Anti AI Guys Keep Making AI Sound So Fucking Cool

being as i am an idiot, and having been one my whole life, i just wanna say that i find it very easy to do nothing, and go nowhere. i eat chocolate late at night in the dark. i stand in the garden also. and i’m often waiting for something to happen. and i’m stupid.

Triangle Tuesday 3: The orthocenter, the Euler line, and orthocentric systems

Previously, we have looked at two different ways to mark a point in a triangle. First, we drew cevians (lines from the vertices) to the midpoints of the sides and found that they all cross at a point, which is the centroid. Then we tried drawing perpendiculars to the sides from the midpoints, and those all met at the circumcenter. And you could do this with any point on the side of a triangle -- draw a cevian to it, or a perpendicular from it, and see what happens.

This time, though, we're going to do both. That is, we're going to work with the cevians that also form perpendiculars to the sides. These are the altitudes, which run from a vertex to the nearest point on the opposite side, called the foot of the altitude. The three altitudes all meet at a point H, and that's the orthocenter. (The letter H has been used to mark the orthocenter since at least the late 19th century. I believe it's from the German Höhenschnittpunkt, "altitude intersection point.") Anyway, let's prove that the orthocenter exists.

Theorem: the three altitudes of a triangle coincide.

Here's a very simple proof that the three altitudes coincide. It relies on the existence of the circumcenter, which we already proved before. Given a triangle ABC, draw a line through A parallel to the opposite side BC. Do the same at B and C. These lines cross at D, E, and F and form the antimedial triangle (in blue).

Then the altitudes of ABC are also the perpendicular bisectors of DEF. We proved before that perpendicular bisectors all meet at a point, therefore altitudes do as well.

That was easy. Let's do it again, but in a different way. It's not quite as simple, but it includes a large bonus.

Theorem: the three altitudes of a triangle coincide at a point colinear with the circumcenter and centroid, and GH = 2 GO.

Let's take triangle ABC, and let F be the midpoint of side AB. Then mark two points that we already know, the circumcenter O and the centroid G. We'll also draw the median (green) and the perpendicular bisector (blue) that run through F, leaving the other ones out to avoid cluttering the picture.

We already know from our look at the centroid that G cuts segment FC at a third of its length, so GC = 2GF. Let's extend segment OG in the direction of G by twice its length out to a point we'll label H, so that GH = 2GO.

Now consider the two triangles GOF and GHC. By construction, their two blue sides are in the ratio 1:2, and the same for their two black sides. They also meet in vertical angles at their common vertex G. So by side-angle-side, the triangles are similar, and it follows that HC is parallel to OF, and therefore perpendicular to AB. So H lies on the altitude from H to side AB.

By analogous construction, we can show that H also lies on the other two altitudes. So not only have we proved that the altitudes coincide, but also that O, G, and H all lie on a line, and furthermore that G is located one third of the way from O to H, in any triangle.

This proof is due to Leonard Euler, and the line OGH is called the Euler line. Not only these three points but many others as well fall on this line, which we will get to later on.

Let's look at some more properties of the orthocenter and the feet of the altitudes. I'm just going to look at the case of an acute triangle for now, and show how this extends to the obtuse case later.

Theorem: two vertices of a triangle and the feet of the altitudes from those vertices are concyclic.

Proof is easy: the two right triangles AHcC and AHaC share segment AC as a hypotenuse. Therefore AC is a diameter of the common circumcircle of AHcC and AHaC (following from Thales's theorem).

(Incidentally, look at the angle formed by the blue segment and the altitude CHc. It subtends the same arc as angle CAHa, so (by the inscribed angle theorem again) they must be equal. That's not a part of this theorem, so just tuck that fact away for a moment.)

Theorem: a vertex, the two adjacent feet of the altitudes, and the orthocenter are concyclic.

Same idea, but now the right triangles are AHcH and AHbH, and AH is the diameter of the common circumcircle.

(And incidentally, look at the angle formed by the new blue line and the altitude CHc. It subtends the same arc as HbAH, which is same angle as CAHa. So those angles must be equal too. Since both angles between a blue line and the altitude CHc are equal to the same thing, they are equal to each other. Again, not a part of this theorem, just something I wanted to note.)

