The Golden Ratio is the number (1 + sqr[5] ) /2 . "sqr[5]" means the square root of 5. As a decimal, it is 1.61803... to five places. Try punching this out on a calculator, and then punch out the inverse, 2 / ( 1 + sqr[5] ) and you will get the same decimal, but without the "1", 0.61803. This is true to any number of decimal places! This is the only number with this property. It is sometimes written with the greek letter phi. Another way to write it is the following:
phi - 1 = 1/phi
This number phi is the solution to the equation (using ^ to indicate an exponent)
x^2 - x - 1 = 0
Also,this number is related to the way the diagonals of a regular pentagon divide each other. In the regular pentagon shown here, notice that diagonal EC is emphasized with a dashed bold segment and a solid bold segment. If you take the length of EF and divide by the length of FC, you get the Golden Ratio.
If you make a rectangle whose length to width ratio is this same number, it is called the Golden Rectangle. It was used by the Greeks to make an especially "pleasing proportion" in constructing things. You can take a Golden Rectangle, divide it into smaller rectangles that are also Golden Rectangles, and in the rectangle you will have a sort of sequence of squares that get smaller and smaller in a spiral. You can then connect the centers of these squares and you get a spiral curve. This is illustrated below. In the rectangle on the left, AB divided by BC gives the 1.618... ratio and so does AD over DC. The spiral shown in the rectangle on the right is known as the logarithmic spiral and it occurs in many ways in nature. Why should this form of curve, related to the ratio and the series, recur over and over in so many unrelated phenomena in nature? It is logical that it shows the deliberate work of a Creator-God.
This spiral curve is the shape of a very wide variety of things in nature in both the living and the nonliving realms. The most well known example is the shell of the chambered nautilus. The Divine Proportion book relates this Golden Ratio to such things as the possible histories of energy level transitions of an electron in a hydrogen atom, the breeding of rabbits, the tendancy in some plants for the leaves to arrange in a helix, and the double spiral pattern in the florets of a sunflower plant. (The pattern is not perfect probably in many sunflowers, but if you had a very good specimen it would form superposed right and left handed spirals.) This double spiral pattern on the Sunflower is interesting. The seeds grow into the head of the Sunflower along these opposing spirals, so that the seeds are always equidistant from each other. This arrangement is also said to allow for the best use of the surface area, maximizing the number of seeds on the curved surface. The tendancy of leaves on plants to follow the spiral makes for efficency in gathering both light and water. Of course, by natural processes alone, these relationships would be merely accidental mutations, with some help from natural selection.
Furthermore, the Golden Ratio is related to the Fibonacci series. This series is familiar to anyone with a fair amount of math background. It is the following series, where each number is the sum of the previous two numbers. Below is one example of the series.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . 144, 233, . . .
You can also pick any two whole numbers at random, add them together, and start a Fibonacci series that way. For example,
5, 12, 17, 29, 46, 75, 121 . . .
-4, 1, -3, -2, -5, -7, -12, -19, -31, -50 . . .
Taking ratios of successive terms from this series will approach the Golden Ratio, in the limit. So, if you take a calculator, in the first example, divide 34 by 21 and then divide 233 by 144 you will see that the ratio approaches the decimal given above. The same is true for the other two examples.
An interesting example of the Fibonacci series in nature is regarding bees. Some unique facts about Bees are that males are produced by the queen's unfertilized eggs, so they have only a mother, no father. The females, however, have both a father and a mother. Start by imagining one male worker bee, then figure out how many parents, how many grand-parents and how many great-grand-parents he would have. Working this out you can show that the number of bees of each generation follow a Fibonacci series exactly, both for males and females. No this is not the twilight zone, this is the intellegent arranging God has done in the real world.
The Golden ratio and the Fibonacci series in nature are great arguments for intelligent design in nature. This is just a brief start into the subject. In three dimensional geometry, the five regular "Platonic Solids," the tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron relate to the golden ratio in a manner similar to the pentagon shown above. For these solids, the diagonal planes are guess what? Golden Rectanges. There is much more in terms of unique mathematical properties of phi, relationships to musical octaves, and other examples in nature. Sir James Jeans is well known for saying "God is a mathematician." This subject argues for God's existence and his intelligent arrangement of nature to fit a pattern, and at the same time creating us in such a way as to be pleased by it! It argues for an absolute basis for beauty, that beauty is not all only in the mind of the human beholder. A strictly naturalistic thinker has trouble explaining beauty and this kind of order built into things. According to naturalistic evolution, all characteristics should give an organism some kind of survival advantage or an advantage in reproduction. But many beautiful aspects of living things have nothing to do with survival or reproduction. The beauty, symmetry, and order in nature shows the greatness of the God who made it all!
Wayne Spencer
Some sources:
"The Divine Proportion" by H. E. Huntley, from Dover Publications.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html