O'Reilly logo

Fibonacci and Catalan Numbers: An Introduction by Ralph P. Grimaldi

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Chapter 9

Optics, Botany, and the Fibonacci Numbers

As we shall soon see, the Fibonacci numbers even make their way into situations in the natural sciences.

Example 9.1: The following application was introduced in Reference [43] and then solved in Reference [42]. In the science of optics, the branch of physics where one investigates the propagation of light, another instance of the Fibonacci numbers arises. Start by considering a glass plate with two reflective faces, as shown in Fig. 9.1(a). Here Face 1 is the face on the left side of the glass plate, while Face 2 is the face on the right side. In Fig. 9.1(b), we see a single reflection (or change in direction) that occurs at Face 1, when we view this glass plate from the side.

If we now place two such plates back-to-back, as in Fig. 9.1(c), then we have four reflecting faces. Faces 1 and 2 belong to the plate on the left; Faces 3 and 4 belong to the plate on the right. When a light ray falls on this stack of two glass plates, let sn count the number of different paths the ray can take when it is reflected n times, where n ≥ 0. For example, in Fig. 9.1(d), we see the case where there are no reflections, so there is only one possible path and s0 = 1. Part (e) of Fig. 9.1 shows us the two different paths that can occur ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required