Fibonacci Ain’t Nothin’ But a Number

The Fibonacci sequence is probably something you learned about in math class at some point, but I never realized until well after school just how freaking cool it is. Ready to get your mind blown?

Fibonacci numbers are a series generated by adding the last two numbers together to make the next. Starting with zero, the sequence goes:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233″¦
(0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13“¦)

It doesn’t sound very interesting yet, right? Bear with me. Let’s see what happens when you map out the sequence on a grid:

Grid with squares with sides of 1, 1, 2, 3, 5, and 8 units long, arranged in a spiral pattern
Image Credit: Wikimedia Commons, User Borb cc

The boxes make a spiral pattern, which is even more evident when you draw a curved line through the squares in an approximation of a golden spiral:

squares following the same pattern as above but with a curved line spiraling through them from corner to corner
Image credit: Wikimedia Commons, User Dicklyon public domain

Pretty! Turns out it isn’t just a randomly cool pattern; it’s a pattern that has implications in mathematics, computing, and even nature. In mathematics, every positive integer is the sum of numbers found in the Fibonacci sequence without repeating them. For example:

  • 17 = 13 + 3 + 1
  • 73 = 55 + 13 + 5
  • 426 = 233 + 144 + 34 + 13 + 2

Starting with 5, every second number in the sequence is the length of the hypotenuse of a right triangle whose sides are integers. Remember the Pythagorean theorem (x² + y² = z²)? The numbers that make up Pythagorean triples follow a pattern set by the Fibonacci sequence. The sequence also has many applications in computing, including pseudorandom number generators, parallel computing, and search techniques.

Nature, however, is where it’ll really blow your mind. The leaves of many plants are arranged in ratios that derive from the Fibonacci series, as are tree branches. Florets in the center of mature flowers such as sunflowers and daisies are arranged such that they form a different number of spirals depending on which direction you count them in, but both numbers are on the Fibonacci sequence. Pineapples, pinecones, and artichokes are arranged in similar spirals, and ferns uncurl new branches in Fibonacci spirals as well. There are also plenty of other things that closely mimic these spirals, such as nautilus shells, hurricanes, and the arms of spiral galaxies.

Examples of real or apparent Fibonacci spirals - a sunflower, spiral aloe, seashell, spiral galaxy, another flower, DNA, curled fern frond, and hurricane, captioned "Where there is matter, there is geometry" - Johannes Kepler
Found via I Fucking Love Science on Facebook

And while the image below is funny, I’m not aware of any actual evidence that pigeons hang out in Fibonacci sequences on purpose.

Pigeons on a rooftop, spaced out approximately on the same intervals as Fibonacci numbers

Thanks to Twiddle for suggesting this topic. Any other things you’d like to learn about? Leave ’em in the comments below.

By [E] Hillary

Hillary is a giant nerd and former Mathlete. She once read large swaths of "Why Evolution is True" and a geology book aloud to her infant daughter, in the hopes of a) instilling a love of science in her from a very young age and b) boring her to sleep. After escaping the wilds of Waco, Texas and spending the next decade in NYC, she currently lives in upstate New York, where she misses being able to get decent pizza and Chinese takeout delivered to her house. She lost on Jeopardy.

7 replies on “Fibonacci Ain’t Nothin’ But a Number”

Can you explain Avogadro’s Number? (I have no idea if I spelled that correctly). Like, how he came up with it? It’s the only thing I remember from chemistry and all I can envision is some guy bent over a microscope with teeny-tiny tweezers and counting atoms. Or he just made it up…

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