So those are some interesting concyclicities, but now let's look at the pedal triangle of the orthocenter, which is called the orthic triangle.

Oh, hey, it's made up blue lines, just like the ones we were talking about. And we proved that the two longest blue lines make equal angles with the altitude between them. By symmetry, we can prove the same thing about all the angles made by the blue lines. So that means

Theorem: two sides of the orthic triangle make equal angles with the altitude between them.

Another way to say this is that the altitudes are the angle bisectors of the orthic triangle. And I admit that was kind of a roundabout way to introduce the orthic triangle, but I think it makes the proof of this property easier to follow.

Two other properties of the orthic triangle immediately follow from this:

In an acute triangle, the inscribed triangle with the shortest perimeter is the orthic triangle

and

In an acute triangle, the orthic triangle forms a triangular closed path for a beam of light reflected around a triangle

which are two ways of saying the same thing.

But those two properties only hold for acute triangles. What happens to the orthic triangle in an obtuse triangle? Let's push point C downward to make triangle ABC obtuse and see what happens. To make things clear, I'm going to extend the sides of ABC and the altitudes from line segments into lines. Here's the before:

And here's the after:

The orthocenter has moved outside of triangle ABC, and two of the altitudes have their feet on extensions of the sides of ABC rather than on the segments AC and BC. The orthic triangle now extends outside ABC, and certainly isn't the inscribed triangle with the shortest perimeter any more.

But look at it another way. We now have an acute triangle ABH, and the line HHc is an altitude of both the obtuse triangle ABC and the acute triangle ABH. Meanwhile, lines AC and BC have become altitudes of ABH.

So what we have is essentially the same acute triangle with two swaps: point C trades places with H, and Ha trades places with Hb. That means that our two theorems about concyclic points morph into each other as triangle ABC switches between acute and obtuse. Here's an animation to show the process:

And this is why I didn't bother with the obtuse case above -- each theorem of concyclicity is the obtuse case of the other.

So if we can just exchange the orthocenter with one of the vertices, what does this mean for their relationship? If you are given a group of vertices and lines, how can you tell which one is the orthocenter and which one are the vertices? Well, you can't.

Theorem: Given an acute or obtuse triangle ABC and its orthocenter H, A is the orthocenter of triangle BCH, B is the orthocenter of ACH, and C is the orthocenter of ABH.

The proof comes from consulting either of the "before" and "after" figures above. Take any three lines that form a triangle, red or black. The other three lines are then the altitudes of that triangle. The three feet are where a red and black line meet perpendicularly, so they are the same for all four possible triangles, which means all four share the same orthic triangle.

(Of course, if ABC is a right triangle, then we get a degenerate case, as you can see from the gif at the moment when C and H meet.)

Such an arrangement of four points is called an orthocentric system. Of the four points, one is always located inside an acute triangle formed by the other three, and it's conventional to label the interior one H and the others ABC.

Orthocentric systems pop up all over the place in triangles, so expect to see more of them as we go along. Now, let me do one little lemma about altitudes, and then I'll show something cool about orthocentric systems.

Lemma: the segment of an altitude from the orthocenter to a side of the triangle is equal to the extension of the altitude from the side to the circumcircle.

We can show this with just a little shuffling of angle identities. Extend altitude CHc to meet the circumcircle at C'. The angles CAB and CC'B, labeled in red, subtend the same arcs, so they are equal. Triangle ABHb is a right triangle, so angle HbBA, in blue, is complementary to it. The same is true for the right triangle C'BHc, so the two angles labeled in blue are equal. Then by angle-side-angle, triangles BHcH and HHcC' are congruent, and segment HHc = HcC'.

By the same argument, we can show that triangle AHHc is congruent to AC'Hc, which leads us into the next bit.

Theorem: all the circumcircles of the triangles of an orthocentric system are the same size.

The blue triangle has the same circumcircle as triangle ABC. From the foregoing, the blue and green triangles are congruent. Therefore their circumcircles are the same size as well. The same argument works for ACH and BCH.

