13.10.5 (Shoshoni Rectangles) The following problem is taken from Larsen and Mam (1986). Since antiquity, societies have expressed esthetic preferences for rectangles having a certain width (w) to length (1) ratio. For instance, Plato wrote that rectangles formed of two halves of an equilateral triangle are especially pleasing

(for such rectangles w/l = l/&). Another standard, adopted by the Greeks, is the golden rectangle. It is defined by the condition that the ratio of its width to length must be the same to that of the part remaining after cutting off a square with the side equal to its width (i.e., the shaded area in Figure 13.7 is similar to the whole rectangle). Thus a golden rectangle must have w/l = (1 - w)/w, which gives w/l = & - 1; hence w/l = (A - 1)/2 = 0.618.

Both the Greeks and the Egyptians used golden rectangles in their architecture.

Even today the golden rectangle remains an architectural and artistic standard (e.g., items such as drivers’ licenses, business cards or picture frames often have w/l ratios close to 0.618).

The data below show width-to-length ratios of beaded rectangles used by Shoshoni Indians to decorate their leather goods. Use the bootstrap test to check whether the golden rectangle can be considered an esthetic standard for the Shoshonis.

Width-to-length ratios for 20 rectangles found on Shoshoni handicraft:

0.693 0.749 0.654 0.670 0.662 0.672 0.615 0.606 0.690 0.628 0.668 0.611 0.606 0.609 0.601 0.553 0.570 0.844 0.576 0.933