*by Paul Mirocha*

*This discussion is an excerpt from the workshop hosted by The Desert Lab on January 30, 2021: “Fibonacci and Agaves” part of the series *Seeing 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, the Golden spiral–and life, the universe, and everything.**

### What is a Fibonacci, really?

Leonardo of Pisa, called Fibonacci, was a medieval mathematician (1170 – c. 1240). He didn’t design the leaning tower, but, chances are you have heard someone drop Fibonacci’s name, perhaps 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 universe like these numbers so much? 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.

### 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.*

**Odd fact:**The number of petals on a flower is usually a Fibonacci number. What’s with that?

### Then there is this other number

called Phi

Here’s where it gets weird. What mathematicians discovered about the Fibonacci sequence is that the ratios of the successive numbers in the Fibonacci sequence–that is, dividing one number in the sequence by the previous one–getting closer and closer to a special 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). Because Phi is an irrational number, it goes to infinity, never repeating, getting 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

1.618, one of the most famous irrational numbers, is also called the Golden ratio, 1:1.618…. This proportion was so admired by architects, artists, and mathematicians throughout history that it was called “golden” even “sacred.” Some call these numbers and their resulting geometries the basis of our perception of beauty.

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

One reason this ratio is so admired is that it’s found in the golden rectangle. The proportions of the sides of this rectangle are **1 to 1.618**. And why is this proportion so admired?

Once they discovered it in nature, artists, philosophers, mathematicians, and architects used this ratio in their work. They passed this secret formula down through the centuries to create what is considered the most harmonious relationship between visual elements. There’s just something about it. It’s a dynamic form–the brain tries, but just can’t quite map out this rectangle using simple numbers. That’s because it’s based on an irrational number.

So far this has been about straight proportions. But the most interesting part is how the Fibonacci numbers, the Golden Proportion, and the Golden rectangle are all contained in the Golden spiral.

Swiss mathematician Jacob Bernoulli was so impressed with this spiral that he gave it a mystical meaning. When he died in 1705, Bernoulli requested this Spiral be carved on his gravestone, along with the Latin phrase, “**eadem mutata resurgo**”, (“Although changed, I arise again the same.”)

## Drawing the Golden Spiral

I’m going to draw this freehand in pencil for two reasons:

1. drawing using only the hand and brain creates deeper learning. The hand is an extension of your brain.

2. it helps our drawing skills to practice estimating basic shapes. It doesn’t have to be perfect or lines straight. Rough shapes will work just as well or better than using a straight edge and drawing compass.

Here is a quick 1.5-minute preview of me drawing the spiral. Then we’ll do it together. After drawing this, some people report a brief spark of enlightenment.

1. Draw a single tiny square in the lower right quarter of the page. I sheet of printer paper will do. This is the seed of the fractal pattern.

2. Draw another same-size freehand square above and sharing the top side of the first one.

3. Sketch another square to the right, the same height as the two stacked squares. (Hint: this is the elusive Golden rectangle)

4. 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 simple fractal geometry. Each rectangle we create this way will be exactly the same–only the size will change.

5. Draw another square the same height as the previous stack and below it. (Yes, another Golden rectangle) It’s getting lop-sided, but that doesn’t matter a bit.

6. Draw another square below the previous group with one side the same width as the previous Golden rectangle.

6. Draw another square below the previous group with one side the same width as the previous Golden rectangle.

9. Now, if you label the squares by their size based on the first square being 1 unit, 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 length of the sides of each new square will always be in the Fibonacci sequence.

10. 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 other 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. A basic tenet of their worldview was that the universe was countable, definable with whole numbers and fractions. The discovery of the irrationals drove them crazy.

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–the length of the diagonal of a square or hypotenuse of a right triangle– is an infinite, uncountable decimal. Maybe the culture of academia has changed since then–or academicians now have more subtle but, equally effective means to settle disputes.

Eventually, the Pythagoreans accepted this proof and the irrational nature of the hypotenuse became the doorway into a secretive mysticism based on a Cult of Numbers.

To finish this drawing, I sketched the spiral in orange marker. I encourage you to draw this freehand, just by hand and eye. It’s an exercise that will strengthen the visual/spatial parts of your brain, enabling you to observe and see more in the world around you.

To see one reason the Golden spiral so impressed Jacob Bernoulli, notice below that it is a fractal, although the word was not invented yet. A fractal is “self-similar.” As it grows larger, it is always the same. This is true whether it’s a snail shell or a spiral galaxy.

## Epilogue

This 2-hour UA Tumamoc workshop “Fibonacci and Agaves” was held online on January 30, 2021.

The Tumamoc Art&Science initiative was an experiment. It has been closed due to changes in the University of Arizona Desert Lab administration, and the recordings are no longer available. However, the concept is not dead, only waiting for its time to come. In response to requests, I’ve posted the 2-hour video of the class on my Vimeo account. You can find it here as * Seeing Numbers in Nature: Plants Teach Us to Draw the Golden Spiral*.

–Paul Mirocha, February 2024

WOW! As an artist, that needed a tutor to get a D in Geometry, I really appreciate this simple explanation. And as a visual learner, the drawing REALLY helped!! I really wanted to get a better understanding of the Goldne Triangle and have read several articles on it. But actually drawing along with you helped me to understand it in really simple terms. I have a painting that needed this to be added to it and now I think this will really make a difference. But not just in this painting, but in many to come. Thanks again!! I’m so glad I found this!!

