Any compact space is coarse equivalent to a point. ℝ is coarse equivalent to ℤ. The universal cover of a compact manifold is coarse equivalent to the fundamental group.
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Bourgeois society is ruled by equivalence. It makes dissimilar things comparable by reducing them to abstract quantities... anything which cannot be resolved into numbers, and ultimately into one, is illusion; modern positivism consigns it to poetry.
Max Horkheimer & Theodor W. Adorno, Dialectic of Enlightenment
So, one of the people who have stuck with Equivalence for a reeeally long time (it's almost 4 years old! How?) recommended that I write out chapter summaries to remind people of what happened in previous chapters.
The story's gotten really long and more than a little convoluted, so it helped me to write things out, too. (I tried to keep things short, but...well.)
So without further ado, here’s what’s happened!
1. The monsters come to the surface, and start learning about what's different there—particularly, the cold and the sky. Grillby starts some bonfires and sets up a barbecue. Toriel steps in to keep things organized when Asgore isn't up to the task alone. Frisk enjoys an impromptu sleepover and cuddle party.
2. The monsters see their first sunrise. Asgore asks Frisk to be an ambassador, and then Toriel asks if they want to leave. They aren't sure about Asgore's request, but they decide to stay. Some monsters are discovered to be slightly translucent aboveground, at least temporarily. Frisk helps Napstablook find Mettaton so he doesn't get walked through. Undyne, Sans, and Papyrus try to introduce the monsters to the nearby town's city hall. Frisk goes shopping with Toriel in the human town, and decides to make spaghetti for Papyrus.
3. A town has popped up out of nowhere by the time Frisk returns—though they're just magic houses, and will need to be rebuilt if they're going to last for very long. Toriel confesses to keeping Frisk out of the way during the construction for their own safety. Frisk makes spaghetti, but unfortunately it's too solid to agree with Papyrus, or most of the rest of the monsters for that matter. Toriel makes them monster pie in consolation.
4. Time passes, and the monsters rebuild the city the real way this time. Toriel starts doing PR for the human town. Alphys confesses her involvement in Mettaton's rise to stardom, and accompanying secret confinement. She explains that he's planning to live more openly. Undyne comes home upset, and explains that some secret sect of monsters want to ban humans from entering the city. Frisk becomes worried that they'll be seen as a kidnapping victim, or else be forced to leave their friends.
5. Asgore offers Frisk formal Ebott Town citizenship, and Frisk immediately accepts. Asgore hires Toriel for her services after she delivers an ultimatum. Frisk accompanies Asgore and some of their other friends to the underground to do some monster-based PR work. Parts of the underground are falling into disrepair with so many of their inhabitants gone. Frisk has an odd conversation with the River Person steering their ferry, and receives some magic mushrooms and some odd advice.
6. Frisk offers the Snowman Piece back to the Snowman, but they ask them to keep it with them until they're safely moved to the surface. The plight of monsters who don't want to move to the surface right away, or who can't, is discussed. Frisk finds the Nacarat Jester fallen down in the abandoned Snowdin Town, and calls in Sans as backup. Papyrus and Asgore (and, a little bit, Sans) take the lead in getting Frisk home. Between the abandoned regions and what happened to Nacarat, Frisk starts to regret the damage they've done to the monster community by breaking the barrier.
7. Toriel bakes with Frisk to cheer them up, but Frisk starts having nightmares and questioning their right to stay in the monster town. Frisk refuses Toriel's offer to stay home with them, and decides to visit Napstablook instead.
8. Frisk hangs out with Napstablook and learns about the latest developments in his music. Mettaton comes, and the three of them have a conversation about pronouns. Mettaton takes Frisk away to his mansion and engages in some aggressive comforting measures, and he makes Frisk promise to tell him if they ever decide to leave the ones they love behind.
9. Undyne POV chapter. Asgore negotiates with "Mr. J," who's been put in the role of a kind of monster liaison; they prepare for an oncoming blizzard and compare notes on magical knowledge, anime, and the Internet. Undyne meets Mr. J's ward (Nat), who is afraid of the snow.
10. Back to Frisk, whose mental health (anxiety in particular) has picked up in light of recent events. Toriel sends them out after the blizzard to play in an attempt to cheer them up. Frisk scopes out the snowman's future home...and sees a human (Nat) climbing the mountain, not so far from where they themselves fell. Frisk chases him, but they're abducted by a River Person before they can reach him. Frisk tries to follow, but collapses instead.
