Sometimes one discovers something by means of visual thinking using background knowledge, resulting in a cogent argument from which one could construct a proof. A nice example is a visual argument that any knot diagram with a finite number of crossings can be turned into a diagram of an unknot by interchanging the over-strand and under-strand of some of its crossings (Adams 2001: 58–90). That argument is a bit too long to present accessibly here. For a short example, here is a way of discovering that the geometric mean of two positive numbers is less than or equal to their arithmetic mean (Eddy 1985) using Figure 6.