Wednesday, April 23, 2008

Fibonacci Rectangle Not a Golden Rectangle?

I recently started studying φ (Phi), the Golden Ratio.

One of the thing I noticed a lot while searching for information on the Web is that people often use the Fibonacci Rectangle and Spiral to demonstrate a Golden Rectangle and vice-versa. Are they that similar? After all, the Fibonacci Sequence's ratio is a) not the same as φ, and b) not constant (see the graph below).

The difference is seen when building the two rectangles using their own respected techniques. The Golden Rectangle is basically built from the biggest square to the smallest with a length to width (l/w) ratio of φ. The extra length to the square (creating the rectangle of φ/1) is itself another Golden Rectangle that you divide the perfect square out and obtain another Golden Rectangle, and so on—going inward, creating the Golden Spiral.

For the curious, here is how you build a Golden Rectangle.

Golden Spiral

The Fibonacci Rectangle is built from the smallest Square outward, following the Fibonacci Sequence.

(0, 1, 1, 2, 3, 5, 8, 13 ,21, 34, 55...)
When n > 1 → Fn = Fn-1 + Fn-2

So you start with a two 1unit² square, then build a 2unit² square under, then 3 units, then 5, then 8.

Fibonacci Spiral

If you build the Fibonacci Rectangle first and stop at a width of 8 units (the 8unit² square), you will get a length of 13 (the square next to it is 5 units, so 8+5=13... oh look, it's the next number in the Fibonacci Sequence). Trying to build the Golden Rectangle with an 8x8 units square will immediatly show you the problem. You end up with a length of 12.94427191 (4 + sqrt(80)) instead of 13, like the Fibonacci Rectangle.

Also, The Golden Rectangle is "Golden" because it's ratio (l/w) is the Golden Ratio. The Golden Rectangle built here is built with the Golden Ratio. The Fibonacci Rectangle has a ratio of 1.625 instead (13/8).

One important thing to note is that the bigger the Fibonacci ratio or Rectangle, the closer to the Golden Ratio you get. The median of the Fibonacci ratios is about equal to φ.

Fibonacci Graph
Graph of the ratios of the Fibonacci Sequence

No matter how close you believe the Fibonacci Sequence is related to the Golden Ratio, you cannot see both Rectangles/Spirals the same way. They are similar, but only the Golden Spiral is truly "Golden".