The Spiral Constant Why should the rank size relationship closely parallel the curves of an equiangular spiral? The key to understanding the linkage lies with the mathematical constant phi derived from (1 plus-or-minus sqrt(5))/2 approx. equal to 1.61803... In addition to its relation to the constant angle of a spiral generated by the Fibonacci series (as in equation 1), the spiral constant, also known as the golden mean, can be derived from many sources in mathematics and geometry. To cite just two examples (Huntley, 1970, 24 25, 42): 1. Any two intersecting diagonals of a regular pentagon divide each other into two segments having the ratio (phi):1; 2. If the edge of a regular decagon is 1, the radius of a circle circumscribed around the decagon is (phi):1 In addition to, and in part because of, its recurrence in geometric figures, numerous aesthetic qualities have been ascribed to the ratio. Indeed, the symbol phi was selected to designate this ratio in honor of Phidias, a Greek sculptor who made use of the ratio in his works. Another example from the Greeks is that the facade of the Parthenon is a rectangle, the sides of which are in the phi ratio. In modern architecture, Le Corbusieur is known for his use of spirals and the golden mean in his work. Tufte (1983, 189) refers to this ratio as a venerable but dubious rule of aesthetic proportion. In this paper, the term spiral constant rather than phi will be used to avoid confusion with Euler's formula, but the symbol will be used (written out as (phi) in the .html on the WebPage). The aesthetic appeal of rectangles, in particular, those with sides in (phi) proportion has not gone unnoticed by designers and marketers of goods. One will find that a large proportion of everyday goods have a rectangular shape approximately in f proportion: houses, rooms, desks and table tops, books, cigarette packages, newspaper pages, briefcases, television sets, toasters, microwave ovens, and even 3x5 and 5x8 index cards (although no one ever accused these last two items of having much aesthetic appeal). It has been suggested that the aesthetic appeal of objects in (phi) proportion derives from our subconscious familiarity with numerous objects in the natural world, especially organic objects, which have dimensions in proportion. Many average size relationships in portions of human anatomy can be approximately described by (phi): the ratio of one's height to the height of one's navel; the length and width of one's head are two examples (Ghyka, 1977, 97). Needless to say, the frequent occurrence of the spiral constant in the most unexpected places in mathematical formulas as well as in the physical world has led, on occasion, to endowment of the ratio with mystic qualities. The ratio has thus been called the Golden Mean, the Golden Ratio or the Divine Proportion. Figures in the proportion of the spiral constant are sometimes given names such as Golden Rectangles and Golden Triangles. Mathematicians do not attempt to answer "why" the spiral constant appears repeatedly and in apparently unconnected places in both the mathematical and physical worlds. The spiral constant, like pi, simply occurs. On to the next section