Fibonacci Day: Adding the Squares

Today is a Fibonacci Day! It is the first, and one is a Fibonacci number. One is in fact two Fibonacci numbers. In the last post, you learned how to square numbers that end in five. To keep this squaring theme, how about we square the Fibonacci numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34...
1, 1, 4, 9, 25, 64, 169, 441, 1156...

We've been doing lots of adding Fibonacci numbers. Let's finish it off by adding the square Fibonacci numbers.

1 = 1
1 + 1 = 2
1 + 1 + 4 = 6
1 + 1 + 4 + 9 = 15
1 + 1 + 4 + 9 + 25 = 40
1 + 1 + 4 + 9 + 25 + 64 = 104

Do you see a pattern? It is a little hard to find, but definitely present. Look at this:

1 = 1 =                                       1 x 1

1 + 1 = 2 =                                 1 x 2

1 + 1 + 4 = 6 =                           2 x 3

1 + 1 + 4 + 9 = 15 =                   3 x 5

1 + 1 + 4 + 9 + 25 = 40 =           5 x 8

1 + 1 + 4 + 9 + 25 + 64 = 104 = 8 x 13

They are the product of the two consecutive Fibonacci numbers! Why on earth would that be? I had recently looked for one on the internet, and found an amazing geometric proof for it.

Have you ever heard of the golden rectangle, or the golden ratio? We touched on the golden ratio when I gave you the explicit formula for Fibonacci numbers (the golden ratio is the same as the greek letter fi). The golden rectangle is a rectangle of which the ratio of the length and width is the golden ratio. Something else whose ratio is the golden ratio is Fibonacci numbers! So, the side lengths of the rectangle are consecutive Fibonacci numbers!

Since we are dealing with squares of Fibonacci numbers, let's make some squares.

We've just taken these squares and organized them in a fashion that makes the side lengths two Fibonacci numbers. We went up to 34 squared, so let's see what the side lengths are.

They are 34 and 55. So, to figure out the area of the whole thing, you can add up the areas of all the squares, or just multiply the 34 by 55. And because of the way it is laid out, you can do it with any Fibonacci numbers! I think that is really cool!

Bonus Pattern: How about we add the squares of consecutive Fibonacci numbers.

1 + 1 = 2

1 + 4 = 5

4 + 9 = 13

9 + 25 = 34

25 + 64 = 89

64 + 169 = 233

The sums of the consecutive square Fibonacci numbers is in fact a Fibonacci number. I don't know a proof for this, but please tell me if you find one!