APPENDIX 3

Golden Mean (Golden Section)

Nautilus

Thomas L. McCartney

GOLDEN MEAN (GOLDEN SECTION)

The Golden Section refers to a rectangle of any size that has a length that is 1.62 (1.618) times that of its width. A practical application to the Golden Section is in making a decision on how to crop a photograph. The closer the cropped print is to the Golden Mean, the more pleasing the print size should be. In photography, for example, with a 3 × 5 inch print, the 5-inch side is 1.66 times longer than the 3-inch side. As a ratio, it is written 5:3. Rectangles with such a ratio are considered to be pleasing to the eye. Prints that are 4 × 5 inches, 8 × 10, and 16 × 20 have a ratio that is less than the Golden Mean, being only 1.25. Once-popular 35 mm film was 24 × 35 mm and had a ratio of 1.46.

The Golden Section has a long history going back to early Greek and Egyptian times and has been used in art and architecture for centuries. The Parthenon and the Egyptian pyramids serve as examples. It is also found in growth patterns of nature, curves of sunflowers, seashells, and galaxies. The Golden Section is also known as the Golden Mean and the Golden Rectangle, and in Egypt it was know as the Sacred Ratio. On the previous page is the cross-section of a nautilus shell, showing the expanded spiral growth pattern.

Fibonacci (1180–1250), an Italian mathematician, had a passion for numbers and discovered a series of numbers that describe not only the expanding growth pattern of things in nature but also the Golden Section. The numbering sequence is amazingly simple—0, 1, 1, 2, 3, 5, 8, 13, 21 …. The numbers are arrived at by adding the sum of the previous numbers, such as 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5 and so on. By taking the ratios of the adjoining numbers, one finds that they follow the golden section closely:

As the numbers continue, they approach the exact ratio for the Golden Section, 1.618. Looking at the series of ratios in the Fibonacci series, prints cropped to 3 × 5, 5 × 8, 8 × 13, 13 × 21, and so on would meet the Golden Section ratio.

In the 1800s, Gustav Fechner, a German experimental psychologist, did a study to find out the size preference of paintings. He asked a number of people to indicate their choices of rectangles of different formats and found that their preference was for rectangles with proportions (ratios) near the Golden Mean. However, he found out later that measurements of hundreds of paintings in museums, on average, had proportions of 4:3 (1.33) for horizontal and 5:4 (1.25) for verticals.

As in many situations, rules should be considered guidelines to assist you in your decisions. The photograph should be sized and framed to a ratio that works well for the print. For example, a panoramic print would not fit well into a Golden Mean ratio.