On Proofs Without Words

Most mathematicians will be familiar with the above picture. This diagram, credited to the Ancient Chinese mathematical text Zhou Bi Suan Jing, is a charmingly simple visual proof of the Pythagorean Theorem, one of mathematics’ most fundamental results. It would be hard to argue that this proof is not convincing. In fact, most standard proofs of the Pythagorean Theorem still use this picture, or variations of it. In this paper, we will examine pictures, such as this one, which claim to prove mathematical theorems and have come to be known as “Proofs Without Words”, or PWWs. This paper will have two main parts. In the first part, we will present collection of PWWs accompanied by explanations. Essentially we will be putting the words back into these “Proofs Without Words” by explicitly stating what our brains are seeing, and how we are supposed to reach the intended conclusions given only the visual clues contained in the figure. In addition, where appropriate, we will include “parallel proofs”, which are more traditional proofs of the same results portrayed by the PWWs. The aim of this is to see the differences between formal logical structure, and the logic that our brains will follow given visual information. The second part will examine what it actually means to “prove” a result, whether Proofs Without Words satisfy this definition, and if they do not, what value they have for mathematics. Any textbook will tell you that a proof is a series of statements that show how a new statement is true by using logic and statements that are already known to