When we teach people how to use cartesian coordinates, we tell them to move x units along the x axis, y units up the y axis, and then take the intersection of the perpendiculars. The other day it occurred to me to wonder what would happen if instead, we drew a line from (x,0) to (0,y) and called the midpoint our destination. So if [a,b] are coordinates in Midpointesian coordinates (corresponding to the cartesian endpoints of the line (a,0) and (0,y)), and the point (x,y) is the corresponding point in Cartesian space, we could have a function f([a,b]) = (x,y). That function turns out quite trivial, but was nonetheless fun to try to visualize before I realized how straightforward it actually is, so I won't spoil it by giving it explicitly :) Then I realized it would actually be a rather useful thing to do. Say you're sketching out the graph of a function, but you don't have any graph paper. Use a compass and ruler to create X and Y axes, and mark at regular intervals. To plot a point (x,y) s.t. f^-1((x,y))=[a,b], draw a line from (a,0) to (0,y) and then use the compass to find the midpoint. Or say you're mapping out things you find at an archaeological dig. Create an X and Y axis at your site, and when you want to find the placement of an item, have a friend walk along each axis holding ends of a tape measure, with you holding it in the middle. When you read half the value seen by the friend holding the other end, they record their positions on the axes. (Or, in fact you could also record [a,b,r] for any line passing through the item, (a,0), (0,y) and showing a distance of r on the tape).