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]