I can’t believe I’ve taught math for almost six years and not done much with the golden ratio, phi (φ) ≈ 1.6180339887, and the associated Fibonacci sequence 0,1,1,2,3,5,8,13,… (add the previous two numbers to get the next number, so 8+13=21 c0mes next)! Especially when so much of my own research in college was connected to phi. You see that spiral up in the blog heading? It’s related to the more famous golden spiral and Fibonacci spiral,
but it’s actually something never-before-seen I discovered about six years ago: the Yates Golden Diophantine Spiral. It has the special property that, when centered in a coordinate plane at (φ,0), it has x-intercepts at precisely ratios of consecutive Fibonacci numbers! For example, at 3/2, then at 5/3, then at 8/5, etc. That’s something those other spirals can’t claim!
And then there’s my research into continued fraction representations of irrational numbers, of which phi is the simplest:
It turns out that every quadratic irrational –like φ = (1+√5)/2– has a continued fraction that repeats periodically at some point (e.g. √7 = [2;1,1,4,1,1,4,1,1,4,…] has period length of three because 1,1,4 repeats). Now this by itself is pretty cool since quadratic irrationals’ decimal expansions continue forever but never repeat! Anyway, seven years ago during the summer, a group of three other undergraduate students, myself, and our faculty advisor were able to find a way to write alternative continued fractions for every quadratic irrational number with just a single number repeating every time (period length one). This is true even when the standard continued fraction has a period a million numbers long, which, when you think of it, is pretty surprising! Our results were published in the Journal of Number Theory. In fact, our first step toward discovering these results was experimenting with some Fibonacci identities like these.
So, it seems like a natural topic for me to include phi and Fibonacci in my classes, especially with all the geometry involved in golden rectangles and spirals, and the quadratic equation that generates the algebraic number phi. But, aside from a short detour my first year teaching Algebra I, where we were talking about patterns including the Fibonacci sequence, and my students asked me to explain my research, I have not done anything with phi in teaching Geometry or Algebra 2.
I’ve developed a project centered around that video, that connects Algebra 2 to Biology and Art. More on that tomorrow. 🙂