In 1884, Edwin Abbott wrote a strange and enchanting novella called *Flatland*, in which a square who lives in a two-dimensional world comes to comprehend the existence of a third dimension but is unable to persuade his compatriots of his discovery. Through the book, Abbott skewered hierarchical Victorian values while simultaneously giving a glimpse of the mathematics of higher dimensions.

In 2007, *Flatland* was made into an animated movie with the voices of Martin Sheen, Kristen Bell and Michael York. Now there’s a sequel called *Flatland 2: Sphereland*, which expands the story into non-Euclidean geometry. It’s a gem: an engaging, beautiful and mathematically rich film that children and adults alike can enjoy.

The first film, *Flatland: The Movie*, takes immediate advantage of its medium by opening with a dazzling flyover of Flatland. The inhabited parts of Flatland look like ornate Islamic tilings, and the uninhabited parts are filled with exotic fractal patterns that could come from the surface of Mercury.

We then zoom into the home of Arthur Square, who is late to take his granddaughter Hex to school at St. Euclid. On the way, Arthur drills Hex — a darling little hexagon, complete with a bow and a big expressive eye — on Flatland’s uncompromising hierarchy. Each subsequent generation, Hex dutifully reports, acquires an additional side, so that Arthur Square’s children are pentagons and Hex is a hexagon. In Flatland, having more sides supposedly means that you’re smarter. Lowly triangles are good only for manual labor; squares are part of the professional class; and creatures with so many sides that they look circular are priests, who, frowns Hex, “just make rules that everyone else has to obey.”

The circles pronounce a frightening decree: Anyone espousing the nonsensical and heretical notion that the third dimension exists will be executed. What, Hex asks, is a dimension? So Arthur explains: A point is zero-dimensional; a point moving straight traces out a one-dimensional line; and a line moving perpendicular to itself traces out a two-dimensional square. Hex immediately makes the forbidden leap: A square that somehow moves perpendicularly to itself, she reasons, would trace out a “super square” in three dimensions. She even calculates how big such an object would be. But her mathematical insights only earn her a scolding from Arthur.

By the end, Hex wins happy vindication. Both she and Arthur get a mind-blowing tour of the full three-dimensional universe from a sphere, Spherius — and they even manage to proclaim their discovery and save their own skins (er, perimeters).

But when they ask Spherius about a fourth or fifth dimension, following their mathematical logic, he’s as skeptical as their compatriots had been about the third.

The sequel joins Hex 20 years later, with her bow long lost and a disillusioned cast to her eye. Although Hex and Arthur’s discoveries have knocked the circle priests from power and brought equal rights to all shapes, Flatlanders still deny the reality of the third dimension. The sphere never returned, and Arthur has died heartbroken and disgraced. Hex is now living in isolation, pursuing her mathematics.

Then a fellow hexagon, Puncto, seeks her out for help with a mathematical problem he can’t get anyone else to take seriously. He’s an engineer for the Flatland space program, and his data haven’t made any sense. By his calculations, the angles on some very big triangular paths that Flatland’s rockets will follow to other planets add up to more than 180 degrees. Everyone has been telling him that he must just be making a mistake, but he’s convinced there’s a deeper issue. Space itself must be warped, he says. And if space is warped, the rocket they’re about to send out could hit an asteroid in the Sierpinski belt!

Hex and Puncto end up on an otherworldly adventure through multiple dimensions and worlds. Hex stumbles on a key mathematical insight — the key to Puncto’s dilemma — when they visit one-dimensional Lineland. The world appears to be a straight line, but when they travel high above it they discover that it’s a circle. Hex realizes that similarly, Flatland itself might not be flat, even though it seems so — it could be curved into the third dimension. Perhaps Flatland is on the surface of a sphere: Sphereland! If so, Hex realizes, the edges of a triangle in Flatland would actually curve outward in three dimensions, making the angles a bit more than 180 degrees, just as Puncto had found.

But if Hex is right, the rocket’s path is off, and unless she and Puncto convince the Flatlanders of their discovery, it could crash. Thus begins a madcap race back to Flatland, complete with other mathematical revelations along the way.

Calling the films educational somehow seems an insult. They manage to accomplish that so-rare feat of giving viewers a taste of the delight of mathematical discovery while carrying them along through a quirky, multi-dimensional story.