Graphs from Rectangles

At its core mathematics is concerned with the study of patterns of different kinds. What is perhaps surprising is that changing points of view often show that patterns involving things we regularly see around us are not always as fully explored as we might think, especially in situations where the patterns emerge from looking at and exploring the properties of geometrical objects.

The basic objects that we will consider are rectangles drawn in the plane in a special way (Figure 1). Notice that the sides of the rectangles are either horizontal or vertical. Such rectangles are often said to be axis-aligned, since their sides are parallel to either the x- or y-axis when drawn in an (x, y) coordinate system.