When I found out I was going to be on a panel discussing mathematical science fiction I had that moment of panic. I have loved mathematics since I was an early teen, and science fiction since I was about six, but I have no special knowledge about the convergence of the two.
Mathematics is a subject that most science fiction avoids, or when it does use it gets it wrong. For example, I was always a little galled when comic books and now film make the chronic mistake of using the word "dimension" when what they mean is "parallel universe." It is only a small rearrangement of the universe as we know it to say there is another universe parallel to ours. I am not even sure what it would mean to say there is another dimension with a different set of people living in it.
As I grew up I savored TV science fiction that dealt with mathematical ideas more for their rarity than their quality. The Twilight Zone had one episode, an adaptation of Richard Matheson's "Little Girl Lost," in which a little girl had rolled over in bed and fallen into an intersecting (hence not quite parallel) universe. Two Outer Limits episodes dealt with intersecting universes, "The Borderland" and "The Production and Decay of Strange Particles."
I know of five anthologies of mathematical stories. These are all anthologies of stories about mathematics, and containing science fiction stories. I do not guarantee every story is science fiction.
1953—Science Fiction Adventures in Dimension, edited by Groff Conklin. Any 1950s collection of science fiction edited by Groff Conklin is going to be a great book to read. Add to it mathematical subject matter and you have a real find. Unfortunately it is mostly time travel and/or alternate universe stories. It could have more diversity.
1958—Fantasia Mathematica, edited by Clifton Fadiman. This is a mix of science fiction, poetry, and essays on a mathematical theme.and is the classic anthology of mathematical science fiction. Nobody talks about mathematical science fiction without mentioning it. Seen today the stories may be a little simplistic, but everything is fun. Fadiman gives everything a playful approach.
1962—The Mathematical Magpie, edited by Clifton Fadiman. Fadiman followed up his classic anthology with a second joyous helping. Again it is mostly stories (many of which are science fiction) with essays and poetry mixed in. It is just slightly less of a classic than Fantasia Mathematica.
1987—The Mathenauts, edited by Rudy Rucker. Rucker gets a good six sentences into the book before mentioning the two Fadiman anthologies. His book is all science fiction stories. Basically it is a collection of the best mathematics science fiction stories from the intervening quarter century. It is not as playful as the Fadiman books, but it has the goods.
1999—Imaginary Numbers, edited by William Frucht. This is apparently much closer to the spirit of the Fadiman anthologies with a collection of stories, poems, and essays.
While I am listing related books with mathematical science fiction, I should point out that there is a classic fiction work of science fiction with several authors following up on it. In 1884 Edwin A. Abbott published Flatland: A Romance of Many Dimensions. This rather unique book envisioned a two-dimensional world encountering our world of three dimensions. In the process it tells by analogy what it would be like for us to become aware of a world of four dimensions. There have been several follow-up stories by other authors. Three of book length have been Sphereland (1965) by Dionys Burger, Planiverse (1984) by A. K. Dewdney, and Flatterland (2001) by Ian Stewart. Rudy Rucker's anthology Mathenauts contains his own short story "Message Found in a Copy of Flatland."
In addition to the above, among the most popular mathematical science fiction works (of varying lengths) I know of are:
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