ABSTRACT
You probably recall doing various constructions with a compass and a straightedge in high school geometry. We will imagine idealized tools. Thus, the straightedge is an unmarked ruler, which in principle can be as long as necessary. With it we can draw line segments of arbitrary length, perhaps passing through a particular point or connecting two given points. Likewise, the compass is as large as necessary; with it we can draw arcs and circles and duplicate distances. For instance, if A and B are marked on one line and point C is marked on another, the compass allows us to mark a point D on the second line so that the distance between C and D is the same as the distance between A and B. Of course, in actually carrying out these constructions with real rulers and compasses, there is always error involved. However, we are concerned with idealized constructions-perfect constructions with no error.
