MAM2000 (Essays/Dimension)

# 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. [an error occurred while processing this directive]