Here is a timeline of a mathematical fantasy:
1884. Flatland: A Romance of Many Dimensions, a satirical novella by Edwin Abbott.
1965. Sphereland: A Fantasy About Curved Spaces and an Expanding Universe, a novel by Dionys Burger
2007. Flatland: The Movie, a short animated film
2012. Flatland²: Sphereland, a short animated film
Abbott's book received mixed reviews, especially for its treatment of women, even though in mathematical circles it became a notable classic. For example, James R. Newman commented on his copy,
The dust jacket of my copy (a 1941 reprint) records opinions that the book is "desperately facetious," "mortally tedious," "prolix," a "soporific"; also that it is "clever," "fascinating," mind broadening," and worthy of a place besides Gulliver.
Sphereland dispenses with much of Flatland's obsolete social sting, bringing the Flatland's society more in line with the spirit of the 1960s, and devoting more room to the mathematical aspects of Flatland's geometry and space. The book did not make a lot of impact even among mathematicians, although reviews were mostly positive.
The movies are rather loosely based on the books - "inspired by" may be a better description. Flatland: The Movie was a spectacular achievement: the movie won several prestigious awards and was, I heard it said, a commercial success, despite very limited channels of distribution. Beautiful graphics, smoothly orchestrated animation, a very well known cast of characters, and ingeniously realistic plot made a mathematical fantasy an eye and mind catching affair and a favorite with geometry teachers.
Now comes the sequel. There is real continuity. The managing crew is the same: Seth Caplan (producer), Dano Johnson (writer, director of animation), Jeffrey Travis (executive producer). The voices behind the returning heroes (Kristen Bell, Michael York, and Tony Hale) joined by Danny Pudi, Kate Mulgrew, Danica McKellar. The plot takes off just 20 years after the events of Flatland: The Movie; it's almost tangibly , engagingly realistic.
Hex, the mathematician granddaughter of the now late Arthur Square - the 2D thinker and author whose memoirs inspired the prequel movie - failed to convince her fellow countryman of the existence of the third dimension and became an outcast. Puncto - another hexagonal scientist - is stumped with attempts to plan a safe flight route for a planned (2d) exploratory space travel, for radars indicate the presence of triangles with the angle sum in access of 180°. Puncto seeks Hex's help who reluctantly agrees. After a twenty years hiatus, Serius - a 3D sphere, the original Arthur Square's Guide in Spaceland - appears and uses the opportunity to appologize for dismissing Arthur's idea of the fourth dimension. (In the meantime he had a visitor and has been enlightened.) Serius takes Hex and Puncto into 3D where, from high up, they observe a Lineland and notice that it is curved and is actually a Circleland. They conclude by analogy that the Flatland is in fact a Sphereland and find an explanation for Puncto's data anamoly. They save the exploratory spacecraft from crashing into Sierpinski belt and find the many dimension Portal located at the center of Flatland. Hex and Puncto decide to continue exploring space together.
Rather obviously, the foregoing writeup in no way does justice to this captivating movie. It's beautiful; it's edifying; it's realistic and enigmatic at the same time. It makes one ponder of the dimensional nature of space. After the 4D-Oversphere has denied its divine nature, Spherius expostulates:
But a higher dimension implies higher knowledge and clarity.
At the space Portal where all spaces and universes cross, Hex observes a ship carrying a parallel Her and her Grandfather and wonders what it was those did differently over there that did not cause A. Square's heart to break.
The movie ends with a touchingly human attempt of a tender kiss between Hex and Puncto. With their eyes closed, they miss each other ... Hmm, but what if they kept eyes open? As a matter of fact, there are so many questions that could be asked about the whole concept of Flatland and of its rendition in the movie. I can imagine several meaningful discussions about rights and wrongs in the books and the movies. Is a Flatland at all feasible?
There is no good reason for me two dwell on that point, it's best left to teachers and their students to wonder about. I am sure they will, while thoroughly enjoying the movie. There is moving wisdom in the last dialog between Hex and Puncto:
"It won't be easy to convince every one about our discovery."
"It never is. But the point of exploring and learning the truth, is to share it."
"That's what Grandpa always said."
And this is what the movie does.