Speaking of sunflowers, Fermat’s spiral, and Fibonacci numbers….

Did you know?

Pictures copyright 2012 Russel Ray Photos

Speaking of sunflowers…..


You don’t remember us speaking of sunflowers?

We spoke of them in yesterday’s Friday Flower Fiesta!

So, as I was saying, speaking of sunflowers….

Did you know that what most people think of as the flower of the sunflower….


….is actually not a flower at all? It’s called an inflorescence, or flowering head, and comprises many hundreds of flowers, called florets. Each floret is capable of being pollinated and creating a seed, which is why you get hundreds of seeds from each inflorescence “flower.” The yellow petals around the flower are not flower petals at all. They are modified leaves, called bracts.

What I find extraordinarily interesting is how those little florets are arranged.

DISCLAIMER: I love mathematics, which is why Numb3rs is one of my favorite TV shows.

Those little flowers, those florets, are arranged in a certain pattern, described by Helmut Vogel in June 1979 (“A better way to construct the sunflower head,” Mathematical Biosciences 44 (44): 179–189). The arrangement of the florets is called “Fermat’s spiral” and involves Fibonacci numbers (made famous in the book and movie “The Da Vinci Code”) and the golden angle of 137.508 degrees, the approximate ratio of Fibonacci numbers, a sequence of numbers whereby the following number in the sequence is the sum of the preceding two numbers. The plain vanilla Fibonacci sequence begins with 0, 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584….

A sequence doesn’t have to begin with 0, 1 though. Pick any two numbers to start. Figure out the remaining nine numbers for this sequence:

4, 2

What does this mean to the sunflower and to us? Well, if you look at the inflorescence, you can see the spirals of the florets. Here’s my best picture showing the spirals:

Sunflower showing Fermat's spirals

Even more interesting is that the spirals go clockwise and counterclockwise, and the number of clockwise and counterclockwise spirals can be predicted using Fermat’s spiral and Fibonacci numbers.

Count the spirals going in one direction. Let’s say that we have 34 spirals going clockwise. In the other direction, counterclockwise, we’ll have either 21 spirals or 55 spirals, depending on the size of the inflorescence. See where I got those numbers? Look at the Fibonacci sequence of numbers up above. Find 34 and look for the numbers on either side of 34, which are 21 and 55!

You can even predict the pattern of the spirals if you simply count the number of florets. For example, if there are 500 florets in the sunflower inflorescence, here’s the predicted pattern:

Predicted sunflower spirals with 500 florets

Fermat's Spiral

Of course, the actual number of spirals and how beautifully they are arranged depends on Mother and Father Nature cooperating. If the plant suffers for water or nutrients, or little bugs eat part of the plant — things like that — the number and beauty of the spirals could be substantially different.

Does all of this have a practical application? Yes! Fermat’s spiral has been found to be the most efficient layout for concentrated solar power plants, due to the curvature of the Earth, distance from the sun, etc.

Here are some other sunflower pictures from my photograph collection. Not until two years ago while roaming the campus of San Diego State University (just a mile from where I live) did I know that sunflowers naturally come in colors other than sunny yellow.







What other famous plant can you name where the flower actually is not a flower but instead is an inflorescence surrounded by bracts?

Hint: It’s very popular around Christmas time.

That’s right, boys and girls, the poinsettia!

The big, beautiful “petals” are not petals at all, but are modified leaves called bracts. Look at the following picture and you can see all the teeny, tiny, itsy bitsy yellow flowers in the middle of the bracts:


Pictures copyright 2012 Russel Ray Photos

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Pictures copyright 2012 Russel Ray Photos


22 thoughts on “Speaking of sunflowers, Fermat’s spiral, and Fibonacci numbers….

  1. quirkybooks

    I don’t get on too well with maths. Probably due to the fact that my math’s teacher at secondary school use to bring his guitar in to school to play in class and I didn’t learn much. I love the flowers though. You have captured the detail amazingly. Thanks for sharing.


  2. valeriedavies

    That was absolutely fascinating…. what an Intelligent design!!!!

    My Christmas poinsettias I’ve managed to keep “flowering” and flourishing for two Christmases now….


  3. firstandfabulous

    Ok. Here we go: 4, 2, 6, 8, 14, 22, 36, 58, 94, 152, 246. I think.
    I just read a children’s book called Math Curse by Jon Scieszka (Amazon). Ever hear of it? If not, you might get a kick out of it. The teache’rs name in the book is Fibonacci. Now I know why! Funny book. Interesting facts here, Russel.

    Liked by 1 person

  4. John Smith

    Awesome. Okay so how can I use this formula to create the graphical dots in adobe illustrator. I’m looking to create a logo using this dots pattern, but have no idea where to start?


  5. Pingback: SNIPPETS (5-13-18) | Russel Ray Photos

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