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3-adica (2018)
Greg Egan
(click on names to see more mathematical fiction by the same author)

Sentient characters in a horrific video game combining Jack the Ripper and vampires seek to escape to another game called 3-adica where things are strange but peaceful.

This is one of a series of stories by Egan in which role playing games created for the entertainment of paying clients are populated by virtual characters built out from the minds of a collection of human "contributors". Unbeknownst to either the creators of the game or the clients playing it, some of those characters have become sentient and begun remembering the real world. Fearing that the game would be rebooted and that they would be deleted if this was discovered, they secretly explore their virtual world and plan to escape to one where they can be free.

In this story (Egan calls it a "novella"), Sagreda and Mathis are attempting to get to 3-adica, a game which takes place in a world based upon the 3-adic numbers. Although the couple's plan does not proceed without any problems, I hope it is not too much of a spoiler for me to say that two of the sentient characters do escape to 3-adica and so we get to see something of what life is like there.

The story 3-adica is interesting even if you do not know about the math behind it, but it is even more enjoyable if you do because then you can appreciate Egan's attempt to describe beings living in a 3-adic universe. Most people who have earned a degree in mathematics are familiar with the p-adic numbers, but they are little known to the rest of the world. (As far as I know, this is the first work of fiction to even mention them. Please correct me if I'm mistaken!) So, let me say something about the p-adic numbers, which are defined for any fixed choice of a prime number p. In many ways, they are not at all strange. As a set and in terms of their arithmetic, p-adic numbers are just like the usual rational numbers. That is, they are whole numbers and fractions of whole numbers, and the rules for adding or multiplying them are exactly the same as what you learned in school. The thing that is different about them is their topology. More specifically, the metric which determines the distance between any given pair of numbers is different. This is where the p-adic numbers begin seeming surrealistic. Here is how Sagreda explains the notion of distance in the 3-adic numbers (the p-adic numbers in the case p=3) to other characters who are helping them with their quest:

(quoted from 3-adica)

“It's taking us to a world where the distances between numbers aren't the same as they are here.”

Lucy frowned, but her expression was more intrigued than dismissive.

“Here, you can put all the numbers on a line,” Sagreda said. “Like the house numbers on a street. And the distance between two houses is just the difference between their numbers: number twelve is two houses down from number ten . . . most of the time.” Whatever the historical truth, this version of Victorian London hadn't made up its mind whether to number houses consecutively along each side of the street, or to adopt the even/odd rule that was more familiar to Sagreda's contributors.

“So you're going to a world where the houses are higgledy-piggledy?” Lucy guessed.

“Maybe, though that doesn't quite cover it.” Sagreda walked over to the desk, took a sheet of writing paper and started scrawling ovals in ink. “In 3-adica, the numbers are like eggs in a sparrow's nest. Zero, one, and two are all in the same nest, and the distance between any pair of them is exactly one.”

“From one to two is one,” Lucy said. “But from nothing to two is . . . also one?”

“Exactly,” Sagreda conf irmed. “The laws of arithmetic haven't changed: two minus zero is still two, not one.But the laws of geometry aren't the same, and the distance is no longer the difference.”

“But where's three?” Lucy demanded. “Where's seventy-three?”

“Each egg I've drawn,” Sagreda said, “is really a nest of its own. The zero-egg is a nest that contains zero, three, and six. The one-egg is a nest that contains one, four, and seven. The two-egg is a nest that contains two, five, and eight.” She scribbled in the new numbers.

“I can see what you've written clear enough,” Lucy acknowledged, “but I don't know what it means.”

“To be in a smaller nest with a number puts you closer to it,” Sagreda explained. “The distance between zero and one is one, because that's the size of the smallest nest they're both in, but the distance between zero and three is smaller, because they share a smaller nest. In fact, the distance between zero and three is one third, as is the distance between five and eight, or four and seven.”

“And you keep on with that nonsense?” Lucy asked.

Indeed, many students encountering the p-adic numbers react as Lucy does, thinking that it is nonsense. Others love this new metric on the set of rational numbers precisely because it is strange. But, there is a value to the p-adics beyond their weirdness. The irrational numbers (which together with the rational numbers form what we call "the real numbers") are actually built out of the topology of the rational numbers. Since their topology is different, following the same procedure using the 3-adic numbers instead of the rational numbers leads to different results. As Egan explains in this non-fictional essay designed to accompany 3-adica, there would not be a square root of 2 in 3-adica. The alternate topology of the p-adic numbers is useful for number theorists, helping them to prove new theorems about the rational numbers and their relationship to the irrational numbers.

This story was originally published in the September/October 2018 issue of Asimov's Magazine and was republished in the Egan anthology Instantiation. At present, a PDF of the story can be downloaded for free from this URL. It's sequel, Instantiation, is also a work of "mathematical fiction".

More information about this work can be found at
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Works Similar to 3-adica
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. The Clockwork Rocket [Orthogonal Book One] by Greg Egan
  2. Border Guards by Greg Egan
  3. Axiom of Dreams by Arula Ratnakar
  4. The Adventures of Topology Man by Alex Kasman
  5. Instantiation by Greg Egan
  6. Mathematical Revelations by Helen De Cruz
  7. Luminous by Greg Egan
  8. Schild's Ladder by Greg Egan
  9. The Arrows of Time [Orthogonal Book Three] by Greg Egan
  10. Dark Integers by Greg Egan
Ratings for 3-adica:
RatingsHave you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on this Webpage.
Mathematical Content:
4/5 (2 votes)
Literary Quality:
4/5 (2 votes)

GenreScience Fiction,
TopicGeometry/Topology/Trigonometry, Algebra/Arithmetic/Number Theory, Real Mathematics,
MediumShort Stories, Available Free Online,

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Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.

(Maintained by Alex Kasman, College of Charleston)