I've always been a sucker for mathematical SF, like Rudy Rucker's

White Lightwith its exploration of literal infinities, or the kids learning to enter the fourth dimension in Henry Kuttner's classic short "Mimsy Were the Borogoves" (homaged in Greg Bear's "Tangents"). So I couldn't resist Ian Stewart's latest bookFlatterland: Like Flatland, Only More So.Edwin A. Abbott's original fable

Flatland: A Romance of Many Dimensionsappeared in 1888 and ever since has fascinated mathematicians and SF authors who can't resist adding sequels. Natives of two-dimensional Flatland are simple geometrical figures. Our narrator is a gentlemanly square subtly named A. Square. His servants are irregular triangles, and (this being before feminism) women are mere lines.When A. Square is visited by a sphere from 3-space, he can "see" it only as a 2D cross-section, a circle. Nevertheless he catches on quite rapidly, acquires a new perspective on his flat world, rushes around preaching the gospel of higher-dimensioned space, and naturally gets locked up as a loony.

The mathematician C.H. Hinton, who wrote lots about the fourth dimension, followed Abbott with

An Episode of Flatland(1907). Here the Flatlanders discover they live on the rim of an enormous circle, their flat planet Astria. So they can hike and sail all the way around the circumference to attack their ancestral enemies from behind. This melodramatic plot, as Martin Gardner reported, is a bit two-dimensional.Dionys Burger's sequel

Sphereland(1965), is told by A. Square's hexagonal grandson and features adventures in higher and lower spaces, building up to the difficult idea of a expanding three-dimensioned universe. As in the originalFlatlandand all too much early SF, the whimsical story is only there as a vehicle for educational ideas.Computer scientist A.K. Dewdney took a practical, engineering view of Flatland and had lots more fun in his non-fact book

The Planiverse(1984). His "Arde" is the first 2D world with a plausible physics and ecology. We learn that the Arde building trade has a hard time of it since a nail driven right through a 2D plank automatically separates it into two. Electrical circuits are hugely complex because without a third dimension to separate them, all crossed wires short-circuit at the crossing point. Balloons are easy, though, since any loop of airtight string encloses a volume – that is, an area – of 2D gas. Dewdney even designed a workable flat steam-engine.Ian Stewart's contribution to this strange little genre,

Flatterland, typically begins with awful puns. (This is the man who presents geometrical problems via a family of worms with names like Anne-Lida and Wermentrude.) Clearly A. Square is Albert Square fromEastenders, with other males of the family being Grosvenor, Berkeley and Leicester. Since the women are lines, they just have to be called Victoria and Jubilee.Stewart duly hauls precocious young Victoria Line, the original Square's great-great-granddaughter, through a variety of physical and mathematical spaces that mostly hadn't been thought of when the original

Flatlandwas written. All very educational, I'm afraid, but great fun in a mind-bending way.By coincidence, another partly mathematical story just arrived and is also a sequel: Donald Kingsbury's

Psychohistorical Crisis(2001 USA). To practised SF readers the setting seems awfully familiar. The Galactic Empire has fallen, as predicted through subtle "psychohistory" maths by the unnamed Founder who tried to cut short the resulting 30,000 years of barbarism by setting up a foundation (not actually called that) of scientific excellence on the faraway world Faraway ...Yes, it's a sequel to Isaac Asimov's original Foundation trilogy (ignoring his later, inferior additions), but unauthorized by the Asimov estate. Hence all the names had to be changed. For example, that clownish-looking supervillain the Mule subtly becomes "Cloun-the-Stubborn".

Kingsbury, a maths professor, wants to argue with the way Asimov handled psychohistory. Pop-science lectures by the Founder himself develop the basic idea of psychohistorical prediction, while the overall storyline respectfully puts the boot into Asimov for getting large-scale implications wrong.

This sounds distinctly heavy, but it's actually an enjoyable book, better crafted and written than the fifty-year-old stories which it discusses. Asimov's characterization was always, dare I say it, a bit flat. Sequels by other hands are usually dismal affairs:

Psychohistorical Crisiswas a pleasant surprise. And with a spice of mathematics, too.

David Langford is planning his unauthorized sequel "Harry Potter and Fermat's Last Theorem" which – argh! ouch! I was joking!