Crop circles: Theorems in wheat fields
Since the late 1970s, farmers in southern England looking out on their wheat fields in the morning have sometimes been startled to find large circles and other geometric patterns neatly flattened into the crops. How these crop circles were created in the dead of night at the height of the summer growing season remains a puzzle, though hoaxers have claimed responsibility for some of them.
Several years ago, astronomer Gerald S. Hawkins, now retired from Boston University, noticed that some of the most visually striking of these crop-circle patterns embodied geometric theorems that express specific numerical relationships among the areas of various circles, triangles, and other shapes making up the patterns (SN: 2/1/92, p. 76). In one case, for example, an equilateral triangle fitted snugly between an outer and an inner circle. It turns out that the area of the outer circle is precisely four times that of the inner circle.
Three other patterns also displayed exact numerical relationships, all of them involving diatonic ratios, the simple whole-number ratios that determine a scale of musical notes.