This pattern was irresistible to the applied mathematician! I can’t recall the mathematics that’s embedded in the pattern here, whether it’s something to do with the Golden Ratio or the Fibonacci sequence, but there are obviously multiple intersecting “curves” in the pattern of this cactus' external structure.
Turns out it’s both!
“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,’
https://mathworld.wolfram.com/GoldenAngle.html ] 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 cactus has done: packed its dew-gathering structures in as compact, and efficient, an arrangement as possible on a two-dimensional surface. Mathematics and economics. I think I found my proper calling(s)…
A simpler 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,”
https://mathworld.wolfram.com/Phyllotaxis.html
See also:
»
https://www.math.smith.edu/phyllo/
»
https://365project.org/brittgow/365/2011-02-01
Species page at PhytoImages,
http://phytoimages.siu.edu/cgi-bin/dol/dol_terminal.pl?taxon_name=Mammillaria_spinosissima&rank=binomial
Taken at the SIUC Plant Biology Greenhouse,
https://www.plantbiology.siu.edu/facilities/plant-biology-facilities/greenhouse/index.php
One year ago (“Necklace on dresser”):
https://365project.org/rhoing/365/2012-02-20
Two years ago (“Let there be light!”):
https://365project.org/rhoing/365/2011-02-20
I never realized cactus and sunflowers would share in this pattern. It is a hard pattern to photograph. This looks great.
It was a challenge to capture this pattern. This was the third day I tried and I was pretty pleased with how it came out. Thanks for your comment!