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.
Next Section: Other Derivations of the Ratio of the Spiral Constant