So here is an orthocentric system with its four circumcircles.

The four circumcenters O, Oa, Ob, and Oc form another orthocentric system, congruent to the first one.

If you found this interesting, please try drawing some of this stuff for yourself! You can use a compass and straightedge, or software such as Geogebra, which I used to make all my drawings. You can try it on the web here or download apps to run on your own computer here.

happy holiday everyone!

On this day, July 27th in 1987, a single was released that would change the world forever.

It's Rick Astley's debut single, Never Gonna Give You Up!

I have tried for years to discover something, anything, about this card with no success.

This is a good example of what are called "Barnum statements" -- things which are true of almost everyone, but which seem insightful when you read them because they are true of *you*. Furthermore, each result contains many statements (I counted 56 in mine), so that if any one of them matches your current preoccupations or self-concept, you'll forget the ones that didn't apply or only slightly applied.

A fun exercise when you encounter something like this is to tally up all the statements and see how many were really true. In my results, I counted 20 statements that were true about me and 25 that were false.

This doesn't mean that the results are useless, since you may not have thought about your life and emotions in these terms before, and it is still useful to consider e.g. whether you harbor resentment about someone in your life, or whether peace is truly achievable with your current approach. It just means that the usefulness of the results depends entirely on you and your self-knowledge, and the test has no special insight about you.

Quotes from my results under the cut for spoilers.

[disliked olive green] ...inhibiting limitations. Difficult circumstances limit your opportunities for experience and your freedom of action. You feel deprived because you have to do without some of the things that would make life pleasant. You expect far too much understanding for your needs from other people, and as a result, you often feel disappointed. You might ask yourself how much understanding and empathy you extend to others. You would like to be free of your...

I think it's quite clever in the way it mixes different motivations and traits. Here, the main topic is "inhibiting limitations", which makes me think of anxiety and an aversion to breaking rules. But then the next two sentences are about being constrained by non-omnipotence and unable to afford everything you want, which is a universal complaint among finite beings, but not one that seems particularly related to inhibitions. Then, the next two sentences are about wanting others to understand you but not being very understanding of others, which again is very common among humans, but which has even less to do with inhibitions.

As a result, someone whose main problem is any one of {anxiety, poverty, ingratitude} might feel like this paragraph applies to them "uncannily well". And there are 12 such paragraphs! It's no wonder that so many people upthread have felt understood.

where’s that quiz where you choose lke 4 colours u like and 4 u dont and it hands your ass on a plate

sygol framed poll (handle with care), 2024 mixed media on tumblr post

for /-yr/ i like the song La Monture from Notre-Dame de Paris

for the fun-to-say pile: méli-mélo, micmac, assujetissement, eussent été, farfelu

i'm now looking at my list of least favorite french words to pronounce and going "too many r's" for about 40% of them and "skill issue" for most of the rest. some of these are actually very fun to pronounce i just couldn't wrap my tongue around them a year or so ago, but now i can i guess??? so that's very exciting. makes me hope that someday i'll be able to pronounce the rest of them. this is a bit pie in the sky because i really don't see myself ever getting there with procureur du roi but you never know. and luckily the french abolished the monarchy so it's not like i'll ever have to use that phrase in modern conversation.

anyway here are the words i actually love pronouncing now: décaféiné diététicien filleul pneumonie

i now feel normal/neutral about these words that used to be hard for me: automne, condamner douloureux électricité, énergie inférieur, supérieur, etc. itinéraire lourdeur salmonellose sclérose subodorer succincte

words that are definitely within the realm of my current capability but i haven't practiced them enough: bugle hiérarchisation méditerranéen phtisie

words that are still the bane of my existence but i live in hope: [yʁ] plus at least one other r or [y] sound: chirurgie, fourrure, marbrure, moirure, nourriture, ordures, peinturlurer, procureur du roi, prurit, purpurin, sculpture, serrurerie, structure, sulfureux, tournure all words beginning with ur-, hur-, or sur- other difficult sequence of r's and vowels: construire and other -truire verbs; lueur and sueur; utérus too many r's: marbre, martre, meurtre, opprobre, proroger, réfrigérateur, rétrograde, rorqual difficult sequence of vowels and/or semivowels: coopérant, extraordinaire, hémorroïdal, kyrie eleison, météorologique, micro-ordinateur, micro-organisme, mouillure, quatuor, vanillier not pronounced the way i would expect from the spelling: indemne, penta-, punk just hard for some reason: humour

first time using lasso tool, i had a lot of fun with this

and there are eight billion of them!

