Fibonometry

The term fibonometry was coined by John Conway and Alex Ryba in their paper titled, you guessed it, “Fibonometry”. The term describes a freaky parallel between trigonometric formulas and formulas with Fibonacci (Fn) and Lucas (Ln) numbers. For example, the formula sin(2a) = 2sin(a)cos(a) is very similar to the formula F2n = FnLn. The rule is simple: replace angles with indices, replace sin with F (Fibonacci) and cosine with L (Lucas), and adjust coefficients according to some other rule, which is not too complicated, but I am too lazy to reproduce it. For example, the Pythegorian identity sin2a + cos2a = 1 corresponds to the famous identity Ln2 – 5Fn2 = 4(-1)n.

My last year’s PRIMES STEP senior group, students in grades 7 to 9, decided to generalize fibonometry to general Lucas sequences for their research. When the paper was almost ready, we discovered that this generalization is known. Our paper was well-written, and we decided to rearrange it as an expository paper, Fibonometry and Beyond. We posted it at the arXiv and submitted it to a journal. I hope the journal likes it too.


Share:Facebooktwitterredditpinterestlinkedinmail

2 Comments

  1. Leo B.:

    sin(2a) = 2sin(a)cos(b)

    b -> a

  2. tanyakh:

    Thanks, Leo, I fixed it.

Leave a comment