Mind Games By Scott Kim

FLATHOUSE 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. bed d. plant b. patio chair e. refrigerator c. dining table f. book

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.W hy is the weather vane similarly illogical?

SLICES The Flatlanders live in a two- In each case the shape moves dimensional world that is much straight up at a constant speed like the surface of a pond. without rotating. Can you iden- Since they are trapped inside tify each of the 3-D shapes? their world, they cannot see an entire three-dimensional solid all at once. Instead they 1. see only a 2-D cross section of a solid where it crosses the Flatland planes of their world. For in- 2. An Animated Romance of Many Dimensions stance, an illustration in the original novel shows a sphere Flatland: The Movie (flatlandthemovie.com) is a 30-minute animated ascending through Flatland as 3. film inspired by Edwin A. Abbott’s classic 1884 novel, Flatland. Re- the inhabitants would see it—a leased on DVD in June and set in a world of just two dimensions, circle that shrinks to a point sc o tt k i m (7). wo r l d p o u ct ion s ; c ou rtesy : © 2007 f l at the story follows Arthur Square and his granddaughter, Hex, as they and then disappears. try to understand the third dimension. The movie, aimed at teachers At right are pictured cross 4. and students, is the work of producer Seth Caplan, director Jeffrey sections of other shapes as they Travis, and animator Dano Johnson and features the voices of major would appear as they ascended

actors, including Martin Sheen, Kristen Bell, and Michael York. through the planes of Flatland. 5. Clo ck wi se F r o m l eft

Hypercube corner three edges come together. How perspective. How many square faces does many corners does the hypercube have? the hypercube have? How many square How many edges meet at each corner? faces meet at each corner? 2. The cube has 12 edges. At each edge 4. The hypercube has something the cube two square faces meet. How many edges does not have: hyperfaces that are 3-D does the hypercube have? How many cubes. The two obvious hyperfaces are square faces meet at each edge? the big outer cube and the smaller inner 3. The cube has six square faces. The two cube. (The red cube appears to be smaller Here are pictures of a 3-D cube and its 4-D obvious faces are the big outer square only because it is farther away.) There are cousin, the hypercube. By understanding and the smaller inner red square. (The red also distorted cubes connecting the inner the family resemblances between these square appears to be smaller only because and outer cubes—the distortion is due to two shapes, we can start to comprehend it is farther away.) There are also distorted 4-D perspective. How many hyperfaces the fourth dimension. squares connecting the inner and outer altogether? 801. The cube has eight corners. At each squares—the distortion is due to 3-D ANSWERS ON PAGE 84 Mind Games Solutions

Flathouse

1. To face each other, one of the two characters must rotate 180 the other side of a wall. The hinge is a rubbery black triangle that degrees. But this means that one character will be upside down, expands to fill in the gap when the door is opened; food is stored in with the eye below the mouth. In order to see eye to eye, the mouth several narrow pockets. and eye must change places. In Flatland: The Movie, the eye mi- f. Near the lower left corner is the library, containing a red easy grates to where the mouth used to be, and a new mouth opens up chair and two shelves of books. Books have white pages that look on the other side of the eye. This happens so quickly that the odd like strands of beads; a page in Flatland is a one-dimensional line, transformation is barely noticeable. instead of a 2-D rectangle. 2. Here are notes on how the objects work. A number of other authors have tried imagining Flatland in detail. a. The bed is in the upper left corner of the house; it is made of In The Planiverse, a novel by A. K. Dewdney, Flatland has gravity, two disconnected wooden parts that form a frame. The frame has which means that when two creatures meet, one must climb over openings on both sides, so Mr. and Mrs. Square can get into bed the other in order to pass by. from either side. In place of a mattress, there is a scattering of white 3. For a weather vane to point in the direction of the wind, it must puffballs that cushion the sleeper on all sides. be able to rotate around its supporting pole. But that sort of rotation b. There are three white patio chairs in the gray room to the right. would take the arrow outside the planes of Flatland. 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 Slices chairs, which float freely in space. c. The dining table is near the vertical center of the image and to the 1. Cone 2. Square 3. Octahedron right and looks like half of a captain’s wheel. Each alcove encloses pyramid (double one of the three family members, along with his or her dinner plate. square d. There is a potted plant near the middle of the image, facing down pyramid) 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 Discover8-30-07.pdf 8/30/2007 3:02:02 PM 4. The letter A 5. Tetrahedron (triangular pyramid, passing through Flatland edge first)

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.

C

M HYPERCUBE

Y 1. The hypercube has 16 corners. At each corner 4 edges meet. CM 2. The hypercube has 32 edges. At each edge 3 square

MY faces meet. 3. The hypercube has 24 square faces. At each corner 6 CY square faces meet. CMY 4. The hypercube has a total of 8 cubic hyperfaces.

K

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