Wednesday, May 10, 2006

the golden ratio, φ

The Da Vinci Code makes quite a bit of the so-called Golden Ratio, φ, which is mathematically equal to (√5+1)/2. [In the DVC, it is described as being 1.618, which is accurate to 4 significant figures, but not exactly.]

It comes up all over the place in nature, but not as often as claimed in The Da Vinci Code. Wikipedia's article on φ is pretty helpful on clearing up some of the fallacies about it. It doesn't mention the "sacred feminine" at all, and I've never come across references to φ in that context outside the Da Vinci Code.

But why should it matter?

The book of nature is written in the language of mathematics.
Galileo Galilei

As Galileo said, mathematics is a very profound and useful tool for describing the world we live in. One of the simple tools used in mathematics is the Quadratic Equation, an equation which looks like this:

ax² + bx + c = 0

where a, b and c are any number at all. The simplest non-trivial quadratic equations have a, b and c all as either 1 or -1. This gives us 8 possible equations.

x² + x + 1 = 0 (equation 1)
x² + x – 1 = 0 (equation 2)
x² – x + 1 = 0 (equation 3)
x² – x – 1 = 0 (equation 4)
-x² + x + 1 = 0 (equation 5)
-x² + x – 1 = 0 (equation 6)
-x² – x + 1 = 0 (equation 7)
-x² – x – 1 = 0 (equation 8)

Of these 8 equations, only four have real solutions (equations 2, 4, 5, 7). The solutions to those four equations are:

equations 2 and 7: -φ or φ – 1 (which is also 1/φ) equations 4 and 5: φ or 1-φ (which is also -1/φ)

Because φ is so heavily involved in the solutions to those equations, that means it has a few useful properties:

  • If you square it, you add 1
  • If you divide 1 by it, you take 1 off
  • If you cut a line into two parts, one bigger than the other by a factor of φ, then the ratio of the total to the longer section will also be φ.

Those properties of the number mean that it does come up in nature and in geometry a fair bit, but not as much as claimed in the DVC.

Nothing mystical, just maths.

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