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Equivalence of Two Fibonacci Number Interpretations

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

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

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Contributed by: Robert Dickau (March 2011)
Open content licensed under CC BY-NC-SA

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References:

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd ed., Reading, Massachusetts: Addison-Wesley, 1994.

M. Gardner, Mathematical Circus, New York: Alfred A. Knopf, 1979.

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