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Flatland: The Movie (flatlandthemovie.com) is a 30-minute animated film inspired by Edwin A. Abbott’s classic 1884 novel, Flatland. Re-leased on DVD in June and set in a world of just two dimensions, the story follows Arthur Square and his granddaughter, Hex, as they try to understand the third dimension. The movie, aimed at teachers and students, is the work of producer Seth Caplan, director Jeffrey Travis, and animator Dano Johnson and features the voices of major actors, including Martin Sheen, Kristen Bell, and Michael York.

FlatlandAn Animated Romance of Many Dimensions

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1. on the left is an image from Flatland: The Movie showing Arthur Square (the blue square) at home with his wife (the ma-genta square). notice that Arthur and his wife are both facing to the right and that each has an eye and a mouth. what problem does this present if they want to kiss each other? How do you think the filmmakers solved this problem?

2. Find the objects listed below in the picture at left. Can you explain their peculiar appearances and how they work?

a. bedb. patio chairc. dining table

3. in some cases, the filmmakers took artistic license, including objects that are familiar to us but which would be illogical in a two- dimensional world, like the chimney. in Flatland, this chimney would not work correctly because it lacks a hollow channel through which smoke can flow. why is the weather vane similarly illogical?

The Flatlanders live in a two-dimensional world that is much like the surface of a pond. Since they are trapped inside their world, they cannot see an entire three-dimensional solid all at once. instead they see only a 2-D cross section of a solid where it crosses the planes of their world. For in-stance, an illustration in the original novel shows a sphere ascending through Flatland as the inhabitants would see it—a circle that shrinks to a point and then disappears.

At right are pictured cross sections of other shapes as they would appear as they ascended through the planes of Flatland.

in each case the shape moves straight up at a constant speed without rotating. Can you iden-tify each of the 3-D shapes?

Here are pictures of a 3-D cube and its 4-D cousin, the hypercube. By understanding the family resemblances between these two shapes, we can start to comprehend the fourth dimension.1. The cube has eight corners. At each

corner three edges come together. How many corners does the hypercube have? How many edges meet at each corner?2. The cube has 12 edges. At each edge two square faces meet. How many edges does the hypercube have? How many square faces meet at each edge?3. The cube has six square faces. The two obvious faces are the big outer square and the smaller inner red square. (The red square appears to be smaller only because it is farther away.) There are also distorted squares connecting the inner and outer squares—the distortion is due to 3-D

perspective. How many square faces does the hypercube have? How many square faces meet at each corner?4. The hypercube has something the cube does not have: hyperfaces that are 3-D cubes. The two obvious hyperfaces are the big outer cube and the smaller inner cube. (The red cube appears to be smaller only because it is farther away.) There are also distorted cubes connecting the inner and outer cubes—the distortion is due to 4-D perspective. How many hyperfaces altogether?

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MINd GAMESBy Scott Kim

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1. To face each other, one of the two characters must rotate 180 degrees. But this means that one character will be upside down, with the eye below the mouth. in order to see eye to eye, the mouth and eye must change places. in Flatland: The Movie, the eye mi-grates to where the mouth used to be, and a new mouth opens up on the other side of the eye. This happens so quickly that the odd transformation is barely noticeable.2. Here are notes on how the objects work.a. The bed is in the upper left corner of the house; it is made of two disconnected wooden parts that form a frame. The frame has openings on both sides, so Mr. and Mrs. Square can get into bed from either side. in place of a mattress, there is a scattering of white puffballs that cushion the sleeper on all sides.b. There are three white patio chairs in the gray room to the right. Chairs are shaped like squares, enclosing the sitter on three sides. Since there is no gravity in this world, people sit in rather than on chairs, which float freely in space.c. The dining table is near the vertical center of the image and to the right and looks like half of a captain’s wheel. Each alcove encloses one of the three family members, along with his or her dinner plate.d. There is a potted plant near the middle of the image, facing down and to the right. Two others are at the lower right, facing up. notice how the round bowls pinch the plants to keep them in place.e. The refrigerator is just to the left of the single potted plant, on

the other side of a wall. The hinge is a rubbery black triangle that expands to fill in the gap when the door is opened; food is stored in several narrow pockets.f. near the lower left corner is the library, containing a red easy chair and two shelves of books. Books have white pages that look like strands of beads; a page in Flatland is a one-dimensional line, instead of a 2-D rectangle.

A number of other authors have tried imagining Flatland in detail. in The Planiverse, a novel by A. K. Dewdney, Flatland has gravity, which means that when two creatures meet, one must climb over the other in order to pass by.3. For a weather vane to point in the direction of the wind, it must be able to rotate around its supporting pole. But that sort of rotation would take the arrow outside the planes of Flatland.

MINd GAMES SOLuTIONS

DISCOVER® (iSSn 0274-7529) is published monthly by Discover Media llC, 90 Fifth Avenue, new York, new York 10011. Volume 28, number 11; copyright 2007 Discover Media llC. periodical postage paid at new York, new York, and at addi-tional mailing offices. in Canada mailed under publication mail agreement 41245035, p.o. Box 875, STn A windsor, on, n9A 6p2. GST Registration #817800345RT0001. SuBSCRipTionS: in the u.S., $29.95 for one year; in Canada, $39.95 for one year (u.S. funds only), includes GST; other foreign countries, $44.95 for one year (u.S. funds only). Back issues available. All rights reserved. nothing herein contained may be reproduced without written permission of Discover Media llC, new York, new York. poSTMASTER: please address all subscription correspondence, including change of address, to DiSCoVER, p.o. Box 37808, Boone, iA 50037, or call toll-free 800-829-9132; outside the u.S.A., 515-247-7569. printed in the u.S.A.

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By the way, if a 4-D solid traveled through our world, we would see a 3-D shape that appeared in midair, changed shape, shrank to a point, and vanished. The Flatland: The Movie DVD includes animations that visualize how a 4-D hypercube would look if it passed through our 3-D space.

1. The hypercube has 16 corners. At each corner 4 edges meet.2. The hypercube has 32 edges. At each edge 3 square faces meet.3. The hypercube has 24 square faces. At each corner 6 square faces meet.4. The hypercube has a total of 8 cubic hyperfaces.

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1. Cone 2. Square pyramid

3. octahedron (double square pyramid)

4. The letter A 5. Tetrahedron (triangular pyramid, passing through Flatland edge first)