# Almanac: The Eye Of A Daisy

*To see a World in a Grain of Sand **And a Heaven in a Wild Flower **Hold Infinity in the palm of your hand **And Eternity in an hou*r

-William Blake

The 1.2-mile trail-and-boardwalk path at the Silvio Conte wildlife refuge in Hadley is one of this area’s gems, not least because it is totally accessible for wheelchairs, walkers, and strollers. I was there early one day last week to avoid the stifling midday heat and among many delights were ox-eye daisies blooming in the fields.

Although they’re technically an invasive species (introduced centuries ago from Europe), I have a soft spot in my heart for daisies for several reasons. As far as I can tell they’re not out-competing other native wildflowers, they don’t harm other plants, and they’re just darned pretty. Daisies were also the flowers my mother chose to hold as she walked down the aisle toward my nervous father-to-be at their wedding. But I also love daisies because in their yellow eyes one can see a clear, but still-unexplained, message from a mathematically-minded Creator.

The showy part of daisies are the white “petals” around the inner yellow “eye.” But take a closer look at that eye and you’ll see it’s composed of hundreds of tiny individual flowers arranged in two sets of overlapping spirals, some going clockwise, some counter-clockwise. These spirals can also be seen in related members of the large composite family of flowers, which includes sunflowers, coneflowers, and asters.

The spirals themselves look cool, but that’s just the start. If you have patience and a keen eye, you can count the *number* of clockwise and counterclockwise spirals in the daisy’s eye. (I find the counterclockwise spirals easier to count than the clockwise spirals, but give it a shot.) You’ll discover that, almost always, there are 21 clockwise spirals and 34 counter-clockwise spirals. If you count spirals in sunflower heads you’ll usually find they come in sets of 13/21, 34/55, 89/144, and, in particularly huge specimens, 144/233.

The peculiar thing is that these numbers are anything but random. They are all part of a relatively simple series of numbers that starts with 1 and continues with each new number being the sum of the previous two numbers. Hence the series goes like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 *ad infinitum*. The series is called the Fibonacci sequence in honor of the guy who first wrote about it in the West—a medieval mathematician from Pisa, Italy, named Leonardo Fibonacci. In 1202 Fibonacci wrote *Liber Abaci*, or Book of the Abacus, in which he introduced the base-10 Arabic number system, numerals, and methods of computation to a Europe still mired in the clumsy Roman numeral system. This was a revolutionary step forward and you might think that this, rather than an odd series of numbers, would be why he is remembered today.

But as simple as it seems, the Fibonacci sequence has proven to be a rich vein of mathematical ore. To name just one peculiarity, the ratio between any two Fibonacci numbers approaches to an ever more exact degree the ratio 1:1.618, which is called the Golden Mean, or Golden Ratio. The Greeks used this ratio a lot in art and architecture because they found it to be aesthetically pleasing. If you divide a Golden Rectangle (created with sides conforming to the Golden Ratio) into smaller Golden Rectangles and draw a curve through them, you form a spiral found in many sea and snail shells, most notably the nautilus shell.

Once you start looking for them, you start to find Fibonacci numbers everywhere in nature. The opposing spirals in many pinecones, for example, come in pairs of 3, 5, 8, and 13. The eyes of pineapples form three sets of spirals commonly occurring in sets of 8, 13, and 21 spirals. Fibonacci numbers also can be found in the growth rings of some animal horns.

So what the heck is going on? Why do these numbers crop up, like a secret code, in wildly divergent life forms? This turns out to be an ongoing biomathematical mystery. The Fibonacci Journal has been publishing papers since 1963 exploring both theoretical and observational explanations for the phenomenon. Explanations I’ve come across involve things like the optimal patterns for packing small elements (e.g., florets) into round or circular shapes, or the ways that cells divide and die as tissues grow. The basic assumption is that the spiral patterns exist because they have proven to be maximally efficient in terms of energy use or survival/reproductive advantage across billions of generations. But, to date, the precise mechanisms underlying Fibonacci spirals have yet to be elucidated.

Galileo, in *The Assayer*, compared the universe to a “grand book” written in the language of mathematics. That things like the fundamental relationship between energy and matter (E = mc^{2}) or the nature of gravity (F = G[m_{1}m_{2}/r^{2}]) can be expressed in such beautifully elegant equations surely proves his point (although, lord knows, not all equations describing fundamental forces are this simple). You don’t need to know anything about math, of course, to appreciate and love nature, but there is something deeply thrilling to discover regularities and hidden patterns beneath the riotous complexity of this extraordinary world of ours, and to glimpse this quiet grandeur in the eye of a common daisy.

*Almanac is a regular *Indy* column of observations, musings, and occasional harangues related to the woods, waters, mountains, and skies of the Pioneer Valley. Please feel free to comment on posts and add your own experiences or observations. *

1,

1+1/(1+1),

1+1/(1+1/(1+1)),

1+1/(1+1/(1+1/(1+1))),

1+1/(1+1/(1+1/(1+1/(1+1)))),

1+1/(1+1/(1+1/(1+1/(1+1/(1+1))))),

1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1)))))),

1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1))))))),

1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1)))))))),

….

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html

P.S. Some local folks (especially Marjorie Senechal, the Louise Wolff Kahn Professor Emerita in Mathematics at Smith College) have thought a lot about the mathematics in these patterns (see, for example, this expository website, which may no longer be actively maintained — it seems to now be a webarchive:

https://web.archive.org/web/20100526030713/http://www.math.smith.edu/phyllo/About/index.html

— and I’m not sure who is behind this website, though it could be her colleagues Pau Atela and Chris Golé), but this situation almost begs a follow-up interview with some of them (if Steve is so inclined)!

My previously posted sequence of continued fractions (it’s a good exercise for interested readers to work out the common fractions over the summer break [hint: some appear in Steve’s article!]) converge to The Golden Section [(1+√5)/2], and while the convergence is as fast as possible, almost paradoxically the convergence is also the slowest possible: in other words, The Golden Section is

The Most Irrational Number, which is conveniently the title of a very recent articlehttps://slate.com/technology/2021/06/golden-ratio-phi-irrational-number-ellenberg-shape.html

by another math friend, Jordan Ellenberg, at Wisconsin.

My “Celestial” take on the situation: Nature may abhor a vacuum, but Nature also avoids overcrowding — a repetitive process (like cell division and plant growth) risks “bumping into itself” when its parameters are ratios of whole numbers (rational numbers), so the there is selective advantage for irrational parameters, and optimally selective advantage

for the most irrational parameters like (1+√5)/2….