girl who is animated, she is a flesh and marrow golem - a division of the world that runs according to the needs of a number of complex cohabitating forms of life, notably coming in the form of a number of sponges each with its own distinguishing properties and materials. The body, that is, that political organisation of sponges, reshapes itself as an organic sculpture. The sculpture looks something like a city that reaches into the sky, and there it is in conversation with a great light that drives it.

Ender's Game (novel)

Is Ender Wiggin (pictured above as the little brother from Malcolm in the Middle) guilty of xenocide?

Actually, let's first answer a different, but related, question:

What game does the title "Ender's Game" refer to?

It's not as simple a question as it seems. There are three games that have a prominent role in the plot, all very different from one another.

The obvious answer is the Battle School zero-gravity game, where teams of competitors play glorified laser tag in a big empty cube. In terms of page count, most of the book is dedicated to this game. It's also the game depicted on the cover of the edition above.

Yet this game vanishes during the story's climax, when Ender is given a new game to play, a game he is told is a simulator of spaceship warfare. This "game" turns out to not be a game at all, though; after annihilating the alien homeworld in the final stage, Ender learns that he was actually commanding real ships against real enemies the whole time, and that he just singlehandedly ended the Human-Bugger war forever via total xenocide of the aliens. This is both the final game and the most consequential to the plot, despite the short amount of time it appears.

There's also a third game, a single-player video game Ender plays throughout the story. The game is procedurally generated by an AI to respond to the player's emotional state, and is used as a psychiatric diagnostic for the players. Of the three games, this is the one that probes deepest into Ender's psyche, that most defines him as a person; it's also the final image of the story, as the aliens build a facsimile of its world in reality after psychically reading Ender's mind while he xenocides them.

Because all three games are important, the easiest answer might be that the question doesn't matter, that the story is called Ender's Game not to propose this question at all but simply because the technically more accurate "Ender's Games" would improperly suggest a story about a serial prankster.

Fine. But why does the title use the possessive "Ender's" at all?

He does not own any of these games. He did not create them. He does not facilitate them. All of these games, even the simulator game, predate his use of them as a player, were not designed with him in mind, were intended to train and assess potential commanders for, ostensibly, the hundred years since the last Human-Bugger war.

It's in this question that we get to the crux of what defines Gamer literature.

These games are Ender's games because he dominates them into being about him. He enters a rigidly-defined, rules-based system, and excels so completely that the games warp around his presence. In the Battle School game, the administrators stack the odds against Ender, thereby rendering every other player's presence in the game irrelevant except in their function as challenges for Ender to overcome. The administrators acknowledge this in an argument among themselves:

"The game will be compromised. The comparative standings will become meaningless." [...] "You're getting too close to the game, Anderson. You're forgetting that it is merely a training exercise." "It's also status, identity, purpose, name; all that makes these children who they are comes out of this game. When it becomes known that the game can be manipulated, weighted, cheated, it will undo this whole school. I'm not exaggerating." "I know." "So I hope Ender Wiggin truly is the one, because you'll have degraded the effectiveness of our training method for a long time to come."

In this argument, Anderson views the game the way games have been viewed since antiquity: exercises in acquiring honor and status. This honor is based on the innate fairness inherent to games as rule-based systems, which is why in ancient depictions of sport the chief character is often not a competitor but the host, who acts as referee. In Virgil's Aeneid, for instance, the hero Aeneas hosts a series of funeral games (the games themselves intended as an honor for his dead father). Despite being the principal character of the epic, Aeneas does not compete in these games. Instead, he doles out prizes to each competitor based on the worthiness they display; his fairness marks him symbolically as a wise ruler. The Arthurian tournament is another example, where Arthur as host is the principal character, and the knights (Lancelot, Tristan, etc.) who compete do so primarily to receive honors from him or his queen.

