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:
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:
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.
And while the image below is funny, I’m not aware of any actual evidence that pigeons hang out in Fibonacci sequences on purpose.
Thanks to Twiddle for suggesting this topic. Any other things you’d like to learn about? Leave ‘em in the comments below.