Have a Golden Day!

Judy G.

Thank you! I want to represent these concepts in a tattoo, all I know is that it feels magical. I’m horrible at math and geometry! ☺️ So thank you for helping me wrap my head around it!

Thank you for this, reading Jim Carrey’s book “Memories and Misinformation” lead me here. He mentioned Fibonnaci spirals, I Google imaged it, that led me to, “What is The Golden Spiral?”. Very kind of you to publish this for people. Grateful for the lesson.

Very well done to help visualize this sequence.

Hey, I did experience that little spark of enlightenment!

So simple, but actually putting pencil to paper for myself made this concept come to life.

I too will look at my surroundings, everyday life, with a little bit more enlightenment now.

Thank you.

Thanks Shawn–that is so encouraging to have someone have the same experience that I did with drawing making a concept come to life…

Paul

FANTASTIC!

Here’s a math critique, but let me start by saying you’ve done great art and pretty close math.

You’ve actually drawn a Fibonacci Spiral, which is close to a Golden Spiral. The ratio of any two numbers in a Fibonacci Sequence is not precisely the Golden Ratio. Still, as the Fibonacci Sequence gets to greater and greater numbers, the ratio of Fibonacci Numbers approaches the Golden Ratio in the limit. And you can’t tell the difference with your naked eye, so this is just fine for the arts.

Good blog.

Thanks for your comments and correction. Yes, the Fibonacci and golden spirals are slightly different, and I will make this distinction in future. The intention of this post was more to the point of understanding abstract mathematical ideas by experiencing the physical feel of drawing the shapes freehand. That way one uses both abstract and visual parts of the brain together, making for deeper understanding,

In my understanding, the distinction is almost academic because they look the same when you are drawing them freehand. Just as the ratio between to adjacent fibonacci numbers gets closer to the golden ratio (1/1.618…) as the sequence gets higher, the fibonacci spiral gets closer and closer to the golden spiral as it grows. Since the golden ratio is an irrational number, the two spirals will never exactly coincide.

There’s something profound and beautiful about that image of the two spirals expanding together infinitely yet never reaching congruence.

Thank you so much for making these instructions available. I have to do an art project and the golden spiral is supposed to be in it.

Although I had to start over 3 times (haha, 5mm finally fit on the paper) I felt quite relaxed and joyful.

As mentioned by the previous commenter, this is not quite accurate; The Golden Spiral is a logarithmic spiral. This is the one you find approximations to, in nature. The Golden Ratio Spiral is an approximation of the logarithmic spiral, which can be drawn using the Golden Ratio boxes, each measured to Appx 1:1..618. What YOU have constructed is the Fibonacci Spiral, which is quite different because it starts from the centre, and each additional square has a slightly different proportion to the next, in line with the Fibonacci Series. Hence, what you have drawn cannot be fractal as it has an interior starting point very different to the outer parts. The centre is effectively a semicircle – this cannot even be regarded as an approximation to the logarithmic spiral. Hence, the animation cannot be the same shape. I know all the similar terminology is confusing, but best to get it straight before attempting to educate others…

Yes, thank you for correcting the use of more accurate terms. Please see my previous response for more.

Paul

Can you make this recording available to be purchased and viewed now ?

Felice

The video of the class is no longer available through the Desert Lab site, but I added a link to the video in an epilogue to the post. See it here: https://vimeo.com/915406797

Dear Paul,

Thank you very much. Your work is very meaningful to me. Since I was a child I have been seeing the numbers and fascinated by what I see, enjoying nature so much and wanting to learn more. I can’t wait to watch your video.

Felice

Felice, Can I connect with your site or blog if you have one?

–Paul

A year ago, as a newbie artist, I started creating designs using lines, circles, triangles and spirals using black markers on paper. I filled the paper with these designs. It was very calming to get lost in creating these pieces of art. I framed two of them ( 24 x 18) and hung them in my home. They were lovely. Then I entered one in an art exhibition and it was accepted based on it’s unusual and pleasing composition. It’s crazy, but looking at my piece hanging in the museum was the first time I noticed the unusual design I had created with in a large circle filled with black dots that I started in the center spiraling outward. It looked like a flower. I tried to repeat the shape/pattern but I couldn’t “copy it”. It was as if I was flowing naturally the first time I was creating, caught up in the spiral movement of my hand. Now that I read your article, I feel like my spiral was meant to be. Also, I am a dancer and have used spirals in my choreography. When I teach, i sometimes end the lesson with everyone holding hands in a line while I lead them into a spiral, pause for a meditative moment and reverse the movement to “uncork” the spiral. Very powerful for the group. The most emotional moment came with 150 people holding hands weaving like a serpent around the room and forming a spiral causing us to be able to look closely at each other. Each person had come that day to remember a loved one who had passed. That spiral reminded us that we were united in our loss and remembering.

Chris, I’d like to see that marker design you speak of. In some ways, there’s nothing mystical about sacred geometry. It’s just baked into organic reality, which includes our minds as the natural path of least resistance for contunual growth and creativity. I like your description of the dance–it’s a natural way for a large group to connect in a limited space in the most efficient way. The movement just flows like that and creates this feeling you describe. The spirals in a sunflower head work tin a similar way, it’s the most effective way to pack a lot of florets in the flower head as it changes and grows over time. If you want to send an image, I’m also at paul@paulmirocha.com.