Lesser Known Facts About Sunflowers and Pinecones (Pt. 2)

In my last post I brazenly announced that in the sequel I would explain why there are usually two sets of spirals on a pinecone, and both are Fibonacci in number. When I went to write this all down, it turned out to be much harder than I expected! So much so that I decided to write up the details for publication, since they don’t seem to appear anywhere. Interested readers can find a preprint of this paper here. This post will borrow heavily from that paper, but here I can only give a general sense of what’s going on.

The Golden Ratio, often denoted with the Greek letter φ,  is the positive number whose square is one larger than itself.  It is often cited as a sacred quantity that appears in all kinds of places, from the Parthenon in Greece to DaVinci’s Mona Lisa. Most of these claims have been debunked (see,  for example, George Hart’s wonderful video here). However, it’s relationship to plant growth is undeniable. Many, many people have observed that in pinecones and sunflowers (or anything else exhibiting phyllotaxis), the angle between successive seeds/petals/buds is very often 360/φ degrees.

Another observation about phyllotaxis that has been repeated countless times is that the number of spirals is (usually) a number from the famous Fibonacci sequence,  usually either 3, 5, 8, 13, 21, or 34 (although you can certainly find larger numbers in nature).

The more technical papers on this subject focus on a possible biological mechanism for plant growth, and then give a mathematical optimization argument for why one or the other of these two observations follows as a result. Here I’ll do something a bit different and try to explain why, if you assume growth is governed by the Golden Ratio, then there must be a Fibonacci number of spirals. Then I’ll explain why two families of spirals in opposite directions are usually evident, both Fibonacci in number. Finally, I’ll talk about why flat phyllotaxis patterns, such as in sunflowers, seem to have different numbers of spirals depending on how far from the center you look.

We’ll start with cylindrical examples of phyllotaxis, like in the Brussels sprout stalk shown here.

To create the idealized model shown next to it, I placed “sprouts” at points around a vertical cylinder as follows. Begin at the bottom and place the first point. Then rotate around the cylinder 360/φ degrees, and move up the cylinder a distance h. Place the second point there, and repeat. To our eye, there are obvious spirals apparent in this model.  The explanation for these perceived spirals is simple: our brain creates them by connecting each point to its two closest neighbors, as shown in the third picture above.

Let’s label the points in the order they were added, starting from zero. As we go around the stalk adding seeds we eventually come to a point that is closer to point 0 than any other point. The distance from point 0 to point n is the same as the distance from point n to point 2n, and the distance from 2n to 3n. Visually, these points all lie on the same spiral. For example, in the following image, point 0 is connected to point 8 and point 8 is also connected to point 16, etc.

Note that the distance from point 0 to point n is the same as the distance from point 1 to point n+1. Hence, there is another spiral going through points 1, n+1, 2n+1, etc. Similarly, there is a third spiral going through points 2, n+2, 2n+2, etc. Continuing in this way, we find spirals that also start at points 3, 4, 5, all the way up to n-1. Since the first spiral started at point 0, we must have a total of n spirals. This observation is worth repeating: if point n is the closest to point 0, then there will be n spirals. Hence, if we are interested in the number of spirals, we have to get a handle on which point is closest to point 0.

The complete mathematical argument is more technical than would be appropriate here. Interested readers should look at the preprint mentioned above. The upshot is that which point is closest to point 0 will depend on the vertical spacing h, but will always be a Fibonacci number! The smaller h is, the larger the Fibonacci number we see, and hence more spirals. The following animation demonstrates this effect. In this video, the points are placed by an algorithm, and the spirals are calculated from those placements. In other words, I didn’t write a program to create 3, 5, 8, 13, and 21 spirals. I only varied the placement of the points, and at various times these numbers of spirals just appeared.

Often when we look at objects that exhibit phyllotaxis, we actually see two sets of spirals. The most visually dominant will be the ones described above, coming from connecting each point to its closest neighbors. The other set, spiraling the other direction, come from connecting each point to its second closest neighbors. As you may have guessed, the number of these spirals is determined by the number of the second closest point to point 0. The same results mentioned above show that this number will always be the preceding element of the Fibonacci sequence.

Now that we have a better understanding of cylindrical phyllotaxis spirals we are ready to take on flat phyllotaxis patterns, like the sunflower shown here.

In these patterns, we tend to see a different number of spirals depending on how far away from the center you focus. Regardless,  these numbers are still usually Fibonacci. Here’s why. Consider a narrow ring of this picture between concentric circles. Then the distance between seeds in this ring  is comparable to the distance between seeds on a cylindrical phyllotaxis pattern of a similar radius. If we look at a bigger ring, farther out from the center, then the picture is more like that on a wider cylinder. Just as the number of spirals on a cylinder increases with a smaller vertical spacing between successive points, it also increases when you make the cylinder wider. Hence, you’ll see more spirals further away from the center of the sunflower, although the number will always be Fibonacci. For example, in the image above there are 21 dominant spirals toward the center (shown in red), and 34 toward the outside (shown in blue).

