Proofs without words

"Mathematical proof can convince, and it can explain. In mathematical research, its primary role is convincing. At the high-school or undergraduate level, its primary role is explaining."

Reuben Hersh in [3]

The diagrams presented in this page are usually referred to as Proofs Without Words. These proofs are diagrams or drawings suggestive enough in order to easily show us how to extend the pattern to the general case. They help us "to see WHY" a certain property holds and, in many cases, they also indicate how we may prove it. As R. Hersh advocates in [3], "In mathematical research, the role of the proof is to convince. It is an effective proof when it convinces qualified judges. In the classroom, on the other hand, its role is to explain. The suggestive use of proofs in classroom has the primary goal of promoting the understanding of students and not to please abstract standards of "rigour" or "integrity".

In the early days of our civilization, when there was no proper language to describe general mathematical ideas, proofs without words were the proofs. Although proofs without words are not real proofs, they are important as they stimulate mathematical reasoning through visual stimuli. Polya used to advise "draw the figure".

Visualization has as important role in Mathematics (see [4]) and gives the original meaning to the word "theorem". Indeed, the Greek root of the word theorem yeore\(\mu\)Ó means "what you see" and not, as understood today, what is proved (in addition, the word yeorein means "to see"). It also reminds us that mathematical discovery is not usually made in a deductive way, and recognizes the "eye" as a legitimate tool of discovery and inference.