger are troubling inconsistencies of the same sort, which already appear in the first 80 chapters that are universally acknowledged to be Cao's.  The root of the problem is that Cao was an inveterate reviser, who by his own admission (in the body of the novel itself, naturally) had already rewritten various parts of the entire story several times.  Over time, he must have changed the fates of many characters across the entire breadth of the book.  He was not, however, the most careful reviser, however, and scattered in the thousand-plus pages are numerous continuity errors.  Chief among these were the various poems.  They could not be rewritten nearly as easily or as transparently as prose, so in many cases, Cao merely left them the way they were (possibly intending to return to rewrite them, should the opportunity arise), preserving the older versions of characters (in Hawkes's words) "like flies in amber."

Such observations have led Hawkes, Minford, and Anthony Yu (who authored the tremendously literate translation of Journey to the West, remember) to conclude that despite the questions raised by some of the unfulfilled prophecies, the last 40 chapters in Gao's edition appear to complete Cao's general intent, if not his exact wording, and that Gao likely did just edit some collected fragments, rather than creating the completion out of whole cloth, as used to be the prevailing opinion. Of course, that editing could have been quite substantial, especially if the parts that Cheng and Gao collected were substantially incomplete in patches. But the debate continues.

Its Place in Chinese Literature

All of these needlesome questions notwithstanding, Red Mansions engrosses more of the Chinese reading public than ever.  What accounts for its endless fascination?

Some of it is surely what my mother complained about: a kind of worship cult that has grown up around it.  Because it is continually written about, readers conclude, there must be something for people to be writing about.  We always want to know what all the fuss is about.

But it seems to me that there is more to it than mere reputation.  There is an air of mystery pervading it, both in the story itself and in the story of its creation.  And despite its occasionally glacial pace and fascination with 18th-century Chinese high-class culture, it confronts questions about the meaning of life and reality more directly than any other prominent piece of Chinese literature.  To read Red Mansions is to expose oneself to contradictions of experience and truth. One can decide that they are merely a matter of perspective, but I think it is hard to argue that they are immaterial—fictional or otherwise.

Remember that Red Mansions itself states baldly that it is fiction. There are parts of it that clearly belong to the realm of magical realism: monks disappear into the mist almost in front of one's eyes, characters somehow discover truths that they should not be able to know, and even some lives are lost by some kind of sympathetic magic. Yet this mysticism runs headlong into the crushingly realistic depiction of the juxtaposition between rich and poor, and the cataclysmic fall of the Jia family.

Cao even alludes to this duality in the names of two families in the novel: the aforementioned 賈 Jiǎ family, central to the story, and another, more peripheral family named 甄 Zhēn. There is even a Baoyu in the Zhen family, who closely resembles Jia Baoyu. Nor are these two names chosen by accident, for they are exactly homophonous with the characters 假 jiǎ "false, not real" and 真 zhēn "real." But aside from this obvious piece of symbolism, what exactly does Cao tell us?

As it happens, Cao was born into the lap of luxury, but when he was about 13—the same age as Jia Baoyu at the start of the novel—the old emperor (who had grown up with Cao's grandfather and always supported the Cao family) died and the new emperor, intending to make a political example and distance himself from his predecessor, had the Cao family's holdings stripped. By all accounts, Cao's own family's decline mirrored the Jia family more closely than the Zhen family, who never make much of a deep impression on the story. Does the Jia family in the story merely represent an exaggerated version of Cao's own family?

There is a suggestion that Cao knew, or was told, that it would be impolitic for him to make the Jia family's decline too obviously an unmitigated disaster, that his family (already poor) might have more miseries visited upon it by the powers that be if he were not to soften the blow. Seen in that light, the use of the Jia name might be a way to deflect additional persecution over what could be seen as overly frank criticism of the emperor.

Even then, however, the mere presence of that kind of symbolism (for there is more of it, usually less obvious, scattered throughout the names in the novel) makes it almost irresistible to treat the novel as a roman a clef, which we could interpret as a kind of biography of Cao, if we could only discover the key. Contributing to that sensation is the fact that the earliest versions of the novel include annotations by some commenters who are clearly closely connected with Cao, and which indicate that many of the characters were closely modeled on real people. I think that certainly accounts for a large part of the novel's appeal to readers, year after year after year.

To be sure, there are plenty of episodes that Cao has clearly put in as comic relief or dramatic color. And yet even the characters that Cao puts forth here, these one-offs, are memorable in their short appearances because Cao endows them with recognizable human weaknesses and biases. They do not serve solely to further the plot—in fact, they frequently don't advance the plot at all—but in addition (or instead) remind us of people we all know, until it almost seems as though Cao knows our friends better than we do.

The bulk of the rest of it, of course, is the love triangle between the three main characters. It is not, plotwise, a very complex story, and flatly described, it would not be very compelling. But though it is occasionally sentimental and overwrought, it is nonetheless told with such richness and verisimilitude that generations of readers have found it memorable. And in this novel, it is tied together with notions of predestination and of former lives, which I think Western and even modern Chinese readers associate with some distant ineffable Eastern mysticism.

