Fibonacci’s Area Problem

Afew years ago, a curious series of numbers got a lot of attention. Books were written about them, they played a part in solving crimes on TV, and people who really weren’t all that interested in math knew at least the nickname of Leonardo of Pisa.

The first few digits of the Fibonacci Series are 1,1,2,3,5, and 8. Each successive number is the sum of it’s two predecessors (5+8=13, 8+13=21, 13+21=34, etc), which makes it a pretty easy series to grasp. As the numbers get larger, the ratio between them approaches the irrational number “phi” (Φ, the “golden” ratio), which makes them a favorite of those who like to see order (or patterns) in everything. So, it has something for everybody!

My favorite little Fibonacci factoid is what some refer to as the “area problem.” It’s not really a problem, but it’s interesting …

Consider a square measuring 21 by 21 units with an area of 441 square units. Dissect it in segments with measurements equal to the two previous Fibonacci numbers (8 & 13), rearrange the pieces, and presto! It is now a rectangle measuring 13 units by 34, with an area of 442 square units. (This is where the geometry teacher steps back from the chalkboard and sweeps their arm for dramatic effect).

FIBONACCI-AREA

Upon closer inspection, though, we can see that there is actually a fifth piece – a tiny sliver measuring one square unit that fills in the diagonal gap between those four pieces. It’s hard to see, but it’s there … mystery solved!

But what if we do the same exercise with a square measuring 13 by 13 units (169 square units)? Those pieces rearrange into a rectangle measuring 8 units by 21, which is an area of only 168 square units. That’s because the four pieces now overlap in the middle, which is where that extra square unit went.

As a formula, the difference between the square and the rectangle looks like this (This formula is known as “Cassini’s Identity” – just in case you’re the curious, ‘google it for more info’ type – the parentheses in the left portion of the formula aren’t necessary, but I put them in for the sake of clarity):

\left( F _{n-1} F_{n+1} \right) - \left( F_n \right)^2 = \left( -1 \right)^n

Every time you square a Fibonacci number greater than 1, it differs from the product of its neighbors by one unit – and only one unit … every time (There’s a great page outlining proofs of this over at cut-the-knot.org – you should check it out). For numbers in odd positions in the series (2,5,13,…) the square is one square unit larger than the rectangle; for numbers in even positions (3,8,21,…) the square is one square unit smaller.

Pretty interesting, no?