From “Spirals: The Math in Snails and Sunflowers” in Patterns in Nature: Why the Natural World Looks the Way It Does by Philip Ball:
“Of all the patterns and forms of nature, the spiral has probably held the greatest appeal for mystics and dreamers. It is revered by adherents of ‘sacred geometry,’ who consider the patterns and forms of nature to embody spiritual truths of the cosmos. Spirals are found in ancient and indigenous art ranging from the carvings on the Bronze Age stones of Newgrange in Ireland to the paintings of Australian Aborigines.
“Nothing better exemplifies the apparent mystery and profundity of the logarithmic spiral than its manifestation on the heads of flowers such as sunflowers and daisies. The seeds of a sunflower head are arrayed in rows that trace out not just a single logarithmic spiral but two entire sets of them, rotating in opposite directions. The pattern that results has profound mathematical beauty: crystalline precision combined with organic dynamism, creating shapes that seem almost to shift as you stare at them….
“If you count the numbers of spirals in each set, you find that they only take certain values…. For smaller sunflowers there might be 21 spirals in one direction, 34 in the other. For very large heads, there might be as many as 144 and 233. But only these pairs of numbers — never, say, 22 and 35. Why are some of these numbers favored over others?
“No one is yet sure why the sunflower seeds adopt this arithmetical arrangement. One longstanding idea is that it enables the florets or seeds or leaves to pack most efficiently as they bud from the tip of the growing stem…. This is simply a geometric problem: if you want to arrange objects in an array spiraling out from a central source, what should be the angle between one object and the next? It turns out that the most efficient packing, which gives the double-spiral Fibonacci pattern of phyllotaxis, is one for which this angle is about 137.5 degrees — known as the Golden Angle.”
This is the first of two posts with photographs of white asters — most likely, Shasta Daisies (Leucanthemum × superbum) — that I recently took at Oakland Cemetery’s gardens. Many of these Shastas appeared in large clumps — spanning fifteen to twenty feet horizontally — and (as you can see from the first three photos) were quite content to grow in the shade of an old Oak Tree, while edging their way toward sunnier positions on one of the garden’s sidewalks.
As is true for most of the flowers in the Aster family Asteraceae, the central disc of these daisies actually consists of many tiny, individual flowers — which gave rise to “Composite” or “Compositae” as an earlier name for Asters. While working on some of the close-up photos in this series, like this one…
… I became a bit obsessed with how the orange-yellow disc looks, where (below in a zoomier view), you can see how the center of the center is packed with flowers but the outer edges are not.
In my imagination (such as it is!), I thought maybe some little bees had come around, picked the flowers from the outer rings, and gave them happily to their other bee friends. Hey, why not? But then it occurred to me that they probably wouldn’t have managed such nearly perfect circles as they picked the flowers, so that might not be an accurate observation.
I wanted to learn more about why the central discs looked like this, and after a few abortive attempts, hit on a question I could ask one of my AI Assistants:
When I look at photographs of a daisy’s disc florets, it appears that some of them are empty, especially around the outer edge of their circle. Why do they look like that?
The response I got included several possibilities — including “removal” by insects (haha!) and wind or rain damage — but the most plausible explanation was that the disc fills with flowers from the center outward, and those in the outer rings had not yet matured. Armed with this knowledge, I went back a few days later and checked some of the same flowers again to see if the discs had filled in — but it was too late and the white Shastas were already beyond their flowering stage. Perhaps next fall, I’ll try that again.
That the central disc fills with flowers from the center to the outer edge was equally fascinating to me, and digging into that I learned a little more about what happens. The tiny florets actually grow in two concentric spirals — with one spiral running clockwise and the other running counterclockwise. Look again at the zoomed-in photo and you can clearly see the spirals. And once you see them, you’ll see them every time you look closely at a flower like this.
This arrangement is not only not random, it runs in a mathematical sequence among the flowers in the Aster family. Starting from the center outward, the number of individual florets follows the Fibonacci Sequence — where each subsequent number is the sum of the previous two numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Most of the smaller Asters — like the Shastas in this post — have 34 or 55 individual florets (yes, I counted them!) in the outer ring. Sunflowers — also members of the Aster family — are often used to explain this mathematical sequence in nature, so if the subject interests you, search for terms like “Fibonacci sequence and sunflowers” or phyllotaxis (which encompasses the study of natural shapes, merging botany and math) on YouTube and you’ll find quite a few fun explanations.
Thanks for reading and taking a look!