Equivalence of Two Fibonacci Number Interpretations

n

1

figure

1

type

dominoes

reflections

both

This Demonstration illustrates the equivalence of these two interpretations of the Fibonacci numbers

F

n

:

• the number of ways to tile a

2×(n-1)

checkerboard with

2×1

dominoes• the number of ways a ray of light can pass through two panes of glass with

(n-1)

reflections

In the figure, when viewing reflections, the top row of the checkerboard pattern represents one pane of glass, and the bottom row represents the second pane of glass; reflections can occur on any of the three "surfaces" of the panes.