11 thoughts on “Lesser Known Facts About Sunflowers and Pinecones (Pt. 2)

  • This is too ironic! I JUST submitted artwork to Bridges looking at the patterns of modularly colored Phylotactic Spirals, with an increasing number of colors. I colored the dots in the order they are generated from inside out. I too thought I was on to something new because I can’t find anything relating to the patterns I was seeing. I can send you some images if you’re interested. Great article

  • Also, Michael Naylor has a 2002 Mathematics Magazine paper on spirals arising from different philotaxis ratios, which is one of the citations in my paper. If you haven’t read that, you definitely want to look it up.

  • Also, hopping back to your previous post (which I found after this), if you’re inspired to track it down, you might get a kick out of an old biology paper I found that takes the spiral shape you discuss there and uses it to derive formulas to tell you where different Fibonacci spiral families appear and disappear in a seed head. Here’s the bibliographic data: M. Kunz and F. Rothen, Phyllotaxis or the properties of spiral lattices
    III. An algebraic model of morphogenesis. J Phys I France, 75 (1992),
    2131–2172.

  • Aw, thanks! That was a great conference, wasn’t it?

    Your preprint looks cool, and I’m hoping I can find time to read it carefully soon. Part of what is compelling about the Fibonacci seed head pattern is that so many different ideas intersect there.

  • The geometric explanation of this subject has been known for a long time, behind the premise “if you assume growth is governed by the Golden Ratio” is where the subject is. But as a review is fine, I’m surprised to see how every spring I return to this subject 🙂

    “the upshot is that which point is closest to point 0 will depend on the vertical spacing h, but will always be a Fibonacci number!”
    It also depends on the radius of the cylinder. It would be interesting to know what is the mathematical relationship between h and r in order to predict the fibonacci number without need a geometrical relationship.

    “To locate a point on a cylinder relative to point 0, we rotate around
    the cylinder some amount and go up some amount. If we do more than
    one full rotation, then it may be shorter to rotate a smaller amount the
    other way. We will call the smallest amount you have to rotate to get
    from point 0 to point i the net rotation. We will measure net rotation as
    a fraction of a full rotation, so that it will always be between −1/2 and
    1/2 (with the sign determined by which direction you have to rotate).
    For example, if some point is constructed by doing 5.83 rotations from
    point 0, then the net rotation of that point is -0.17. Similarly, if we do
    6.27 rotations to get from point 0 to point i, then the net rotation of
    point i will be 0.27. The net rotation of the ith point, which we denote
    ω(i), is easily calculated by subtracting from i/φ the nearest integer.”

    I have not understood the concept of net rotation, could you explain it to me please? Why is it relevant? the 1/2 and -1/2 would not be pi and -pi? Or have you normalized the space without specifying it in the paper? What means 5.83 rotations? Why the net rotation is -0.17? Why 6?

    “In almost everything written about
    the growth of sunflowers and pinecones (i.e. anything governed by
    phyllotaxis), the author notes two facts:
    (1) As the plant grows, the angle around the central axis between
    successive seeds is determined by the Golden Ratio.
    (2) If you count the number of apparent spirals, the answer is almost
    always an element of the Fibonacci sequence, usually either
    3, 5, 8, 13, 21, or 34 (although you can certainly find larger
    numbers, for example, in sunflowers).”

    You mention papers but you do not link them 🙁
    1) It is not determined, it is approximated. The difference is great, nature does not use mathematics because they are human tools. A plant does not use the phi number at any time, rather it is the result of how they grow, not viceversa. Using numerical methods that simulate chemical reactions, phyllotaxis can be achieved without using phi.
    2) I have seen for some time now somewhere that 90% of the plants in the northern hemisphere have numbers from the fibonacci series, also others from the Lucas series.

    Please, see:
    http://math.arizona.edu/~anewell/publications/201newell.pdf

    • Hi! Thanks for all of your comments. Here are a few quick responses:

      –I know everything I said is well known, but I haven’t found a reference. Do you know of one? The best I could find were explanations that use continued fractions, but I wanted to present something that would be understandable to a more general audience.

      –As I said in the passage you quoted, net rotation is “measured as a fraction of a full rotation.” It’s not an angle measure. You can easily convert though, buy multiplying by 2pi (for radians) or by 360 (for degrees). When I gave the example of doing 6.27 rotations around the cylinder, that would be rotation by an angle of about 39.39 radians.

      –If you go around a circle 6.27 times you’ll get to the same point as you would if you went around just .27 times. That’s the essence of the net rotation measure. It’s necessary for the arguments in the last section of the paper. Similarly, if you go around a circle 5.83 times, you’ll get to the same point as if you went around the other way 0.17 times.

      –I didn’t bother listing references because there are too many. Just google “Phyllotaxis Fibonacci” and you’ll find lots of them.

      –I made no claims about plants “using phi” or “knowing mathematics”. Mathematics is, of course, a perfect abstraction of an imperfect world! The whole point of the post/paper is to show that if you ASSUME growth according to the golden ratio, then Fibonacci numbers are a necessary consequence.

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