Tag Archives: fibonacci numbers

Nature’s Geometry: Succulents—SALE!

Nature's Geometry: Succulents, by Russel Ray

I had 400 copies of my book printed in October 2019.

Based on my sales at cactus & succulent club meetings and shows, I should be out of them by now.

However, the pandemic canceled meetings and shows throughout the nation.

Meetings where I was scheduled to do a 45-minute Powerpoint presentation and sale books were canceled at

    1. South Coast Cactus & Succulent Society (Los Angeles, California)
    2. Gates Cactus & Succulent Society (Riverside, California)
    3. San Francisco Cactus & Succulent Society
    4. Sacramento Cactus & Succulent Society
    5. Long Beach Cactus & Succulent Society (California)
    6. Los Angeles Cactus & Succulent Society
    7. San Diego Cactus & Succulent Society
    8. Tucson Cactus & Succulent Society (Arizona)
    9. Atlanta Cactus & Succulent Society (Georgia)

I have been selling them for $35 at my Etsy shop, $30 at plant shows, and $25 at meetings.

Effective today, I am now selling them for $25 at my Etsy shop.

Nature’s Geometry: Succulents

I’d rather have my book in YOUR hands than sitting in boxes in my garage. I don’t know how long I will sell them at this price, so get yours today!
Etsy also takes credit cards. Free USPS Media Mail shipping to U.S. locations.

Facebook review of "Nature's Geometry: Succulents"

San Gabriel Valley C&SS review of "Nature's Geometry: Succulents" presentation

Facebook review of "Nature's Geometry: Succulents"

Nature's Geometry: Succulents back cover

Thanks for stopping by! See you next time!

Did You Know?—Spirals in plants

Did you know?

Fibonacci Numbers
1 – 1 – 2 – 3 – 5 – 8 – 13 – 21 – 34 – 55 – 89 – 144 ….

One of the cool things about spirals based on Fibonacci numbers (i.e., the golden spiral) is that in many plants, one can see both clockwise spirals and counterclockwise spirals.

The number of spirals in each direction in a mature plant almost always are consecutive Fibonacci numbers.

In the following picture of a mammillaria seen at the Los Angeles County Arboretum on February 13, 2020, there are 34 clockwise spirals and 21 counterclockwise spirals.

Mammillaria at the Los Angeles County Arboretum

Nature’s Geometry: Succulents—Orostachys spirals

Nature's Geometry: Succulents by Russel Ray

I’m always on the prowl for plants that exhibit relationships derived from the Fibonacci sequence of numbers. The sequence was published by Leonardo Pisano Bigollo (ca. 1170–1250), better known as Fibonacci, in 1202 in his book, Liber Abaci. (In his book, Fibonacci also introduced Arabic numerals to the Western world. If not for him, we might still be using Roman numerals!)

Fibonacci introduced the Fibonacci sequence of numbers to solve a problem on rabbit breeding. Apparently, rabbit overpopulation was a serious problem in Italy in his time. Here’s the problem:

Beginning with a single pair of rabbits (one male and one female), how many pairs of rabbits will be born in a year, assuming that every month each male and female rabbit gives birth to a new pair of rabbits, and the new pair of rabbits itself starts giving birth to additional pairs of rabbits after the first month of their birth?

Rabbit bathFibonacci determined that the first pair of rabbits would have 377 pairs of rabbits, or 754 rabbits during the year (assuming no rabbit deaths!). If you’ve ever taken care of rabbits for an extended period of time, you know that 754 rabbits is a gross undervalue!

Without going into a great deal of mathematics, the relationship between individual numbers in the Fibonacci sequence creates what are called golden segments, golden ratios, golden squares, golden triangles, and golden spirals.

I became fascinated with the Fibonacci sequence in 1972, and how they manifest themselves throughout nature and the universe.

Golden spirals are my favorite, and I recently discovered the Orostachys genus, species of which are absolutely gorgeous in their display of spirals. Here are two pictures of Orostachys spinosa, a plant that now is high on my list of must-have plants:

Orostachys spinosa

I’m well on my way before I even start!

Flowers in Russel's cactus garden

Ever since I joined my first cactus & succulent club in February 2017, I had been wanting to do a presentation of nature’s geometry using the Fibonacci sequence of numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…..). It’s an additive sequence, meaning that the next number in the sequence is the sum of the two previous numbers.

