Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

GEOMETRIC ILLUSTRATIONS AND PARADOXES

A picture is worth a thousand words.

–A Chinese proverb

In the previous chapter, we encountered numerous Fibonacci and Lucas identities, and summation formulas. In this chapter, we add a geometric dimension to some of them, and to some new ones, by illustrating them pictorially.

To begin with, consider the identities

6.1

6.2

6.3

6.4

These can be confirmed easily; see Exercises 6.1 through 6.4. We can obtain their Lucas counterparts by changing to . The techniques work for any sequence that satisfies the Fibonacci recurrence, but always adjust the least value as required.

6.1 GEOMETRIC ILLUSTRATIONS

Interestingly, we can illustrate geometrically these Fibonacci identities (and hence their Lucas counterparts), as Br. A. Brousseau did, among others, ...