Beyond being “whatever mathematicians do”, mathematics is the study of abstract structures: integers or points in space, for instance, or sets, and sets of sets. The beauty of mathematics is revealed when we develop a deeper understanding of such structures. To cope with mathematical abstractions, people have used practical representations as a way to visualize mathematical ideas ever since they first realized they had ten fingers.

Tracing the history of technology in mathematics, we see the invention and use of the abacus, the quipu, Napier’s bones, and the slide rule, all of which were improved ways of manipulating numbers. Drawings have always been used as a tool to express geometric ideas; Archimedes was slain as he pondered a drawing in the sand. The ruler and the compass were more than just drawing tools to the ancient Greeks. Their use inspired deep mathematical questions as to what could and could not be drawn with them–questions that eventually led to the …