image

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

6.2 equation

These can be confirmed easily; see Exercises 6.1 through 6.4. We can obtain their Lucas counterparts by changing c06-math-005 to c06-math-006. 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, ...

Get Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.