*Flatland*, by Edwin A. Abbott. Originally published in 1884. Free. Google Play. Project Gutenberg (numerous formats).

*Guest review by Jaime Green*

When I was about twelve years old, I saw the shadow of a four-dimensional cube. It was in a special exhibit at the New York Hall of Science on geometry and topology, the math of surfaces and shapes. And for some reason, it's stuck with me. A three-dimensional model built of angles and lines, square and trapezoids, like a metastasized cube. It was hung in mid-air, just above my tween eye level.

I can't quite picture the three-dimensional shadow now. So many years have passed. It's also the sort of thing that hovers just beyond your understanding, so it's fitting that the memory flits out of reach, too. Much of trying to understand advanced math can be like that. You understand the analogy, or you can read the equation, but you can't really picture the concept. You know it's there. Maybe if you sit very still, maybe if you hold your breath, something will flash into your mind. Imagine a four-dimensional sphere. Just try.

The best pleasures of Edwin A. Abbott's classic book, *Flatland*,
are the moments in which Abbott draws you a little closer to that flash of
insight, impossible as, for a layperson, it may be. Through analogy and
explanation, Abbott makes you feel like you are a hair's breadth away from
understanding, from being able to see in four dimensions. It's a book about
math and yet, even despite its short length, it's much more.

Written in 1884 by an English schoolteacher and theologan, *Flatland
*is by pretty equal division half satire of Victorian society and half
philosophical exploration of geometric dimensions. It was not a huge success in
its initial publication, but in the years since it has come to be a beloved
classic, the kind of book--if my experience is any indication--math teachers
recommend to their eager students. Other authors have written imitations and
sequels – *Sphereland, The Plainiverse*, and *Flatterland*, among
others – and there have been several filmed adaptations. Now it is available a free ebook, with Abbott's lovely schematic
illustrations intact.

The narrator of *Flatland *is a square, who lives in an eponymous two-dimensional world with other geometric shapes. The
first part of the book, called “The World,” is taken up by a detailed
evocation of the social structure of Flatland. A shape-person's
number of sides and symmetry corresponds with his advancement. Thieves and vagabonds are isosceles triangles.
Lawyers and doctors are squares. The aristocracy is made of many-sided
polygons, and at the top of the social hierarchy, the priests are all circles.

Well, those are the men. All women in Flatland are lines. As they have no insides, they have no capacity for analytical thought or, it seems, any memory or long-term thinking at all. Their pointed ends are dangerous and nearly invisible, and so women must announce their presence by a gentle waggling to make themselves visible when seen end-on, or by the constant soft utterance of a "peace-cry."

As I read, I couldn't decide whether this was satire or some really hardcore misogyny and social conservatism. Across a gulf of more than a century, I had trouble divining Abbott's authorial intent through his writing. The Internet tells me Abbott was a social reformer, and that the square's endorsement of Flatland's absurdly rigid social strictures is satire. And yet the square, our narrator and guide, is a sensitive, inquisitive, and responsible shape. Without the cues of a silly or maleficent character endorsing the satire, I was never sure where we stood. I couldn't tell if the social world of Flatland was absurd because Abbott meant it to be so, or because he wrote this book over a hundred years ago.

Abbott's catalogue of details about Flatland's society and history felt dry and weakly imagined. The square goes into great depth about the impact of irregularity on Flatland's living shapes, the methods of recognition (necessary since any shape seen edge-on appears as a line), the detailed history of the Color Rebellion. I was left longing for the explanations I really wanted, the good sci-fi realities of the world. How do shapes move and manipulate objects if they don't have hands? How do squares and straight lines reproduce? How could 120,000 triangles gather around a speaker? Since they only see in the flat plane of their world, shouldn't it be impossible for them to see around or over someone standing in front of them? Abbott asks the reader to brush away all curiosity about these logical concerns. That's a tall order for a demanding reader used to more recent science fiction and science writing.

But then the good stuff starts. In Part Two of *Flatland,* called "Other Worlds," a sphere travels from Spaceland (our three-dimensional world) into Flatland. It offers the square a vision of – and a visit to –
worlds of other numbers of dimensions. First the square dreams – it's implied
that the sphere somehow conjures this vision – of Lineland, where one-dimensional
beings shimmy next to each other in their straight line world, and can't begin
to comprehend a second dimension. Then the sphere brings the square into
Spaceland, pretty much blowing his mind. The square returns home grasping
fleeting memories of cubes, and struggling with the impossible mission of explaining
the brave new three-dimensional world to his flat comrades.

The real joys of *Flatland* come from accompanying the
square into 3-D space. By extrapolation of his struggle to understand, we can
edge up to imagining geometry in four dimensions. It's mental acrobatics akin to trying to lick your elbow: striving for a thing that's there but
beyond reach. I found myself setting down the ancient Kindle on which I read
this book and straining to picture geometry in four dimensions. Square a line:
you get a square. Square a square: you get a cube. Square that again. I strain and
strain and can almost get there. The sphere can see our square's insides from
outside Flatland. Could a four-dimensional being see mine?

As a philosophical and mathematical exploration, the second
half of *Flatland *is as effective and compelling a piece of writing as it
ever was. Abbott's target for the very serious satire of the first half, on the other hand, has
faded into history, and we're not the readers it was written for. So maybe
skip ahead to Part Two. You may not come away with a clear vision of a
four-dimensional cube, but you'll get tantalizingly close. And that new way of
thinking about the very space you move through stays with you more than a clear
and easy answer ever would.

*Jaime Green is a graduate student in Columbia's MFA writing program. Her work has appeared in *The Awl, Spezzatino, The Hairpin, *and* Parabasis*. She is writing a book about the possibility of life in the universe.*