Good as Gold: The Fibonacci Sequence, the Golden Ratio, and More

Updated: Dec 16, 2020

The Fibonacci sequence is one that's all too familiar in the math world. It's one of the first examples many people see of the beautiful side of math, and it's a great one. If you've already seen the Fibonacci sequence, don't worry! I got some really neat extensions you probably haven't seen before coming. If you haven't, we can't have that, so you're in for a treat!

The Fibonacci Sequence (and Bunnies!)

The Fibonacci sequence starts with 1,1 and each following term of the sequence is simply the sum of the two previous terms. So the third term is 1+1 = 2 and the fourth term is 1+2=3, and so on:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765...

Leonardo Pisano Fibonacci himself was a man that lived as early as the 1100s and 1200s. At the time, the Arab and Indian world was thriving in mathematics, and the European world was behind. Fibonacci was one of the first to introduce Arab and Indian mathematics to Europe, and in doing so, brought with him not only the Fibonacci Sequence, but also Hindu-Arabic numerals. Europeans were still using Roman numerals for the most part, which gave them no real insight into how numbers work, and Hindu-Arabic numerals (i.e. today's system: 1,2,3,4,5...) were revolutionary to European math. The influence Fibonacci had on European math is fascinating, and you can read more about it here.

Leonardo Fibonacci. Photo courtesy of Getty Images.

Back to the sequence.

One of the ways I've often seen the Fibonacci Sequence introduced is with bunnies. Say we start with one pair of bunnies, one male and one female. Every pair of bunnies mates at one-month-old and produces a pair of bunnies (one male, one female) at two-months-old and every subsequent month. The same rules apply to all pairs produced, and our bunnies never die. How many bunnies do we have at the end of each month?

Let's ignore the questionable family practices and the immortality of the bunnies for a second. This problem, however strange, actually produces some really cool math.

1. At the start, there is 1 pair of bunnies.

2. At the end of the first month, there is still 1 pair of bunnies. They mated, but there are no new bunnies.

3. At the end of the second month, there are 2 pairs of bunnies because the original pair produces a pair.

4. At the end of the third month, there are 3 pairs of bunnies because the original pair produces a second pair.

5. At the end of the fourth month, there are 5 pairs of bunnies because the original pair once again produces a pair, and the second pair born two months ago does as well.

6. At the end of the fifth month, there are 8, then 13, then 21... why? See if you can figure out why the Fibonacci sequence is emerging here before you keep reading.

The bunny problem setup. Photo courtesy of Maths.Surrey.Ac.Uk

Here's why: We're essentially adding the last two terms! We're summing up the number of pairs of bunnies we had a mere month ago (they're all still here) with the number of news pairs of bunnies we produce this month (it's the same as the number of pairs of bunnies two months ago because each pair born at least two months ago is old enough to produce a pair). That means we're adding the previous two terms of the sequence to make the new term, so it's just the Fibonacci sequence!

Even though this problem is a little ridiculous... here brothers and sisters mate and one male and one female are produced every time, this still can be modified to model real-life relatively accurately. After all, n bunnies do produce roughly n bunnies if a pair produces a pair, so a similar pattern can emerge if we do this with very large amounts of bunnies.

If we focus on the number of females, it takes a little bit of the ambiguity away and makes it a bit more familiar. Here's a problem created by the English puzzle whiz Henry E. Dudeney called Dudeney's Cows:

If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

This should look very familiar, so try solving it!

Section 1.3 on this great article talks about honeybee's family trees. (It turns out male bees only have a mother!) This is really fascinating approach to Fibonacci numbers, so I'd recommend reading it.

Fibonacci in Nature

This is one of the reasons Fibonacci shows up practically everywhere in nature! It seems animals (and plants) prefer it.

Let me show you some examples of plants that prefer Fibonacci numbers.

Most flowers have three, five, eight or even thirteen petals:

Count the petals! Photos courtesy of Go Figure Math

More than petals, we can look at spirals!

Here's a pinecone:

A spiraled pinecone. Photo courtesy of Phys.Org

Notice that we can look at two different sets of spirals, all the spirals that go clockwise (red) and all those that counterclockwise (green). It turns out that the number of red and green spirals are usually Fibonacci numbers (and usually adjacent Fibonacci numbers like 8 and 13 or 13 and 21).

We can also find spirals in other places like the center of a sunflower:

Sunflower seedhead. Photo courtesy of The Smart Happy Project

Or even on a plant stalk:

Some plant stems. Photo courtesy of Laura Resta.

This was one of the observations of the mathematician and astronomer Johannes Kepler of the 1500s and 1600s. He noticed that if you start on a leaf and rotate one, two, three or five turns around the stalk, there is a leaf lined up with the first one (and usually it's also the second, third, fifth, etc leaf too)

There are way more natural Fibonacci numbers that I'm showing you here, so go out in nature and find some! They're on vegetables, flowers, trees, and more... tell me what you find in the comments!

The Golden Ratio

As we get higher and higher up along the Fibonacci sequence, the ratio of adjacent terms gets closer and closer to a number called phi (ϕ) or the golden ratio.

The number is 1.618033988... and unlike one of the other well-known irrational numbers, pi (π), there's actually a formula for this one:

That picture ended up a little bigger than I thought... but now you really know it!