The program chairperson for the first club told me in April 2018 that they had just had such a speaker a year ago, that he was very good, and that I would be hard-pressed to follow him. I told him that, since 1973, I had been studying the Fibonacci numbers and how they are expressed in nature, so I wouldn’t be hard-pressed to follow anyone. However, having a speaker just a year ago meant that having another speaker on the same subject the following year probably would not go over well. I got the name of the previous speaker, and it turns out that “a year ago” was defined as “five years ago.” More importantly, though, I since found out that the program chairperson for that club doesn’t schedule anyone who is not already on the cactus & succulent club speaking circuit. That, of course, begs the Catch-22 question, “How do you get on the speaking circuit to begin with?”

Turns out that one has to make contacts in a club which will take a chance on you, and that happened in June 2019 for me with the Palomar Cactus & Succulent Club of Escondido, California.

I recently found out two other ways: (1) write many articles throughout the years and have them published in reputable books and magazines, and (2) to publish a book, which I did in October 2019:

"Nature's Geometry: Succulents" front cover

Nature's Geometry: Succulents back cover

My book is for sale at my Etsy shop, $30 with free shipping to United States locations: etsy.com/shops/russelrayphotos

Many decades ago, being an author of a book was a pretty good indicator of expertise. In today’s world of self-publishing, not necessarily. That’s a problem that I dealt with with my own head when looking at publishers.

I wanted desperately for the Texas A&M University Press to publish my book since Texas A&M University is my alma mater, Class of ’77. However, the cost of having the Press publish it would have been about $44 per book, so I would have wanted to sell it for $50 to make at least a little money. The only books that are 174 pages that sell for that much money are academic books by academic publishers. So that was out.

The other problem was the time frame. It would have taken up to 18 months to publish the dang thing.

So self-publishing it was…. inexpensive and as fast as I wanted it to be. I chose BookBaby because they are a print-on-demand service. I can have one book printed, hundreds of books, or thousands of books. Of course, the more books one has printed, the less expensive the cost per book.

I could choose to have BookBaby completely involved in everything, or nothing. I chose nothing because I have been doing writing, editing, graphic design, book and magazine design and layout, and publishing all my life.

If I used all of their services, the cost would have been right up there with the Texas A&M University Press, and the lead time would have been up there, too. By buying an ISBN number from BookBaby and then using their printing services, I kept the cost low and the lead time short.

I am extremely happy with BookBaby’s printing, paper, and binding.

Having a book published immediately got me on the cactus & succulent club speaking circuit. I’m also exploring many other speaking circuits, including horticulture clubs, gardening clubs, community retirement homes (I had no idea that so many of them have regular programs; two already have expressed an interest), and city and county libraries, many of which have up to five programs each week.

The reception of Nature’s Geometry: Succulents also has me looking at doing another book. My two immediate choices are Nature’s Geometry: Flora and Nature’s Geometry: Fauna. However, my main goal is to stay on the cactus & succulent speaking circuit where I already know my intended audience and their likes.

I have noticed that a great majority of the cactus & succulent speakers give presentations on their travels to foreign countries. The Atacama Desert region of Chile and the Oxaca region of Mexico are two of the most popular. That, though, caused me to think that maybe, just maybe, the southwestern United States has a lot to offer.Fishhook barrel cactus

There are a lot of cacti & succulents that grow in our region, many of which are found only here. Carnegiea gigantea (the saguaro) comes immediately to mind, but there also is Ferocactus wislizeni, the Southwestern barrel cactus (picture at right), Agave utahensis (which, you might guess, occurs in Utah), Ferocactus cylindraceus, the California barrel cactus, and Yucca brevifolia, the Joshua tree.

Knowing this, I have decided to do a second book, tentatively titled
SSS: Southwest Succulent Staycation. For my purpose with this book, I will define “southwest” as California, Nevada, Utah, and Arizona.

United States SSS map

During 2020, I will be endeavoring to go to all the national parks, national monuments, botanical gardens, zoos (many zoos in the southwest also are botanical gardens), state parks, and cities with a population of at least 50,000. There will be exceptions, I’m sure. I already have a few million photos from excursions in the southwest, so I’m well on my way before I even start!

The nice thing about this second book is that, for people back east, going to the southwestern United States can be very much like going to a foreign country, so I might be able to get on cactus & succulent speaking circuits outside of my home territory of the southwest.

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.

Sunfower

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

Did you know?

Pictures copyright 2012 Russel Ray Photos

Speaking of sunflowers…..

What?

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….

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.

Sunflower

Sunflower

Sunflower

Sunfower

Sunfower

Sunfower

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:

Poinsettia

Pictures copyright 2012 Russel Ray Photos

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Pictures copyright 2012 Russel Ray Photos