Tuesday, February 28, 2012

The Golden Ratio φ and the Fibonacci Numbers

 While mathematicians can be proud of π (or rather τ, because π is wrong), and physicists can claim the fine structure constant α-1 (≈137), I would like to claim φ for the computer scientists.
Being the golden ratio and all, it's pretty awesome. As Vi Hart explains in her video, the angles between leaves on a plant, the number of petals on a flower, and the number of grooves on a cone all follow the tenets of "the most irrational number" φ, which is bounded by the ratios of adjacent Fibonacci numbers. φ is also the only number for which φ2= φ+1 or changed slightly, φ2-φ-1=0, which we can solve using the quadratic formula. We get
 Plants weren't the only organisms to use Fibonacci numbers. In colonies of bees, drones will have 1 parent (the queen), whereas queens will have 2 (the queen and a drone). So a drone's family tree has 1 parent, 2 grandparents, 3 great-grandparents, and so on. That's pretty trivial, but how would you explain that a DNA molecule is approximately 34 angstroms long by 21 angstroms wide? Or that the ratio of lengths of the two subdivisions of the bronchi is 1:1.618. If you want something a little more touchable, the proportions of the human body, are all in line with the golden ratio. From the organization of our face (i.e. the distances between our eyes and chin) to the length of our limbs, to the perfect human smile, the golden ratio plays a well-noted role in our body's measurements.

Perhaps it shouldn't be surprising then, that there is a theory that the universe is shaped as a dodecahedron, which is in turn based on φ. Or that even in the nanoscale world, the frequencies of magnetic resonations are in the ratio of φ. The "magic" of this number hasn't escaped human attention. Leonardo da Vinci, Michelango, and Rapheal all used the golden ratio in their work. (I don't know about Donatello) Dimensions in architecture from the Pyramids to the Parthenon to the U.N. building have all been inspired by φ.
Within optimization techniques, the golden section search uses φ to find extremum of unimodal functions; within financial markets, φ is used in trading algorithms. In my humble opinion, I'm still skeptical of φ for all its glory, because there is an underlying force that simply results in φ (as Vi Hart considers in her third Plant video). But the golden ratio will certainly see much more use in human civilization because of its sheer usefulness. For more on the ubiquity and beauty of φ, I strongly suggest the following video:
"This pattern is not just useful, not just beautiful, it's inevitable. This is why science and mathematics are so much fun! You discover things that seem impossible to be true and then get to figure out why it's impossible for them not to be." - Vi Hart
Oracle ThinkQuest's The Beauty of the Golden Ratio


  1. Aren't these blog posts supposed to be related to the course???

    1. I take it you weren't in class on Monday! The golden ratio came up in one of the theorems and I asked Brian to post some of the things he had been reading/watching about it recently...

    2. Oh, sorry! My bad...