The Annotated Flat-land: A Romance of Many Dimensions , Edwin A. Abbott , Introduction and Notes by Ian Stewart Perseus, Cambridge, Mass., 2002. $30.00 (239 pp.). ISBN 0-7382-0541-9

In his 1885 classic, *Flatland* (Roberts Brothers), Edwin A. Abbott was intent on making his three-dimensional readers understand that a fourth dimension may be real despite limitations of human intuition. Abbott wrote about a world peopled by two-dimensional creatures that cannot imagine a third dimension. One of these creatures, A. Square, is visited by a sphere that moves him into the third dimension, where he can view Flatland from “above.” When A. Square returns to Flatland and talks about the reality of the third dimension, he is imprisoned for life.

Today the fourth dimension may be familiar, but not many of us have an intuition for the kinds of spaces encountered in physics and mathematics. Ian Stewart’s *Flatterland* begins where Abbott left off. Vikki, a teenage descendant of A. Square (“A” now stands for Albert) finds a copy of the banned book *Flatland* and scans it into her computer before her father destroys the book. Following instructions left by Albert, she summons the Space Hopper, who guides her out of Flatland and introduces her to some beautiful worlds in mathematics and physics.

A worthy successor to Abbott, Stewart served as the Gresham Professor of Geometry at the University of Warwick, England, in a long line of distinguished scholars that includes Robert Hooke. For furthering the public understanding of science, he received awards from the Royal Society and from the American Association for the Advancement of Science.

*Flatterland* contains many surprising connections and “aha” moments. On one of her first adventures, Vikki finds that unit cubes in *n* dimensions can be used to understand a class of Richard Hamming’s codes for correcting errors in a sequence of binary digits. On another visit, she uses color as a fourth dimension to see how a knot in three dimensions can be untied in a fourth spatial dimension.

Vikki also finds out about self-similar fractals and their fractional dimensions. As the Space Hopper explains, a fractal’s dimension is determined by how many copies of the fractal are required to make a larger version of the same fractal. For example, the Sierpinski gasket is constructed by starting with a triangle, cutting the largest possible hole in it that is a similar upside-down triangle, and continuing in this way. Because the first hole has the size and shape of a half-size gasket, you can make a gasket with length-2 sides by joining, corner to corner, 3 gaskets with length-1 sides. The number 3 defines the gasket’s fractal dimension as log3 / log2. Similarly, in a plane, you need four length-1 triangles to make a length-2 triangle, so the plane’s dimension is log4 / log2 = 2.

It is amazing how clearly Stewart explains normally obscure geometrical concepts. Teachers at any level would gain by studying his pedagogical techniques. A masterly example is his treatment of projective planes, which begins by asking how seven varieties of grapes can be arranged in plots. Each plot has exactly three varieties, any two plots have exactly one variety in common, and any two varieties lie in exactly one common plot. By drawing a graph that shows the grape varieties as points and the plots as connecting segments, Stewart devises a beautifully clear explanation of the 7-point projective plane. He then proceeds to the 13-point projective plane, and to some yet unanswered questions about projective planes.

In a place called Platterland, Vikki encounters non-Euclidean geometry, and in Cat Country, she learns about Schrödinger’s poor cat and about an electron’s dual wave-particle nature. The narrative slightly missteps here. An electron hits a metal plate, which subsequently emits a photon, a reversal of photon and electron roles in the photoelectric effect. But there follows a good discussion of macroscopic decoherence times as related to Schrödinger’s cat. Later in the book, a few other particle-physics inaccuracies occur, but the excellence of the exposition far outweighs those minor flaws.

After visiting the Paradox twins to learn about inertial and accelerated frames in relativity, Vikki enters the amazing domain of the Hawk King, which contains not only black holes and white holes, but wormholes that can serve as time machines. To use a time machine, Vikki and the Space Hopper have to enter a black hole. To emerge, they use a white hole and a time machine that traverse a pair of closed timelike curves! Stewart is at his best in describing closed timelike curves and how to avoid the logical paradoxes their existence appears to imply.

Vikki goes on to learn about big-bang nucleosynthesis and quantum foam. The Space Hopper explains, “That’s quantum foam—particles springing into and out of existence, creating space and time along with themselves. Try some, it’s quite tasty” (page 282). In teaching Vikki about supersymmetry, the Space Hopper offers an excellent discussion of the viable 10-dimensional superstring theories and why they may be different views of a single 11-dimensional “M-theory.”

One may recall that Abbott’s *Flatland* portrays women as line segments and men as polygons of higher status. In Stewart’s *Flatterland*, Vikki is a line in Flatland before she goes to other spaces. I won’t give away the clever ending, but suffice it to say that Stewart portrays women as more than merely equal to men in ability and social status.

In *The Annotated Flatland*, Stewart augments *Flatland* with an introduction and extensive notes, both to explain the historical context and to update and expand selected topics. The notes are wonderfully clear and succinct. For example, in slightly more than half a page, a note on π summarizes Archimedes’s use of polygons on a circle to bracket the value of π, mentions the 1761 proof that π is irrational and the 1882 proof that π is transcendental, and defines irrational and transcendental numbers. A longer note discusses Euclidean construction of regular polygons (those familiar Flatland shapes), and even mentions Gauss’s proof of when such constructions are possible. In a note on cellular automata, Stewart observes that if Flatlanders had cellular-automata brains, their intelligence would not be limited by their dimensionality. I found the notes in *The Annotated Flatland* to be gripping and enjoyable.

*Flatterland* and *The Annotated Flatland* are instructive, stimulating, and great fun. I highly recommend both books to physicists and students who enjoy having fun with mathematics.

**Leonard Parker**, *a professor of physics at the University of Wisconsin–Milwaukee, does research in gravitational physics. His current work is on the relationship of quantized fields in curved space-time to the recently observed acceleration of the expansion of the universe.*