Fibonacci Ratios

We've noticed that as the Fibonacci Ratio Sequence continues, it gets closer and closer to . We also saw that the even terms of the sequence tend to be greater than , and the odd terms tend to be less than . If we plot these points on a graph, we get the following:

Note that, even though the graph shows that the Fibonacci Ratio Sequence actually becomes around the 7th or 8th term, this is not the case. It gets closer and closer, but the Ratio Sequence never actually reaches .

We can also show this relationship using the Golden and Fibonacci spirals, as shown below:

Notice that the farther you get toward the wide end of the spirals, the closer together the two spirals become. How can we explain this?

Every Golden Spiral has to fit perfectly within a Golden Rectangle, and if you subtract sections from the end of a Golden Spiral, it will still fit into a Golden Rectangle. Therefore, the rate that the Golden Spiral gets wider is constant. In other words, the spiral gets wider at the same rate everywhere. At what rate does it increase? Well, the rate at which a spiral gets wider is the same as the ratio of the rectangle in which it fits. Since every Golden Spiral fits into a Golden Rectangle, the rate at which it gets wider is .

The Fibonacci Spiral does not work the same way. Every Fibonacci Spiral fits into a rectangle of different proportions. Therefore, the Fibonacci Spiral increases at a variable rate. In other words, it does not get wider at the same rate at all points on the spiral. We can find the rate at which it gets wider at different points on the spiral by looking at the rectangles the spiral fits in.

The shorter dimension of this rectangle is 89, a Fibonacci number. The longer dimension is 89 plus 55 (the next Fibonacci number lower). When you add these two consecutive Fibonacci numbers together, you (obviously) get another Fibonacci number (144), which is the next one above 89. Since every Fibonacci Spiral rectangle is constructed in this manner, they all have dimensions which are consecutive Fibonacci numbers.

What we are looking for is the ratio of the sides of the rectangle that a Fibonacci Spiral fits in. Since all these rectangles have sides which are consecutive Fibonacci numbers, the ratio of the sides of these rectangles is the Fibonacci Ratios Sequence. Therefore, the rate at which a Fibonacci Spiral gets wider is given by terms of the Fibonacci Ratios Sequence.

So now we know that the rate at which the Golden Spiral gets wider is , and the rate at which the Fibonacci Spiral gets wider is given by the terms of the Fibonacci Ratios Sequence. As we just learned, as the Fibonacci Ratio Sequence continues, if becomes closer to . Therefore, if you graph the two spirals together, they will come closer and closer together as you continue. However, just as the Fibonacci Ratios Sequence never really reaches , the Fibonacci Spiral never really becomes a Golden Spiral.

Random Fibonacci number from the 1st through the 200th:

All contents, unless otherwise specified, are © 1999 by Matt Anderson, Jeffrey Frazier, and Kris Popendorf.
Created by Team 27890 for the 1999 competition.