In 1977, Dewdney was inspired by an allegory of a two-dimensional universe, and decided to expand upon the physics and chemistry of such a universe. He published a short monograph in 1979 called Two-Dimensional Science and Technology. This was reviewed by Martin Gardner in his July 1980 "Mathematical Games" column in Scientific American, and shortly after this, all copies of the monograph were sold out. In 1981, following the success of the monograph, Dewdney published A Symposium on Two-Dimensional Science and Technology, which contained suggestions for how a two-dimensional universe would work from scientists and non-scientists on varied subjects. Dewdney wrote The Planiverse as a frame story in which to display the scientific and technical features from these previous works, as well as an allegory for his search for a reality deeper than that of scientific enquiry, and his subsequent conversion to Sufiism.
In the spirit of Edwin Abbott Abbott's Flatland, Dewdney and his computer science students simulate a two-dimensional world with a complex ecosystem. 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 "YNDRD," referred to by the students as Yendred, a highly philosophical Ardean, as he begins a journey across the western half, Punizla, of the single continent Ajem Kollosh to learn more about the spiritual beliefs of the people of the East, Vanizla. Yendred mistakes Dewdney's class for 'spirits' and takes great interest in communicating with them. The students and narrator communicate with Yendred by typing on the keyboard; Yendred's answers appear on the computer's printout. The name Yendred (or "Yendwed", as pronounced by one of the students, who has a speech impediment) is simply "Dewdney" reversed.
Written as a travelogue, Yendred's journey through the West takes him through several cities. He visits the Punizlan Institute for Technology and Science, where Arde's technology is explored in great detail. For example, all houses are underground, so as not to be demolished by the periodic 2D rivers; nails are useless for attaching two objects, so tape and glue are used instead; most Ardean creatures cannot have deuterostomic digestive tracts since they would split into two; even games such as Go have one-dimensional Alak analogues. An appendix explains various other aspects of two-dimensional science and technology which could not fit into the main story.
The underlying allegory culminates in Yendred's arrival at the watershed of the continent and the planet's only building above ground, where he at last finds Drabk, an Ardean who professes 'knowledge of the Beyond,' and teaches Yendred to fly. Yendred finds that to keep contact with Earth is no longer of benefit, and contact with Arde is lost.
Kirkus Reviews considered it "an ingenious intellectual exercise—amusing, edifying, sometimes tedious"  At Tor.com, Jason Shiga found it to be a "tour de force followup" to Flatland, and found the appendix to be the "most impressive section" of the book.
- Begley, Sharon. 1982. "Life in Two Dimensions." Newsweek. January 18, pp. 84–85.
- Dewdney, A.K. 1979. "Exploring the Planiverse." Journal of Recreational Mathematics. 12:16–20.
- Dewdney, A.K. 2000. "The Planiverse Project: Then and Now." The Mathematical Intelligencer. 22:46–51.
- Gardner, Martin. 1980/2001. "The Wonders of a Planiverse." Scientific American, July 1980; reprinted with appendix in The Colossal Book of Mathematics (New York: Norton).
- Sandberg-Diment, Erik. 1984. "Review of Dewdney 1984/2001". New York Times, November 6.
- Author's bibliography
- Kontrol An online action game and 2D universe simulation inspired by The Planiverse
- "Allegory through the computing class: Sufism in The Planiverse by A. K. Dewdney". Sufi. 1991. pp. 26–30.
- THE PLANIVERSE: Computer Contact with a Two-Dimensional World, reviewed at Kirkus Reviews; published March 16, 1984; retrieved June 5, 2017
- Revealing Demon: Volume 3, and Five Other Sci-Fi Books About Math, by Jason Shiga; published September 15, 2016; retrieved June 5, 2017