"Our opponent is an alien starship packed with atomic bombs. We have a protractor."
Anathem is a novel by Neal Stephenson about a world with a monastic organization which, instead of studying God, studies science, mathematics, and philosophy. Members of a mathic order, or "avout", take vows to spend either 1, 10, 100, or 1000 years in complete isolation from the outside world in monasteries called "maths". The only exceptions to this isolation come when they are called out of the maths to help the sęcular world solve some crisis, and during the "sacks", when the sęcular world assaults the mathic orders for being too powerful. Like many of Neal Stephenson's other books he Shows His Work with several appendices of Socratic dialogues and at least a cursory understanding of geometry and platonic philosophy being helpful. Like Cryptonomicon, the book isn't really about the plot; just as Cryptonomicon was really about cryptology and The Baroque Cycle was really about modern economics, Anathem is really about Platonic epistemology. Applied and weaponized Platonic epistemology.
Aerith and Bob: Erasmas, Orolo, Arsibalt, and Jules Verne, who is descended from our Earth.
After the End: The calendar sets the year 0 as the "Terrible Events," a near-extinction level nuclear/nanotechnological war. This was more than 3500 years ago. See also Scavenger World.
Bizarre Alien Biology: The aliens can marginally breathe Arbran air and cannot digest the food, because they are from other universes where the atoms are shaped a little differently, making them all chemically incompatible with one another.
Book Ends: We are introduced to Erasmas as he is serving as "amanuensis" (recorder, or maybe witness) for a conversation between Fraa Orolo and an artisan from outside. The end has him acting as amanuensis again, in a very different context.
Call a Rabbit a "Smeerp": Many technologies and cultural references have been renamed - for instance, cell phones are "jeejahs", Arbre's version of the net is called the "ret" (starting from "reticule" instead of "network"), scientists are called "theors" and martial arts are called "vale-lore" or "vlor". Quite thoroughly justified, seeing as Arbre is within a chain of Hylaean Theoric Worlds. These are similar to one another by definition, and one of them contains Earth, with rabbits and Socrates and Occam's Razor, as opposed to smeerps and Thelenes and Gardan's Steelyard.
Call a Smeerp a "Rabbit": With living creatures, as Stephenson says in the introduction that he has, for instance, chosen to refer to Arbran vegetables which are similar but not identical to carrots simply as "carrots."
Cool and Unusual Punishment: A common punishment is to have avout memorize chapters in "The Book." Each chapter is more challenging than the last. While one chapter is simply memorizing the numbers in Pi, others are subtly wrong in their internal logic (nursery rhymes that don't quite rhyme correctly, for example), making them difficult and taxing to memorize. Legend says only three people ever completed the twelfth chapter, and they all ended up completely insane. The punishment is purely optional, however. Avout can choose to leave the mathic world.
Info Dump: Like all Neal Stephenson novels, Anathem contains many of them.
Kangaroo Court: Erasmus is forced to partake in a public interview much like a trial to explain and defend his actions. His interlocutor is heavily against him and poses all of his questions to in an effort to make Erasmus look foolish. Erasmus immediately realizes that he'll have to draw on all his powers of argumentation to get out of the interview unscathed.
Lost Superweapon: World Burner and Everything Killers, oh and the ability to manipulate the multiverse with nothing more than a little chanting.
Mind Screw: Plotwise, the book is fairly straightforward, until Erasmas and Fraa Jad board the alien spaceship, at which point multiple timelines/universes get involved and it's anyone's guess what the hell is going on. It gets better by the conclusion, though.
Orion Drive: a spacecraft that travels between different dimensions and uses an Orion-style propulsion system. This ship, the Daban Urnud, is discovered by observing the nuclear explosions used to modify its orbit.
Persecuted Intellectuals: Mathematicians (called "Avout") are confined to monasteries (called "Maths"), and only allowed contact with the outside world once per year, decade, century or millennium. On three occasions the Maths were invaded because the Avout invented technologies considered too dangerous ("new matter", genetic engineering, and magic).
Really 700 Years Old: Fraa Jad and, it is hinted, at least some of the Thousanders of the Three Inviolate maths, due to a praxis (read: technology) that allows them to be able to live around their stores of nuclear waste. Involves quantum immortality.
Recursive Canon: In addition to the Literary Agent Hypothesis being in use, earlier chapters of the book are discovered by concent officials at one point in the story, and they directly quote parts of the story as evidence in a hearing, which, of course, is written down in the book...
Red Shirt: The Valers avert this in Mahsht. Bonus points for doing it while actually wearing red shirts.
Rock Beats Laser: (see page quote). Works until both sides break out the femtotechnology, planet burners, and everything killers.
Scavenger World: Played with and justified. Arbre has been at a relatively high level of technological development (on and off) for at least four thousand years. Background events suggest that the natural resources of the planet are almost completely depleted. The cars are powered by a processed fuel that originates in genetically engineered trees, and strip-mining abandoned cities for raw materials is a major industry. Several key pieces of high technology are built from designs that are hundreds or thousands of years old. This includes the space suits.
Shown Their Work: a given for Neal Stephenson novels. Three appendices provide intricately-constructed mathematical underpinnings for plot events or, occasionally, just some exposition. The credits for the book are voluminous and kept online for ease of update.
Strawman Political: Poor Fraa Lodoghir, who exists for the sole purpose of asking stupid questions on behalf of the Procian school of thought. Perhaps the first recorded instance of a Strawman Quantum Theorist. Although his final conversation with Erasmas suggests this was Obfuscating Stupidity and he knew precisely what Fraa Jad was capable of and did.
Translation Convention: The book is written by its protagonist/narrator in Orth; it's just presented to us in English for our benefit. Nearly all dialogue is actually in Orth, with some small parts in Fluccish, though both languages are presented to us in English. The only actual Orth we get to see consists of character/place names. Also, when one character who speaks French shows up, his French is written in a Funetik Aksent, as the narrator has no idea how French is supposed to be spelled.
Unreliable Narrator: It's not that he's lying, it's just that the records on hand prove him wrong...
Warrior Monk: Lio and the avout of the Ringing Vale, though they're monks of logic and science rather than religion. The science part actually comes in pretty handy when they're called on to do a special ops mission in space.
They're the only avout in the world who make a study of athleticism and martial arts.