DATA FOR MAKING A MAP TRIP

The tours Pete traced out on World Wallpaper appeared continuous. Any traveler can start her trip at one location and draw a continuous trip to every location on the Earth. Still, what happens as the traveler approaches the edge of World Wallpaper?

[Meaning?: Does it seem that if the tessellation of the wall by World Wallpaper is sufficiently large, then the traveler will never reach the edge?] [Question?: How can we simulate this being large?]

Is it possible to replace the Mercator map with a wall tessellation that will never violate the continuity assumption of Q_2?

It isn't possible! To see why, consider the data you need for a trip. Picture following a trip as if you are following directions. A real trip, say, by foot or by car, requires serious direction details. Still, these come down to a simple idea. To direct someone somewhere, you give them a series of packets of information, I_1, I_2, ..., I_n. Each packet looks like this: Go length x in the direction w, then look at the next packet of information in the series. For example: I_1 might say, go two blocks on this street; then, follow the instructions in I_2. In real life one might mix the units of distance, sometimes using blocks, other times using miles. For good directions, the data must be consistent. A traveler of great distances in complicated territory must have very precise data.

For example, while you don't need specific details of a trip when you fly across the continent, the pilots need very precise data. Further, pilot data can't treat the earth as a flat map; it must include the height of the plane above the earth's surface. Flying a plane requires a practical understanding of vector calculus. To understand why Pete cannot find a solution to the problem of creating World Wallpaper, we might use vectors even more abstractly than does a pilot.

To display a given direction, you would point that way. Your arm goes the right way; call the direction given by your arm a vector. On paper, represent a vector as an arrow (Figure 11); it can point in any direction and be of any length.

An arrow
Figure 11: A Vector

It does not matter where you put the vector, if it is clear which vector (direction) you mean. Therefore, it can be very valuable to have all vectors start from one place--for example, an origin in the picture. When the tails of all vectors are at the origin, one easily compares their directions and lengths. To make the information more concise, put both pieces of information in one vector: Its length represents appropriate information for how far you are to go. One could use the actual distance as the length. There is a standard name to refer to a sequence of vectors that codes a trip by direction and how far to travel in that direction. We say the sequence parametrizes the trip by arc length.

There are other ways to show how fast they should go in the given direction. You might make the length of the vector equal to the velocity of travel. While this is the common way, it assumes you give directions in packets that cover a fixed length of time. That is, each of the directions I1, I2, ..., covers, say, a minute of the travel. One says a sequence of vectors that codes a trip by direction and velocity parametrizes a trip by time.


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