In Ender's Game, it is the antagonistic figure Bonzo Madrid who embodies this classical concept of honor; the word defines him, is repeated constantly ("his Spanish honor"), drives his blistering hatred of Ender, who receives both unfair boons and unfair banes from the game's administrators, who skirts the rules of what is allowed to secure victory. Bonzo is depicted as a stupid, bull-like figure; his honor is ultimately worthless, trivially manipulated by Ender in their final fight.

Meanwhile, it's Ender's disregard for honor, his focus solely on his namesake -- ending, finishing the game, the ends before the means -- that makes him so valuable within the scope of the story. He is "the one," as Anderson puts it, the solipsistically important Gamer, the Only I Play the Game-r, because the game now matters in and of itself, rather than as a social activity. In the Aeneid and in Arthur, the competitors are soldiers, for whom there is a world outside the game. Their games are not a substitute for war but a reprieve from it, and as such they are an activity meant to hold together the unifying fabric of society. The values Anderson espouses (status, identity, purpose, name) are fundamentally more important in this social framework than winning (ending) is.

Ender's game, as the Goosebumps-style blurb on my 20-year-old book fair edition's cover proclaims, is not just a game anymore. Its competitors are also soldiers, but the game is meant to prepare them for war; the spaceship video game is actual war. And as this is a war for the survival of the human race, as Ender is told, there is no need for honor. The othered enemy must be annihilated, without remorse or mercy.

This ethos of the game as fundamentally important for its own sake pervades Gamer literature beyond Ender's Game. In Sword Art Online (which I wrote an essay on here), dying in the game is dying in real life, and as such, only Kirito's ability to beat the game matters. Like Ender, Kirito is immediately disdained by his fellow players as a "cheater" (oh sorry, I mean a "beater") because he possesses inherent advantages due to being a beta player. In an actual game, a game that is only a game, Kirito's cheat powers would render the game pointless. What purpose does Kirito winning serve if he does it with Dual Wielding, an overpowered skill that only he is allowed to have? But when a game has real stakes, when only ability to win matters, it is possible to disregard fairness and see the cheater as heroic.

This notion of the "cheat power," a unique and overpowered ability only the protagonist has, is pervasive in post-SAO Gamer literature. To those for whom games are simply games, such powers can only be infuriating and obnoxious betrayals of the purpose of games; to those for whom games mean more than just games, for whom games have a primacy of importance, these powers are all that matter.

That's the core conceit of Gamer literature: the idea that the Game is life, that winning is, in fact, everything.

What sets Ender's Game apart from Sword Art Online is that it creates the inverted world where the Game matters above all, but then draws back the curtain to reveal the inversion. The Buggers are, in fact, no longer hostile. They are not planning to invade Earth again, as Ender has been told his entire life. The war, for them, is entirely defensive, and Ender is the aggressor. And due to Ender's singleminded focus on Ending, on winning, on disregarding honor and fairness, he ultimately commits the xenocide, erases an entire sentient species from existence. He wins a game he should never have been playing.

The obvious counterargument, the one I imagine everyone who has read this book thought up the moment I posed the question at the beginning of this essay, is that Ender did not know he was committing xenocide. The fact that the combat simulator game was not a game was withheld from him until afterward. Plus, he was a child.

Salient arguments all. Ones the book itself makes, via Ender's commander, Graff, to absolve him of sin at the end. They're probably even correct, in a legal sense (I'm not a legal scholar, don't quote me), and in a moral sense. In real life, it would be difficult to blame a 10-year-old in those circumstances for what he did. But in the thematic framework of Ender's Game the book, these arguments are completely inadequate.

Ender has been playing a fourth game the entire story. And this is the only game he doesn't win.

A game is defined by its system of control and limitation over the behavior of the players. A game has rules. His whole life, Ender has been playing within the rules of the system of control his military commanders place upon him.