But in fact, for all its romantic filigree, that part of the story is remarkable at its heart for its utter ordinariness. The emotions, though they may be expressed in a foreign and unfamiliar way (especially for Western readers), are still clearly recognizable. Seeing themselves in the novel, readers have for centuries envisioned themselves as Lin Daiyu or Jia Baoyu, much as people in the West have envisioned themselves as Romeo or Juliet, or Puck. It is the ease with which the novel transports readers into its milieu—its seductive immersiveness—that truly makes this novel a cornerstone of Chinese culture.

Mookie Betts's Glove Was in the Field of Play

I got the tl;dr out of the way in the title.

I've written previously about the value of multiple points of view (literal points of view in this case, but I think it's valuable for figurative points of view, too).  Last night, in Game 4 between the Boston Red Sox and the Houston Astros, was another example.

Here's the situation as it was in Houston (the location is kind of interesting, though not really important to the ruling).  It's the bottom of the first, and the Astros are already down 2–0, but they have George Springer on first after a one-out single, and Jose Altuve up to bat.  Altuve hits a deep fly to right, and Red Sox right fielder Mookie Betts reaches up and seems about to make the play, when his glove is closed shut by a fan's hand.  The ball bounces back into right field, where Betts retrieves it and fires it back into the infield.  Altuve ends up on second, and Springer (who presumably had to wait to see if Betts made the catch) stands on third.

Umpire Joe West initially calls a home run, and then appears to indicate interference (as shown here at the 8:48 mark).  The umpires collectively go to the replay, and after a delay of a few minutes, they call Altuve out, and order Springer to return to first.  After Marwin González is hit by a pitch, Yuli Gurriel flies out more conventionally to right and the Red Sox escape without further damage.

In the aftermath of the Red Sox' 8–6 victory, however, there was considerable controversy over whether the interference call was the right one.  The ruling was that because Betts's glove did not exit the field of play—that is, it did not cross the imaginary plane of the outfield fence—he was interfered with.  Had the glove been beyond the fence, then any contact with the fans would not have been considered interference.

The problem is that it's far from obvious where Betts's glove was at the moment of contact.  The Red Sox observed (as did some others) that Betts's body had yet to reach the fence, but the Astros pointed out that Betts was reaching backward for the ball.  Both sides agreed that the ball would have gone into the stands were it not for Betts, and both sides agreed that Betts had a good chance of catching the ball.  (I've seen a few fans claiming that Betts simply closed his glove early, but neither I nor any professional commentator seems to find that credible. See here at the 0:45 mark for a pretty clear video of Betts's glove being closed by a fan's hand.)

Incidentally, whether Betts would have caught the ball doesn't have any bearing on the correct call. West's call was predicated only on whether the fans interfered with Betts's fielding in the field of play. The approved ruling associated with Rule 6.01(e) reads:

If spectator interference clearly prevents a fielder from catching a fly ball, the umpire shall call the batter out.

The comment on that rule goes on to clarify:

No interference shall be allowed when a fielder reaches over a fence, railing, rope or into a stand to catch a ball. He does so at his own risk. However, should a spectator reach out on the playing field side of such fence, railing or rope, and plainly prevent the fielder from catching the ball, then the batsman should be called out for the spectator’s interference.

That's what made the correct interpretation of the replays so vital.

Nevertheless, both sides also thought the replays confirmed their conclusion, each perhaps pretending to a greater certainty than they really felt.  They're really not that conclusive either way, at first glance, and it was important, probably, that the call on the field was interference.  Here's a shot from one angle, for instance (the left-field camera, I think):

Can you tell where Betts's glove is in relation to the fence?  I can't.

Well, we don't have to tell from that shot alone.  Here's a second shot from another angle (maybe the first-base camera):

Hmm, it's not obvious from that shot either.

Once again, though, we don't have to rely on either shot in isolation; fortunately, the two images together will tell us what we need to know.  Both shots show the play a split-second after the fan had made contact with the glove, and with the ball just about to strike the outside of the glove.  The fans are still looking up because they're not trained to follow the ball into the glove, and because that baseball is moving fast, but that white blur is the ball in both photos.

How does this help us?  Well, let's take a look at where the glove is in relation to the wall.   Here are the same two shots, but with the same location marked on the outfield wall padding:

Notice where the glove is in relation to that mark in the two images.  It's to the right of that mark from the point of view of the left-field camera, but it's just about in line with the mark (or maybe a little to the left) from the point of view of the first-base camera.  It's simple triangulation: If the glove is directly above the fence, then it should be in the same position with respect to the mark from both views.  If it's in front of the fence, it should appear further to the right in the first view (from left field), and if it's beyond the fence, it should appear further to the left in the first view.

Since it's further to the right in the first view, the glove must have been in front of the fence at that moment, and the interference call is the right one.  (I was mildly surprised to discover this, by the way.  If I had to guess, I would have guessed that the glove was beyond the fence—but I would have been pretty loathe to guess.)  Without knowing more about the location of the cameras relative to the wall, we can't be sure how much in front it was, but at any rate, the contact was made in the field of play.

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ETA: Here's a third, intermediate view—from the third base camera, I think—further confirming the findings: