Since its publication in 1958, *Fantasia Mathematica* has been the standard by which the few other anthologies of mathematical fiction have been judged. It is our good fortune that, after a long hiatus, both *Fantasia* and its sequel *The Mathematical Magpie* are back in print. However, now that more than forty years have passed, the stories that Random House's legendary editor Clifton Fadiman collected then seem naive and dated. Many of the stories have little literary merit of their own, and the mathematical topics tend to be Möbius bands and four dimensional cubes, topics that now seem somewhat passé. It is time for a new anthology.

William Frucht's *Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems and Musings*, is a worthy, though imperfect, contender to succeed *Fantasia* as the standard of the genre. Its 31 entries, including five poems, are mostly modern and, with only one exception, entirely distinct from *Fantasia*. The writing is consistently excellent.

The first entry, "The Form of Space" by Italo Calvino, is the sad tale of a lover trapped on a trajectory parallel to that of his heart-throb, and also parallel to his rival for her affections. Regular readers of Martin Gardner will recognize Calvino as a member of Oulipo, writers who often incorporate mathematical themes and techniques. Several of Calvino's other works also have a strong mathematical content. "The Form of Space" forms a kind of prose reply to the poem "Mathematical Love" by Andrew Marvell. A single verse of that poem appears in *Fantasia*, and the poem in its entirety is number 15 in *Imaginary Numbers*.

Jorge Luis Borges' "The Library of Babel" is entry 26. The main character is an unnamed librarian in a library that contains all possible books. It is a large library, but not infinite. Borges, one of the Twentieth Century's finest writers, often used mathematical themes, including reflections, patterns, recursion and duality, in his short stories. Some of his other stories are even more explicitly mathematical than "The Library of Babel", but perhaps Frucht picked this one because it, too, mirrors a story in *Fantasia*, "The Universal Library" by Kurd Lasswitz.

The excerpt from Edwin Abbott Abbott's 1884 classic *Flatland* (item 16) and that from A. K. Dewdney's *The Planiverse* (item 16) go very nicely together. One wonders why the editor chose to separate them with a poem.

Stanislaw Lem's comic "The Third Sally, or The Dragons of Probability" is item 13. Lem is the only writer who appears twice. Entry 18, "The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet" is a kind of rewrite of George Gamow's "An Infinity of Guests", which appears in *Fantasia*. Rudy Rucker, in "A New Golden Age", the second short story of the anthology, describes what might happen if mathematics became part of popular culture, like music or art. Lem and Rucker are both mathematicians in their own rights. Other contributors whose names are familiar in mathematical circles include Lewis Carroll, Raymond Smullyan, Piet Hein, Douglas Hofstadter, Edwin Abbott Abbott, A. K. Dewdney and Martin Gardner, among others.

It is not fair to criticize an anthology too harshly for what the editor has left out. The process of winnowing ten thousand pages of items each of which could reasonably have been included is surely an agonizing process. It is easy to say that we would have included a different Borges story or a different Calvino story, or an excerpt from Müsil's *Young Törless* instead of from Zamayatin's *We*, or even some of J. J. Sylvester's mushy poetry. Surely the editor considered all these, and when he had to make his decision, he made it.

It is, however, fair to criticize what he did include. Philip K. Dick's "The Golden Man" (item 8), and J. G. Ballard's "The Garden of Time" (item 31) are both excellent stories, but their mathematical content is elusive. Particularly difficult to discern are the reasons for including the excerpts from an old attempt to get a computer to generate a literary text: "The Policeman's Beard Is Half-Constructed," by Racter. Two of the four excerpts are (in their entirety) "A tree or shrub can grow and bloom. I am always the same. But I am clever." and "Slide and tumble and fall among The dead. Here and there Will be found a utensil."

Perhaps the reasons that some of the items were included would be clearer if the editor had written introductions to more of the pieces. Only the excerpts from books, *Flatland, The Planiverse* and *We* had editor's introductions. Some of the others could have used them as well. The editor could have told us why he included *The Policeman's Beard Is Half-Constructed*, and which of Calvino's other works are also mathematical. He could have mentioned the mathematical careers of Lem and Smullyan and the others.

In his "Acknowledgements" the editor suggests that there may be a second edition someday. Perhaps he will address these small complaints.

In the mean time, *Imaginary Numbers* picks up where *Fantasia Mathematica* and *The Mathematical Magpie* left off. Mathematicians who enjoy modern fiction will almost certainly enjoy Frucht's *Imaginary Numbers*.

**Also Mentioned:**

Fantasia Mathematica: Being a Set of Stories, Together With a Group of Oddments and Diversions, All Drawn from the Universe of Mathematics, Clifton Fadiman, ed. Copernicus, 1997. Softcover, 298 pages, $19.95. ISBN 0387949313.

The Mathematical Magpie: Being more stories, mainly transcendental, plus subjects of essays, rhymes, music, anecdotes, epigrams, and other prime oddments and diversions, rational and irrational, all derived from the infinite domain of mathematics, Clifton Fadiman, ed. Copernicus, 1997. Softcover, 303 pages, $19.95. ISBN 038794950X.

Ed Sandifer (sandifer@wcsu.ctstateu.edu) is professor of mathematics at Western Connecticut State University and an enthusiastic fan of Leonhard Euler.