Friday, December 13, 2019

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 Vd 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 Gd 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 Gd/4d 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.

Friday, September 20, 2019

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):

 
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."

Saturday, July 27, 2019

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.