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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!

The golden ratio actually has a lot of practical applications too. The ratio for a rectangle (the length to width ratio) that artists often consider the most aesthetic is the golden ratio. It's even the ratio of Mona Lisa's face:

Mona Lisa. Photo courtesy of ResearchGate.

And artists also like to put the main point of a painting (the focal point) about 1/ϕ from the side. This is not something we measure out; it's simply something artists (and all people) find most pleasing subconsciously!

One other cool thing that comes from the golden ratio is the golden spiral. You get the golden spiral by taking a golden rectangle and removing the largest square on each side of the rectangle and then the rectangle remaining, etc

One of the essential properties of ϕ is that ϕ-1 = 1/ϕ. So if we say the width above is 1 unit and the length is ϕ units, removing 1 unit from the length in taking out a square leaves a length of ϕ-1 = 1/ϕ, so a remaining rectangle of sides 1 and 1/ϕ. 1/(1/ϕ) = ϕ, so the new rectangle is just a scaled down version of the golden rectangle! (And by the same reasoning, so is every other rectangle formed this way)

To make the golden spiral, simply connect the squares with a curve!

This is one of the most famous figures in all of math! It's also found in nature.

Here's a nautilus shell:

Nautilus Shell. Photo courtesy of NaturPhilosophie

The resemblance is uncanny. Here's a few more examples:

An Equation for Fibonacci?

So far, the only equation I gave you for Fibonacci was in terms of previous terms. If we say the nth Fibonacci number is Fn, we have:

F1 = 1, F2 = 2, Fn = Fn-2 + Fn-1

This definition, in terms of previous terms, is called a recursive equation. You have to calculate all the previous terms to find the one you want.

The other type of equation is called an explicit equation, where you don't need the rest of the terms. You can just plug in n. Is there an explicit definition for Fibonacci?

It turns out there is! This is called Binet's Formula:

That's a lot of square roots for an equation meant to give integers! But it works!

The first term raised to the nth power there is the golden ratio and the second is -1/ϕ.

See if you can use this equation to show why the ratio of the Fibonacci numbers gets closer and closer to the golden ratio as n gets really large. It's a cool bit of math, and it's a fun problem to solve!

Quadratics (Feel free to skip this section if you don't know quadratics yet!)

It turns out that the golden ratio is a root of the quadratic equation x^2-x-1=0, and -1/ϕ, just like above, is the other.

This equation is called the characteristic equation of the recursion equation

Fn = Fn-1 + Fn-2.

It turns out that if we're working with any recursion equation that describes a set Xn where we're given starting values, and Xn = AXn-1 + BXn-2, the characteristic equation is x^2-Ax-B=0, and the two roots of that equation are used in the exact same way as above (but if you have to solve for the coefficients—here, 1/sqrt(5))

This is rather complicated thing to prove and solve, but it's really useful for competitions and it's just a really neat piece of math, so try googling and learning a bit about recursion in general.

A Random Curiosity

It turns out the expansion of 1/89 is the sum:








Try adding the first few terms and confirming it.

Isn't that a bit bizarre?

A Few Problems

Before I go, I'm going to leave you with a few fun properties of the Fibonacci sequence. See if you can figure out why these are true. (Especially since most of us are stuck at home, try to do a bit of fun math!)

1. The sum of the first n Fibonacci numbers is F(n+1) - 1.

2. The sum of the first n odd terms of the Fibonacci sequence (F1 + F3 +...+F(2n-1)) is F(2n).

3. The sum of the first n even terms of the Fibonacci sequence (F2 + F4 +...+F(2n)) is F(2n+1)-1.

4. The sum of the squares of the first n Fibonacci numbers is Fn*F(n+1).

For the answers as well as lot of more fun problems, go here. These are a bit difficult, but I think you might surprise yourself.

One of my favorite problems of all time is a Fibonacci problem (it's a little too complicated to put here, but feel free to ask me in the forums if you want to try it). They're just so neat to solve and they give fascinating results!


Here's some of the amazing articles I used in writing this one that can give you even more information:

My favorite links by far are these two:

Go check them both out! They have so many ideas and anecdotes and cool concepts, and reading one or both of them is a great way to spend your time!

I'll also put two videos here (because I can't remember the last time I posted an article without a video):

(There are two more of her videos in this series, and all three are great!)

This is random, but it's also a lot of fun:

Thanks for reading and have a great weekend everybody! Leave a comment or a forum post if you have anything you'd like to say!



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