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I would like to make this page available to people that have something they want to say but it may not be in publishable form. I expect it to be a readable discussion about some facet of bromeliad culture, perhaps a preliminary report which is intended to lead to a later publication.
As a first step, I'm putting here a recent observation regarding Fibonacci's numbers and Neoregelias. I intend to submit it to the Journal of the Bromeliad Society. But if someone catches something here, we could repair it before it gets into permanent print!!
Rusty Luthe of the James Clerk Maxwell Telescope at Mauna Kea Hawaii mentioned a Neoregelia he was coveting and then dropped the name Fibonacci.. So I immediately grabbed a good-looking neo in my greenhouse, attached numbers to the end of the leaves, photographed it, made a few drawings and came up with this article.
But first, a word about Leonardo. Not Leonardo da Vinci but Leonardo da Pisa. It appears he was either the son of Bonacci (filius Bonacci) or the son of good fortune. In any case he has gone down in the mathematical literature as Fibonacci, originator of the Fibonacci series. He introduced the arabic numeral system and the concept of zero to Europe in the twelfth!! century.
The series sounds kind of silly. You start with 0, then 1. After that each number is the sum of the two previous numbers. This makes the series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55..... Isn't this a silly series?
But, good grief, this series pops up all over nature. When Ruste mentioned the Fibonacci series and neoregelia in the same rush of electrons, I decided it would be interesting to see if there was a relationship here.
First of all, here is a drawing showing the numbering of the tips of the leaves in my neoregelia. There is a legend that shows the numbers of connections for four of the Fibonacci numbers. The center of the plant is indicated.
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Fibonacci Spirals
There are several interesting things. First, the eighteen 3 connections form a tight clockwise spiral. Second the eighteen 5 connections form a looser counterclockwise spiral. And third, the thirteen 8 connections form a looser clockwise spiral. The ten 13 connections form a sort of radial pattern. There would have to be over 26 leaves to show a spiral.
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I hope someone sees something remarkable and understandable about this.
Leaf Angles
And then there is this other idea. A retired physical scientist should not be turned loose with a ruler and a 360 degree protractor. I got to wondering if there was any regularity in the angles of the leaves as they radiate from the center of the plant.
And so I took my drawing of the leaves from the photograph, drew lines to the number and then measured the angles. Using the first leaf as the zero angle, I measured the angles and then took the differences. With 23 leaves, there were 22 differences. They are as follows: 153, 135, 132, 152, 145, 141, 115, 170, 98, 189, 130, 138, 150, 121, 151, 140, 141, 131, 144, 146, 136 and 138. This averages as 147.5 degrees of rotation per leaf.
And it appears to me to have a rather remarkable consistency. Considering the crudeness of the measurements, the numbers agree quite well. And more importantly, there doesn't seem to be any trend!
And unfortunately, I don't know what this means either!
I found a rather good looking Tillandsia lucida that seemed to be a good candidate. So I repeated the steps outlined above with the following results.
First, the picture below is the plant taken from overhead with the numbers on the leaves. The numbers in red are the "extrapolated" values. The observed values are 135.12, 136.39, 130.04, 139.92, 128.06, 143.07, 150.79, 188.98, 133.64, 147.78127.95, 182.63, 138.76, 127.75, 158.73, giving an average value of 144.7. I find the agreement between this and part one, 147.5 rather intriguing, perhaps even startling!
The four drawings that follow connect the leaves that differ by the number in the upper left.
There were four leaves unnumbered in the photo. I predicted what their number would be, based on the average given above. The 2 page shows a clockwise spiral, the 3 page shows a counterclockwise spiral, the 5 page spiral is clockwise and with the addition of the "extrapolated" values, the 8 page shows a counterclockwise spiral.
If you would like to learn more about the Fibonacci series and something about the Golden Section as well, find
