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Three-Dimensional Space

It is a short step from two to three dimensions. From the two-dimensional village layout, we can move to the model of a city, where we have a height for each location as well as a position on the grid. We can augment taxicab geometry with elevator geometry. We specify a position by three numbers, for example, E3N4U9, referring to the ninth floor of a building at location E3N4. We can then determine an algorithm for getting from this location to E7N2U5. Note that in this particular geometry it makes a big difference in what directions one moves. The usual algorithm would be D9E4S2U5. Beginning with D4 gets you to the right level but in the wrong building! The situation would be different for a game played on a jungle gym, with instructions to move from one position to another by going a certain distance left or right, forward or back, up or down. In this case we can carry out the instructions in any order.

Another three-dimensional geometry arises if we want to specify the position of an airplane, giving its longitude, latitude, and altitude. Once again, it makes a difference in which order we give the numbers that indicate a given location or the directions for getting from one point to another.

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