Tag Archives: fibonacci sequence

I guess I’ll find out whether it really likes me

I live in my own little world

Long-time friends and followers know of my infatuation with plants and mathematics, especially the Fibonacci sequence of numbers.

One of the numbers is 13.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89….

The Fibonacci sequence is expressed throughout nature in many ways: golden angle, divergent angle, golden spiral, golden triangle, Fibonacci triangle, and more….

So when I saw this Echinopsis atacamensis ssp. pasacana with 13 ribs….

Trichocereus atacamensis ssp. pasacana

….offered for sale by Gnosis Nursery of Ramona CA, well, I thought it would look good in my gardens with all my other Fibonacci plants.

I planted it yesterday.

I like it, and I think it likes me.

This plant is native to Argentina and Bolivia, and grows at elevations of 6,500 feet to 13,100 feet above sea level on steep slopes. It does not like high humidity, extreme heat, or lasting frost. The mean maximum temperature in its native habitat is about 85°F, and it only gets about 6 inches of rain each year.

I’m at 682 feet high, and we do get extreme heat out here, many consecutive days of 100+°F with a maximum (so far, in 2½ years) of 118°F.  Since January 1, 2019, I have had 46.75 inches of rain.

Consequently, I planted it on the side of the house that gets bright light and shade rather than in full sun. 

I guess I’ll find out whether it really likes me.


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