11. Sans POV. While on patrol, Sans runs into Monster Kid (or technically the other way around), who spotted Frisk while sledding. He uses his shortcuts to teleport over and saves them, but takes them back to Asgore instead of Toriel by mistake. Kid gathers the others, and they try to help Frisk. Word comes in to Asgore that another human child was found on the edge of town.
12. The other child (Nat) is taken to Grillby's and fed and warmed up. Frisk wakes up, and explains what they saw. They wants on going down to talk with the River People, to get to the bottom of things, but most of their friends, and especially Toriel and Asgore, argue they should stay put. Frisk tells Mettaton that they're going to leave that night anyway.
13. Frisk sneaks out of Toriel's house and into the Underground. They convince the River Person steering the ferry to take them to their hometown. They meet a River Person who claims that she was once a human and Frisk that their death is imminent...at least, if they continue living on the surface.
14. The River Person explains the history of the humans who were never found and killed by monsters. Over the years, they were found by the River People, and kept in hiding in their town. However, monster food can't sustain humans' physical bodies, only heal them. Some eventually died, unable to come to terms with the loss of a living body; others survived, but were forever changed, like this River Person herself. Every human who's fallen into the Underground has ultimately grown dependent on the magic there to stay alive. She also explains that this is why the River People have been anonymously petitioning to keep all humans out of the city; they can't explain why without revealing the existence of undead refugees, and fear the deaths of humans after visiting will begin the war once again. Frisk agrees to explain to Asgore the importance of keeping humans outside Ebott Town for as long as possible. The River Person gives Frisk another ration of mushrooms, some frank advice, and a bit of comfort in her own way, before seeing them off back home. Sans catches them on their way back and takes them home.
15. Aboveground, Frisk almost convinces Sans to take them straight to Asgore, but Undyne finds them first and insists on taking them straight to Toriel. Frisk asks if the human child is okay, and Undyne agrees to check in on them. Toriel is very upset with Frisk, and assigns Sans as a bodyguard/babysitter for them. When she tries to ask where Frisk went, Frisk panics and starts a battle with her, but ends it when Toriel refuses to fight. They break down and promise to take Sans with them if they have to go somewhere else again.
16. Frisk takes a long nap and starts figuring out what they can say to who to keep both the humans and the monsters safe, and they and Sans go to visit Alphys. They spend the day watching anime, but Undyne doesn't come home, and Alphys hadn't heard about Frisk running off, so Frisk just asks her to analyze one of their mushrooms. Undyne comes back while Alphys is getting actual work done, and reports to Frisk (and, incidentally, Sans) that the human child was hospitalized the night before. She demands more information, and Frisk explains that they heard rumors that visiting monsters kills humans, and Sans connects the dots.
17. Asgore's POV, slight step backwards in timeline. Asgore leaves for the human village without Undyne (who's out looking for Frisk), and finds out that the human child was hospitalized the night before for hypothermia—a mysteriously delayed reaction from his time on the mountain. Undyne rejoins him in time for them to talk with Mr. J and his ward, and apologize. The child forgives them immediately, but the incident raises questions about Frisk's presence—and their safety—in Ebott Town.
18. Back to Frisk. Sans begins babysitting, and Asgore comes by to tell them they're to stay in hiding until he can straighten things out with the humans...unless they want to go back. When he suggests that it might be a good idea, for Frisk's own safety and the sake of diplomacy, Frisk panics and starts having physical symptoms as well. Sans protects Frisk from Asgore using magic, and calms them down. Frisk then explains why leaving is a terrible idea, and the story they heard from the River People. They tell Toriel as well.
19. Frisk moves back into the Underground ruins with Toriel (and Sans), and starts deteriorating anyway. They have a number of visitors. Sans brings by decorations in preparation for the holidays, as well as some food for Frisk. They're disappointed when they discover that it's human food; their body is craving the magic of monster food. They discuss the future of monsters and humans, and Sans agrees to smuggle Frisk monster food every once in a while.
20. Frisk continues to get worse, and starts having nightmares regularly. They overhear Asgore and Toriel arguing over what to do about Frisk and the humans. Frisk gets a letter from the human child. When Undyne comes to visit, and after Frisk explains some things about Christmas, she discovers that not only is Frisk becoming emaciated, scars from their past battles with monsters are appearing all over their body.
21. Undyne, Toriel, Asgore, and Alphys start intense research to save Frisk. Frisk asks Alphys to stop, because it's having a negative effect on her mental health and Frisk is worried for her safety. They explain to the others what they've asked of her, but falls ill shortly thereafter, and doesn't recover. They're worried about upsetting their friends, but Sans asks them to look after themself instead.
