Black-eyed Susan

These flowers just make me smile … and not just because of the mathematics embedded in the pattern of the “eye,” which is similar to the pattern on my 365.org cactus post, Mammillaria spinosissima. Evident here are “Fibonacci numbers” and the “Golden Angle.”

“In a pioneering work in 1907, German mathematician G. van Iterson showed that if you closely pack successive points separated by 137.5 degrees [the ‘golden angle’] on tightly wound spirals, then the eye would pick out one family of spiral patterns winding clockwise and one counterclockwise. The numbers of spirals in the two families tend to be consecutive Fibonacci numbers, since the ratio of such numbers approaches the Golden Ratio.”

“Such counterwinding spirals are most spectacularly exhibited by the arrangement of the florets in sunflowers. When you look on the head of a sunflower, you will notice both clockwise and counterclockwise spiral patterns formed by the florets. Clearly the florets grow in a way that affords the most efficient sharing of horizontal space. The numbers of these spirals usually depend on the size of the sunflower. Most commonly there are thirty-four spirals going one way and fifty-five the other, but sunflowers with ratios of numbers of spirals of 89/55, 144/89, and even (at least one; reported by a Vermont couple to the Scientific American in 1951) 233/144 have been seen. All of these are, of course, ratios of adjacent Fibonacci numbers. In the largest sunflowers, the structure stretches from one pair of consecutive Fibonacci numbers to the next higher, when we move from the center to the periphery.”
—Livio, Mario. The Golden Ratio. New York: Broadway Books, 2002, p. 112.

I conjecture that that’s precisely what this plant has done packing its seed structures in as compact, and efficient, an arrangement as possible on a two-dimensional surface.

A simpler, more obvious observation? “Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rose have five petals; delphiniums, bloodroot, and cosmos have eight petals; corn marigolds have 13 petals; asters have 21 petals; and daisies have 34, 55, or 89 petals—all Fibonacci numbers.” —Eric W. Weisstein in “Phyllotaxis” at MathWorld—A Wolfram Web Resource

Species pages: [ Missouri Botanical Garden ] [ Wikipedia ]
Images: [ PhytoImages.siu.edu — Not a secure https connection ]

[ IMG_3406S105x70Otm :: 60mm ]

Looking back
1 year ago: “Back to see the kids & grandkids”
2 years ago: “Eastern Tailed-Blue”
3 years ago: “The unmistakable Monarch”
4 years ago: “Holy Spirit Catholic Church, Bowling Green”
5 years ago: “The travel day from *youknowwhere*…”
6 years ago: “New lens … but still with the bugs?”
7 years ago: “Common Whitetail”
8 years ago: “Waiting for sunset”
9 years ago: “Opportunity meets preparation” One of my few PPs
10 years ago: “Double-banded Scoliid”
11 years ago: “Solitude”

15th July 2022 15th Jul 22