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,
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
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.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.