Lesser known facts about Sunflowers and Pinecones (pt. 1)

To say that a lot has been written about the arrangement of seeds on a sunflower would be a gross understatement. Discussions of this topic go back at least as far as D’Arcy Wentworth Thompson’s famous 1917 book, On Growth and Form. However, despite the wealth of available information, I’ve still found a few things that are generally skimmed over, or just not discussed at all.

The relevant technical term that many of my readers will recognize is “phyllotaxis.” A simple google image search for this word will bring up countless photos of not just sunflowers, but also pinecones, pineapples, flower petals, various desert plants, etc. The pattern that is apparent in all of these is the same. For curious readers who have not seen them, I highly recommend Vi Hart’s excellent video trilogy titled “Spirals, Fibonacci, and being a plant.”

Phyllotaxis has been used in the Math/Art world to simulate nature countless times. The most well known recent examples I know of are John Edmark’s Bloom sculptures (click here for video).

In this post, I’ll discuss one particular challenge that arises when modeling a flat-ish sunflower like in the first image above, and then how this same problem gets MUCH more complicated when modeling a more 3-D structure as in the second image.

In modeling any of these spiraling patterns, the idea is simple: First, choose a central point around which to add “seeds.” After each seed is added, rotate around the central point a certain angle, while also moving away from the central point a certain distance. The angle is always the same: about 137.5 degrees. This is known as the “phyllotaxis angle” or “golden angle,” and is related to the famous “golden ratio.” Almost everything that has been written about phyllotaxis is focused on this angle. What I’ll discuss here is the other problem: how far should you move away from the central point as you add each successive seed?

For 2D images, the answer is simple: you should place the nth seed on a circle whose radius is the square root of n. So, for example, the 10th seed should be sqrt(10) units away from the central point, and rotated 137.5 degrees from the 9th point, as in the image below.

Here is what you get after 1000 seeds have been added by this scheme. Notice the similarity to the sunflower photo at the beginning of this post.

So why is sqrt(n) the right distance from the center? One of the key features of this image is that the seeds are fairly evenly distributed. That’s what sqrt(n) gives us. Here’s why. If the seeds are evenly spaced, then each seed will account for the same amount of area of the whole image. For simplicity, I’m going to assume the area contribution of each seed is π units (the same as the area of a circle of radius 1). So, if there are n seeds, then the image has area nπ.

Another way to calculate the total area is to draw larger and larger concentric circles, centered on the central point of the image, which pass through the center of each seed. We will then calculate the area of the ring between consecutive circles, and add up the areas of all the rings.

We’ve placed seed n and seed n+1 on circles of radius sqrt(n) and sqrt(n+1), respectively. The area of the ring between these circles is given by the difference in the areas of the circles:

π(sqrt(n+1))^2-π(sqrt(n))^2=π(n+1)-pi(n)=π.

If we have n seeds total then we’ll have n rings. With an area contribution of π from each, that gives us a total of nπ. Since that’s the same as the total area we got before, we must have placed the seeds correctly!

In 3-dimensions the idea is the same, but unfortunately enacting this idea gets much more complicated. The challenge is to place seeds on a curved “stalk”. The stalk is a surface of revolution, meaning it is symmetric around some central axis. Again, to place seed n+1 after having already located seed n, we rotate around the central axis 137.5 degrees. To find how far away from the center this is, we again imagine concentric circles passing through each seed. We just have to arrange these circles so that the areas of the rings between them are the same. One of these rings is highlighted in grey in the picture below.

In the 2D case, one can locate seed n without locating any of the other seeds: its at an angle of 137.5n degrees around the center, and at a distance of sqrt(n) from the center. In 3D, I don’t know of an easy way to do this. Instead, I used an iterative approach: once seed n-1 is placed, I rotate another 137.5 degrees around the central axis, and move away from the central point, along the surface of the stalk, until the new horizontal ring that is created has the desired (fixed) area. It was quite a challenge to implement this idea in Grasshopper/Rhino to create the above image. I imagine in scripting languages for other CAD packages, the implementation might be very different.

Once the seeds have been properly placed around the stalk, it’s relatively easy to create some interesting sculpture. In Rhino, for example, you can model an individual leaf unit, and then use Grasshopper to copy it to different locations around a surface. Here’s an image of one such model I created using the seed distribution described above. For the purposes of this discussion, the thing to notice is how uniformly the leaves are distributed around the stalk.

In my next post, I’ll continue by discussing a very different aspect of phyllotaxis. Many people have noticed that there are two sets of visually apparent spirals in these patterns, circling in opposite directions. The number of spirals in these two directions is (almost) always given by two consecutive Fibonacci numbers (e.g.  5 and 8, 8 and 13, 13 and 21, etc.). I’ve never seen a good explanation of why this is, other than possibly a passing mention that it has something to do with rational approximations of the Golden ratio.  In my next post I’ll do my best to shed some light on this question.

6 thoughts on “Lesser known facts about Sunflowers and Pinecones (pt. 1)

  • Eii, a few years ago I studied this topic and generalized its application in Grasshopper in the PhylloMachine plugin:
    http://www.food4rhino.com/app/phyllomachine

    As to why the spirals correspond to fibonacci numbers, I couldn’t give you an in-depth explanation. But I can mention some interesting notes. First of all, what we have is that starting from a phyllotaxis point system, fibonacci numbers appear in relation to the minimum distance from one point to another. That is, if we take an ‘i’ point (where i is the index) its neighboring points will be i+F0, i+F2, i+F1, i-F0, i-F2 and i-F1, being F0, F1 and F2 consecutive fibonacci numbers. Why is this so? Well, it may be as simple as sheer geometric coincidence that emerges from phyllotaxis. On the other hand, I think the right question is why phi is so present in nature. My point of view is that it is an evolutionary consequence, a trend of optimization, being the proportion that maximizes access to resources and minimizes cost. Kind of like the principle of minimum action. The relationship of fibonacci numbers to phi can be found on many sides, my favorite is that if you divide F0/F1, the result is close to phi, the higher the number, the closer the phi is.

    • Thanks for your comment. I love playing with PhylloMachine, and highly recommend it to my Grasshopper savvy readers! Yes, in the videos I referenced in my post, Vi Hart goes through a really nice biological explanation for why the golden ratio shows up in plant growth. It is indeed coming from some minimum energy configuration of newly created leaves, like the “principle of minimum action” that you mention. There is a beautiful geometric relationship between Fibonacci numbers, lines in lattices of points, and the golden ratio. As you’ve guessed, it does have something to do with the fact that ratios of consecutive Fibonacci numbers get closer and closer to the golden ratio. I’ll go through all that in detail in my next post!

  • Very nice article! All in all I think its a nice first approximation but I believe the problem is infinitely more complex:
    In pine cones the conical shape is actually created by the scales themselves.Take a look at this image:
    https://media.nature.com/lw926/nature-assets/srep/2016/160104/srep18105/images/srep18105-f1.jpg
    The deal is that scales start growing from below of what we can see on the surface, and still they manage to fit perfectly on the outside. Moreover, they vary in size.

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