ammonite fossil

What is the Golden Spiral? To understand it, we need to draw it.

This discussion is an excerpt from the workshop hosted by The Desert Lab on January 30, 2021: “Fibonacci and Agaves” part of the series on Math in Nature.


How a weird sequence of numbers accidentally discovered by Fibonacci around the year 1200 is the key to the golden proportion, the golden rectangle, and golden spiral–life, the universe, and everything.

These geometric spirals are found everywhere in nature. Sometimes they are not as easy to see until you draw one to understand it.

What is a Fibonacci, really?

Leonardo of Pisa, called Fibonacci, was a medieval mathematician  (1170 – c. 1240). No, he didn’t design the leaning tower, but, chances are you have heard someone drop Fibonacci’s name, especially at an art gallery or cocktail party. You probably heard that it’s part of the sacred geometry that underlies Life, the Universe, and Everything. But did just knowing that impress you? Me neither.

A phrase I like to use at the beginning of a drawing class or workshop is, “You don’t truly understand something until you (try to) draw it.” Let’s do that. This number sequence really does form patterns and geometry found everywhere in nature. Why does the world like irrational numbers? Maybe it’s because nature is always changing and growing and this is the geometry of growth and form.

But first, let’s do the numbers.

Fibonacci

The Fibonacci Numbers

Fibonacci’s famous number sequence was simple: to get the next number in the sequence, add the previous two numbers. It was his answer to a popular number riddle: suppose a pair of rabbits, male and female, are able to mate and every month their offspring produce a pair of male and female rabbits. How many rabbits will there be in one year? 

The answer is 144, a Fibonacci number. As we’ll see, this sequence and the geometry it forms can be found throughout the natural world.

Odd fact: The number of petals on a flower is usually a Fibonacci number. What’s with that?
We’ll deal with it in another post.

Then there is this other number called Phi

The odd thing about the Fibonacci sequence is that the ratios of the successive numbers in the Fibonacci sequence–dividing one number in the sequence by the previous one, converges on a more and more accurate definition another number, 1.618….

8\5= 1.6   
377/233= 1.61802575…

1.618… is called Phi for short. I put ellipsis after it because the decimals go on to infinity (making it awkward to write out in its entirety). Because Phi is an irrational number, so it goes to infinity, coming ever closer to Phi, but never reaching a precise, definable quantity. For your glimpse of this slice of infinity, I calculated Phi out to 100,000 places.

1.618

There is something about an irrational number that fascinates the human eyes and brain. There are many irrational numbers and a few of them are found everywhere in nature. But, you have to know how to see them. Some call these numbers and their resulting math the basis of our perception of beauty.

One of the most famous irrational numbers is 1.618…, also called Phi or the Golden ratio. This proportion was so admired by architects, artists, and mathematicians throughout history that it was called “golden” even “sacred.”

To prove this, I’ll present the following 5 second video clip.

One reason this ratio is so admired is because one of its expressions is the golden rectangle. The proportions of the sides of this rectangle are 1 to 1.618. Artists and architects for centuries have been using this proportion to create what is considered the most harmonious relationships between elements.

This is a golden rectangle: the ratio of its long to short side is 1.618…

That was a lot, and there’s more to say, but with this knowledge drawing the Golden Spiral is more fun.

I’m going to do this freehand in pencil for two reasons: drawing with only hand and brain creates deeper learning. And it helps our drawing skills to practice estimating basic shapes. It doesn’t have to be perfect or lines straight. Rough shapes will work here.

If this seems weird, just humor me. Draw a tiny square in the lower quarter of a piece of office paper.
Draw another same-size freehand square on top of the first one.
Sketch another square the same height as the two stacked squares. (Hint: this is a golden rectangle)
Draw another square on top of your previous rectangle. This makes another golden rectangle. As we add squares, the resulting overall shape will always be a Golden Rectangle. THis is called “self-similarity” and is a simple fractal geometry. Each rectangle we create this way will be exactly the same–only the size will change.
Draw another square the same height as the previous stack. (Yes, another Golden rectangle) It’s getting lop-sided, but that doesn’t mater.
Draw another square with one side the same width as the previous rectangle.
Keep going the same way off to infinity and beyond. You may start to see the spiral hidden in these squares.
I went ahead and drew two more squares using the same pattern as the others.
Now, if you number the squares in order we drew them, you will get: FIBONACCI NUMBERS! As you add squares, the length of each side will be based on the first square being a unit of one. The side of each new square will always be in the fibonacci sequence.
If you draw the diagonal of each square, you can start to see the framework of a spiral.

Note that we are drawing diagonals of squares–generating another significant irrational number found in nature: the square root of two: 1.414… It’s also the hypotenuse of a right triangle, a central concept for the Pythagoreans. So, both of these mysterious numbers are present in the Golden Spiral.

The discovery that the diagonal of a square is an irrational, undefinable number caused a paradigm shift among ancient Greek philosophers. Legend has it that around 500 BCE the Pythagoreans threw Hippasus of Metapontum overboard because he published his proof that the square root of two is an infinite, uncountable decimal. Maybe the culture in academia has changed since then–or academicians now have more subtle but, equally effective means to settle disputes.

I sketched in the Golden Spiral in orange. So we created a fractal spiral based on the fibonacci sequence

To see how the Golden Spiral is a fractal, notice below that as it grows larger, it is always the same. This is true whether it’s a snail shell or a spiral galaxy.

Fibonacci spiral as a fractal

 

We covered a lot here. In future posts, I’ll show how artists and designers have used these simple proportions, leaned from keen observation of nature, to create harmonic compositions and structures. I’ll also show how this spiral is the secret of plant growth.

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