The Divine Proportion Revealed

How Euclid found the Divine Proportion is remarkable. Looking for the magic behind certain geometrical shapes such as the pentagram and pentagon, he divided a simple line into two parts. He said the two parts of the line must be related in a very specific way. In Euclid's own words, A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. Sounds a lot more complicated than it really is, look at the graphic below:

What Euclid was getting at was:

AC/CB = AB/AC

Take the line noted as AC and divide it by line CB. If this number is equal to the number defined by the proportion AB divided by AC, then the number is said to be cut in extreme and mean ratio. The resulting number is the Divine Proportion. Given our discussion about how the Divine Proportion is used to form stems of a plant, used to create the human form, and used to regulate the universe through black holes, this simple formula seems incredulous. How could nature use such a simple idea to create all that there is? But, what is this number exactly? What number satisfies this equation?

Let the length of the shorter section CB be a unit of 1 and the length of the longer section AC be x.

Then:

AC = x, CB =1, and AB = x+1

So if we use the equation above AC/CB = AB/AC and replace with our algebraic equivalents:

Then, by multiplying both sides of the equation by x, and subtracting x and 1 from both sides, we get the simple quadratic equation:

Solving this quadratic equation we get the expression x equals 1 plus the square root of 5 divided by 2:

Using a simple pocket calculator, we can now calculate the Divine Proportion to be 1.6180339887……….this number sequence goes on forever never repeating, never ending.

Now that we have an algebraic expression for the Divine Proportion, we can calculate it to a very high precision. In 1966, M. Berg calculated the Divine Proportion to 4,599 places on an IBM mainframe. Today, your personal computer can do the same in about 2 seconds. In fact, the Divine Proportion was calculated to 10 million places in December 1996 and it took about 30 minutes. The proportion never repeated. The number could have been calculated another 10 million places and it would still not repeat a sequence. The Divine Proportion is indeed a remarkable number.

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