Their control extends even before he was born; as a third child in a draconian two-child-only world, his existence is at the behest of the government. Graff confirms this to Ender's parents when he recruits him to Battle School: "Of course we already have your consent, granted in writing at the time conception was confirmed, or he could not have been born. He has been ours since then, if he qualified." Graff frames this control utterly, in terms of possession: "he has been ours." He does not exaggerate. Since Ender was young, he has had a "monitor" implanted in his body so the army could observe him at all times, assess whether he "qualifies"; even the brief moment the monitor is removed is a test. "The final step in your testing was to see what would happen when the monitor came off," Graff explains after Ender passes the test by murdering a 6-year-old. Conditions are set up for Ender, similar to the unfair challenges established in the Battle School game; he is isolated from his peers, denied practice sessions, held in solitary confinement on a remote planetoid. It's all in service of assessing Ender as "the one."

Ender wins this game in the sense that he does, ultimately, become "the one" -- the one Graff and the other military men want, the xenocider of the Buggers. He fails this game in the sense that he does not break it.

The other three games Ender plays, he breaks. Usually by cheating. In the single-player psychiatry game, when presented with a deliberately impossible challenge where a giant gives him two glasses to pick between, Ender cheats and kills the giant. "Cheater, cheater!" the dying giant shouts. In the Battle School game, Ender is ultimately confronted by insurmountable odds: 2 armies against his 1. He cannot outgun his opponent, so he cheats by using most of his troops as a distraction so five soldiers can sneak through the enemy's gate, ending the game. At the school, going through the gate is traditionally seen as a mere formality, something done ceremonially once the enemy team is wiped out (there's that honor again, that ceremony), but it technically causes a win. Even Anderson, the game's administrator, sees this as a breach of the rules when Ender confronts him afterward.

Ender was smiling. "I beat you again, sir," he said. "Nonsense, Ender," Anderson said softly. "Your battle was with Griffin and Tiger." "How stupid do you think I am?" Ender said. Loudly, Anderson said, "After that little maneuver, the rules are being revised to require that all of the enemy's soldiers must be frozen or disabled before the gate can be reversed."

(I include the first part of that quote to indicate that Ender all along knows who he is really playing this game against -- the administrators, the military men who control every facet of his life.)

Ender beats the war simulator game in a similar fashion. Outnumbered this time 1000-to-1, he uses his soldiers as sacrifices to sneak a single bomb onto the alien's homeworld, destroying it and committing his xenocide. Ender himself sees this maneuver as breaking the rules, and in fact falsely believes that if he breaks the rules he will be disqualified, set free from the fourth game: "If I break this rule, they'll never let me be a commander. It would be too dangerous. I'll never have to play a game again. And that is victory." The flaw in his logic comes not from whether he's breaking the rules of the game, but which game he is breaking the rules of. It's not the fourth game, Ender's game, but the war simulator game, simply a sub-game within the confines of the fourth game, a sub-game the fourth game's administrators want him to break, a sub-game that gives Ender the illusion of control by breaking. When Ender tells his administrators about his plan, the response he receives almost taunts him to do it:

"Does the Little Doctor work against a planet?" Mazer's face went rigid. "Ender, the buggers never deliberately attacked a civilian population in either invasion. You decide whether it would be wise to adopt a strategy that would invite reprisals."

(And if it wasn't clear how much the administrators wanted him to do this all along, the moment he does it, they flood the room with cheers.)

Ender wins his games by cheating -- by fighting the rules of the game itself -- and yet he never cheats at the fourth game, the game of his life.

In this fourth game, he always plays by the rules.

In the inverted world of Gamer lit, where games define everything, including life and death, it's a common, even natural progression for the Gamer to finally confront the game's administrator. Sword Art Online ends when Kirito defeats Akihiko Kayaba, the developer. In doing so, Kirito exceeds the confines of the game, not simply by ignoring its rules and coming back to life after he's killed, but by demonstrating mastery against the game's God. Afterward, Sword Art Online truly becomes Kirito's Game, with nobody else able to lay claim to the possessive. Kirito demonstrates this control at the end of the anime by recreating Sword Art Online's world using its source code, completing the transition into a player-administrator.