22. Toriel's POV. Gerson determines that the problem is that Frisk's body is slowly dying in its delayed attempts to recover from Frisk's starvation and injuries in the Underground. Toriel and the others do their best to take it well. Sans takes her out to Grillby's in the evening, and tries to convince her that what's happening to Frisk isn't her fault. Sans has his own take on the situation, and admits that he thinks Frisk's fate might be inevitable.
23. Frisk becomes bedridden. Mettaton comes to visit with a plethora of (early) Christmas gifts. Frisk apologizes for the fact that they're dying, and says that they don't blame everyone any more than anyone blames them. After Mettaton leaves, Frisk finds a recording of his latest concert; he dedicated the last song of his set to Frisk. Sans asks Frisk to reset the timeline and try to do a speed-run through the Underground, so they don't have time to starve before leaving. Frisk doesn't want to lose the friendships they've made and the monsters' happiness, so they refuse.
24. Sans POV. Frisk falls down. Sans calls in everyone to try to save them, and is about to use a defibrillator that he and Alphys built when Alphys stops him, because she notices that Frisk is still breathing normally, despite heart failure. She argues that the best thing to do is wait for Frisk to wake up, and Sans turns on her, bringing up her past to use against her before Papyrus finally stops him. Alphys breaks down and Sans leaves, and Papyrus goes out to collect him, bring him home, and tell him off, and offer his own words of comfort.
25. Slight jump back in time, Undyne's point of view. Sans leaves Frisk's room, and Papyrus apologizes on Sans's behalf and goes after him. Asgore considers going to meet with the humans, and Toriel tells him he's in no condition to do so (and also to get out of her house). Toriel thanks a tearful Alphys for her help, and then Undyne carries her home, makes her tea, and shouts her into feeling better about things as best she can.
26. Mettaton POV. Late in the day of Frisk's collapse, he (finally) gets word of Frisk's state from Asgore, and is asked to attend a meeting with some of the monsters in charge of human research and negotiations. Frisk's state (basically, magically-extended refeeding syndrome and physical trauma, ending in coma and heart failure) is explained by Gerson and the scientists. The River Person Frisk spoke to explains what she knows about the extended effects of the monster lifestyle on (once-)living humans. The conversation comes around to the point that without doing tests, it's impossible to know the degree to which monster food is (indirectly) dangerous to humans. Sans points out that all they can do is share their knowledge with the humans, and Mettaton volunteers to do some online PR work, and asks about Frisk after the meeting.
27. Alphys POV, a week or two after Frisk's collapse. Alphys prepares to attend an unofficial test that Mr. J has volunteered for, to determine the extent of monster food's healing power and what happens afterwards. Undyne gives her a bit of a pep talk before she goes, and she meets the human child (Nat) and explains what the monster scientists are doing. Undyne introduces her to Mr. J and Nat as her girlfriend, and they chat until the tests start showing solid results. Alphys is spooked by being the one to spot Mr. J's returning (very mild) injury, and leaves the lab, only to be confronted by Sans, who surprisingly enough gives her a pretty satisfactory apology and promises to give her space unless and until she's comfortable being around him again.
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The Three Meanings Of E=mc^2, Einstein's Most Famous Equation
“Even masses at rest have an energy inherent to them. You've learned about all types of energies, including mechanical energy, chemical energy, electrical energy, as well as kinetic energy. These are all energies inherent to moving or reacting objects, and these forms of energy can be used to do work, such as run an engine, power a light bulb, or grind grain into flour. But even plain, old, regular mass at rest has energy inherent to it: a tremendous amount of energy. This carries with it a tremendous implication: that gravitation, which works between any two masses in the Universe in Newton's picture, should also work based off of energy, which is equivalent to mass via E = mc^2.”
When it comes to equations, few can lay claim to being ‘the most famous one’ of all time, but right up there is Einstein’s greatest and simplest: E = mc^2. Yet it doesn’t simply state that mass and energy are equivalent, or that the relationship between them is given by the constant c^2. Sure, it says those things, but there’s also a vital physical meaning behind them. Understanding E = mc^2 has led to a variety of tremendous discoveries and breakthroughs, from nuclear power to the creation of new particles in particle accelerators. It even led directly to discovering that Newtonian gravity was theoretically unsound, ushering in the era of General Relativity, as well as the fact that any theory of gravity needs to include a gravitational redshift/blueshift.
How did it all come about? Find out the three meanings of Einstein’s most famous equation, and what it means for our Universe.
