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:

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".