(Though I wonder, how much of a class reading could one give to this new brand of Gamer lit? If classical games were told from the perspective of the one who controlled them, then is there not something innately anti-establishment in Kirito overcoming the controller? This is the gist of many other death game stories, like The Hunger Games, though none of them may be the most sophisticated takes on the subject, more empty fantasy than anything else.)

Ender never fights or defeats his administrators. He never even tries, other than rare periods of depressive inactivity. He doesn't try even though the option is proposed to him by Dink Meeker, an older student whom Ender respects:

"I'm not going to let the bastards run me, Ender. They've got you pegged, too, and they don't plan to treat you kindly. Look what they've done to you so far." "They haven't done anything except promote me." "And she make you life so easy, neh?" Ender laughed and shook his head. "So maybe you're right." "They think they got you on ice. Don't let them." "But that's what I came for," Ender said. "For them to make me into a tool."

Instead, Ender finds comfort in the control exerted on his life. When sent to Earth on leave, he seeks out a lake that reminds him of living in Battle School.

"I spend a lot of time on the water. When I'm swimming, it's like being weightless. I miss being weightless. Also, when I'm here on the lake, the land slopes up in every direction." "Like living in a bowl." "I've lived in a bowl for four years."

Because of this, Ender never cheats against Graff. He could; Graff states several times that Ender is smarter than him, and the fact that they have Ender fighting the war instead of Graff is proof he believes it. But Ender never considers it. He never considers gaming the system of his life.

If Gamer literature emphasizes the inversion of the world order, where games supersede reality in importance (and, as in Sword Art Online, only through this inverted order is one able to claim real power by being a Gamer), then Ender's Game acknowledges both sides of the inversion. For Ender, the games he plays are not simply games anymore. The psychology game, the Battle School game, the war simulator game; all of these he must win at all costs, even if it requires disrespecting the foundational purpose of these games. But his real life? Ender wants that to be a game, craves it to be a game, can't live unless the walls slope up around him like a bowl, can't stand it unless there is a system of control around him. He does what Graff tells him, even though he recognizes immediately that Graff is not his friend, that Graff is the one isolating him from others, rigging things against him. He does what Graff tells him all the way up to and including xenocide, because Ender cannot tell game from real life. That's the core deception at the end: Ender is playing a game that's actually real and he doesn't know it -- or refuses to acknowledge it, since nobody has ever tricked the genius Ender before this point.

Actually, that's not true. They tricked him twice before. Ender twice attacks his peers physically, with brutal violence. The administrators conceal from him that he murdered both his foes; he simply thinks he hurt them. The only way to trick Ender is to do so in a way that insulates him from the consequences of his actions. The only way he will allow himself to be tricked.

So, is Ender guilty of xenocide?

Under it all, Ender believes he is.

The dying Buggers, after reading Ender's mind, recreate the psychology game in the real world. The story ends when Ender finds this recreation, yet another blurring of the lines between game and reality.

The psychology game is different from the other games Ender plays, because nobody expects him to win it. Its purpose is not to be won, simply to assess his mental health. Yet Ender approaches it like the other games, cheats at it and systematically kills all his enemies until he reaches a place called The End of the World. (Another End for Ender.) His drive to win, to dominate, does not come solely from the pressures of the system around him, but from deep within himself, which is what Ender fears the most. But it is here, at The End of the World, where Ender finds atonement, both in the game and in the game-made-real. In the game, he kisses his opponent instead of killing them, and reaches a resolution he is happy with. He stops playing the game after doing this, though the game seems to continue (when an administrator asks him why he stopped playing it, he says "I beat it"; the administrator tells him the game cannot be beaten). It is through this act of love that Ender can escape the game-like system of control that puppeteers him no matter how smart and clever he is or thinks he is.

In the game-made-real, Ender finds his atonement in the same place, The End of the World. The Buggers left for him here, in this place that they (reading his mind) understood as the location of his mercy and compassion, an egg that can repopulate their species. Through this egg, Ender is given the chance to undo his xenocide. But that chance is also contingent on what The End of the World means to Ender, an end to the game, not simply the games he plays but the fourth game, the game of his life. Ender's Game.