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Planes and Surfaces

Children should become skilled in both following and giving directions. Any child should learn how to direct a person from one part of the school to another and perhaps to describe the neighborhood of the school. Although the algorithm for getting from one street address to another in an actual town might be quite complicated, an ideal town has a simpler structure. We can imagine a sequence of imaginary towns with different dimensional properties — a frontier town all stretched out along a single street or a village laid out on a rectangular plot. A model village could stimulate a good deal of the discussion, while a grid on which children could design their own town would allow for more variation.

No matter what the streets are named, we can still give directions on a grid by saying: "Go right two blocks, then turn left and go three blocks." For persons with a clear orientation, the instructions can be varied: "Go east two blocks, then north three blocks." The first instruction depends on the direction that the person is facing, and the second does not. If the map of a village is hanging on a wall, we can use the natural coordinate directions: "Go right two blocks and up three." Certain pairs of instructions can then be combined: "Go left two, then up three" and "Go left three and down five" combine to give "Go left five and down two." By playing this game with cards, we can easily introduce the operation of adding ordered pairs and even of multiplying numbered pairs by positive integers. If we introduce "put" and "take" operations, we can extend the one-dimensional algebra of signed numbers to an algebra of two-dimensional quantities.

Notice that this algebra of instructions does not require the use of coordinates in the plane. The exercise carries additional value when addresses are given in terms of street numbers or compass directions. For one thing, this avoids the complications caused by negative numbers. To go from E3N4 to E7N2 requires a move of E4S2. The correspondence between this commonsense approach and the algebraic statement (7,2) - (3, 4) = (4, - 2) is something that can come much later in a student's mathematical development. There are a great many people who are confused by negative numbers. They shouldn't be.

"Taxicab geometry" provides an effective variation on the use of directional instructions. Students play the role of dispatchers, telling cabbies how to get from one location to another. "Just go three streets north and two avenues west" would be such a direction. The efficiency of the instructions — and the profit of the cab company — depends on many factors such as one-way streets, accidents, and traffic jams. One can easily imagine a board game that would model realistic city traffic and get students used to the idea of a two-coordinate instruction set.

The surface of the earth is another familiar example of a two-dimensional object. Even though it exists in three-dimensional space, we need only two numbers, latitude and longitude, to specify any location. A dispatcher of ships can give instructions to go 10 miles due east and then 5 miles due north. On the surface of the earth — but not on a flat plane the order of these operations makes a difference: going 5 miles due north and then 10 miles due east can put a ship at a different position! The extent of this difference is an intrinsic indicator of curvature.

In teaching geometry we should not ignore the interactive video game. Today's students take for granted the fact that we can manipulate images on a two-dimensional screen by pushing buttons, turning dials, or twisting joysticks. Programs like LOGO offer students experience in giving simple geometric instructions to move points and objects around on a screen. This gives mathematics teachers a chance to introduce any number of important concepts, including repeated operations to form regular or star polygons and recursive processes for drawing fractal objects or space-filling curves.

Many video games employ wraparound, which introduces interesting ideas in different two-dimensional geometries. Frequently when a point is guided off the left side of a computer screen, it appears at the same height on the right side. This is analogous to the phenomenon on the digital radio dial, which just as well might be thought of as operating on a circle. A segment with its endpoints identified can be treated as a circle. Analogously, if we think of the points on the left side of a computer screen as identified with the corresponding points on the right side then we are dealing not with a flat rectangle but rather with cylinder.

Figure 31. A torus, the mathematical name for a doughnut shaped surface, is a two-dimensional surface in which two closed curves can intersect in just one point and in which a closed curve need not separate its inside from its outside.

But even more can happen. It is often the case that when a point moves off the top of the screen, it reappears at the corresponding position on the bottom, so we get a cylinder with its top and bottom identified. This gives a figure like an inner tube, which mathematicians call a torus. The geometry of a torus is in some ways like that of the plane, but in other ways it is very different. In the plane any polygon that does not intersect itself divides the plane into two pieces. But if we take a closed polygon that goes around the top of the torus, it does not separate the torus into two pieces: its inside is the same as its outside. Related to this phenomenon is the fact that on a torus we can find two closed curves that cross at exactly one point (Figure 31), whereas if two closed curves in the plane cross (not just touch), they must intersect in an even number of points. An unusual object in many ways, the torus is ideal for keeping track of pairs of numbers from circles. [an error occurred while processing this directive]