Last spring I received a letter from a student who wanted to know more about me, and more, especially, about Fibonacci numbers. I wrote him back, and shared a bit more about Fibonacci numbers and their twin sibling, the Lucas numbers. Fibonacci numbers get a lot more attention, but, like real and imaginary numbers, there are many things about them that remain invisible until you put both together.
I’m including the letter below, and I hope you enjoy reading it, or sharing it with students in your life who might be interested in the Fibonacci numbers. Eddie Woo recently included some beautiful Fibonacci-related images in his TEDx Talk. These images, the “magical” connections are the bait we mathematician-educators use to draw people toward our topic. (Mystics do this too.) But you really see the magic and the meaning when you dig into them and do the math.
Here’s the letter.
Thanks for writing. I’m happy to hear about your passion for Fibonacci numbers! They deserve your excitement.
I grew up in Olympia, Washington, and have loved math for as long as I can remember. I’m a big fan of graph theory and combinatorics, which is the mathematics of how things connect, and counting. I’m quite familiar with Fibonacci numbers, though, and I’ve explored and even taught about them.
Here’s something really important about the Fibonacci numbers: they’re only half of the picture. There’s another sequence called the Lucas numbers that go like this:
1, 3, 4, 7, 11, 18, 29, …
(see what’s happening there?)
The Lucas numbers are like the twin sibling of the Fibonacci numbers. They really belong together, and you can see things when you put them together that are invisible with just one.
For example, have you ever added every other Fibonacci number together?
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, …
The sum of every other Fibonacci number is: 1 + 2 = 3, 1+ 3 = 4, 2 + 5 = 7, 3 + 8 = 11.
It’s the Lucas sequence! It was in there all along.
What if we add every other Lucas number together?
1 + 4 = 5, 3 + 7 = 10, 4 + 11 = 15, 7 + 18 = 25, 11 + 29 = 40, …
Notice anything about the sequence 5, 10, 15, 25, 40, … ?
For starters, all those numbers are multiples of 5. If we divide them by 5, we would get… (drumroll) the Fibonacci sequence again.
Once you have these two sequences, you can play away in almost any way you want to, and there’s some connection to find. Take the sums of every third, fourth, or fifth Fibonacci number or Lucas number. Or try the differences! Or square the numbers and add them together. Or cube them and take the difference. You won’t always be able to find a connection, but you’ll be surprised how often it works out.
Here’s one more to notice, just so you see how weird this all is. Write both sequences, one above the other, and then take the product of each number on top by each number on the bottom.
1, 1, 2, 3, 5, 8, 13
1, 3, 4, 7, 11, 18, 29
1, 3, 8, 21, 55, 144, 377
Notice anything about that bottom row? It’s every other Fibonacci number.
This is all fun, but I’m not explaining why this happens. I don’t have a book to provide that does, but if you feel like you can hack it, there’s a great video/essay series from James Tanton that you can get at. He talks fast, and some of the math is a little higher level, but it’ll give you a lot of cool ideas and connections to explore.
As for me, I’m working on all kinds of puzzles and games, most of which are about math but not about the Fibonacci numbers. There are a lot more neat connections to discover, though, and if you’re willing to put in the time to follow your passion, and don’t mind being stuck for a while when the going gets tough, I think you’ll be happy with what you get out of the exploration. Some of the connections related to Fibonacci numbers seem almost like magic. For example, if you want to know, say, the 50th Fibonacci number, just take the golden ration (approximate 1.618), raise it to the 50th power, divide by the square root of 5 (approximately 2.236), and round off your answer to the nearest whole number. It seems like magic, but it all makes sense once you develop the right tools, and put the time in. Once you do, you get to be the magician.
All the Best,