The Planiverse
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The Planiverse (ISBN 0-387-98916-1) is a novel by A. K. Dewdney, written in 1984.
In the spirit of Edwin Abbott Abbott's Flatland, Dewdney and his computer science students designed a vertical 2D world (i.e. East-West and Up-Down, no N-S) and considered the issues of biology and society for the inhabitants.
To their surprise, they find their artificial 2D universe has somehow accidentally become a means of communication with an actual 2D world - Arde. They make a sort of "telepathic" contact with Yendred, a highly philosophical Ardean (or Nsana, as they call themselves), as he begins a journey across the single continent Ajem Kollosh to learn more about a mysterious philosophy the inhabitants of his destination have.
The students and narrator communicate with Yendred by typing on the keyboard, and Yendred describes how he "feels" their thoughts in his head. For Yendred's replies, he thinks an answer, and it appears on the computer's printout.
Written as a travelogue, Yendred crosses the world to reveal its features, explaining to the students diverse topics such as the politics, geography, construction (all houses are underground, for example, so as not to impede movement), tools (nails are useless for attaching two objects, tape and glue are used instead), biology (there is no digestive tract in most Ardean creatures, because of the danger of splitting into two, but evolution devised a solution), astronomy, and even games (such as Alak), all designed for 2D. The book is much more practical and has less social commentary than Flatland.