Thursday, June 29, 2017

Messing With Patterns

This post features no fossils though my initial intention was otherwise.  Fossil foraminifera shells, the golden ratio, and the logarithmic spiral were among the elements in the mix as I began to draft this post but, sadly, it all spun out of control.  I regrouped and this is what resulted, a piece focused only on composite flowers and the Fibonacci sequence with a salute to Alan Turing at the end.

We are a pattern-detecting species and nature obliges by surrounding us with myriad apparent patterns.  Case in point, the beautiful, flower-heavy stand of coneflowers (Echinacea purpurea) in my front yard.  The many blossoms, white and a few pink, are composite flowers; so not surprisingly the coneflower is a member of the daisy family, or more properly the Asteraceae or Compositae family.  The flower head of the coneflower consists primarily of small disk flowers with small ray flowers along the periphery.  These flower heads, as we perceive them, sport clockwise and counter-clockwise spirals emanating from the flower center.  The first picture below shows a blossom in its natural beauty; the second and third present the clockwise and counter-clockwise spirals I detect in this blossom marked in white.

 Such a spiral perceived in the organs (such as leaves and petals) of botanical specimens is called a parastichy; a count of these spirals is referred to as the parastichy number.  Interestingly enough, various definitions of parastichy use the adjectives invisible or hypothetical to describe the spiral patterns, suggesting that the presence of parastichies may well be (in part? mostly?) a function of the predilection of our eyes and brains to join disparate elements into some visual, pattern-filled, coherent whole.

Objects of Wonder, a current exhibit at the Smithsonian's National Museum of Natural History, features unique and seldom-seen treasures from the museum's collections.  A display case promoting the exhibit highlights how scientists find “patterns everywhere," enabling us "to understand the underlying processes that shape our world," part of “a complex and seemingly chaotic universe.”  Among the objects displayed in this case is the flower head of a sunflower (Helianthus annuus); though not one of the museum’s treasures, it is an object of wonder.  The display notes that "[a] sunflower's blossom consists of many small flowers arranged in spirals."  It adds that “[t]his pattern evolved as an efficient way to pack many seeds into a space, keeping them evenly distributed no matter the size of the seed head."

The sunflower, a member of the Asteraceae family and so related to the coneflower, appears prominently in the literature describing mathematical patterns found in living organisms.  So much so, that I consider it a poster child for that concept and particularly for the presence in nature of patterns based on Fibonacci numbers.

Leonardo of Pisa (c. 1170 – 1250), called Fibonacci because he was the son of Bonacci, first presented the series of numbers that Eduoard Lucas (1842 – 1891) named the Fibonacci sequence.  Here is the beginning of the sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .

The first two numbers (1, 1) are given; from there the pattern dictates that each subsequent number is the sum of the preceding two numbers.  The next number in the sequence above would be 144 + 233, or 377.  It’s an infinite sequence and recursive (that is, it has the attribute that any new number added to the sequence is dependent on earlier numbers).

Connection to sunflowers?  Each sunflower blossom appears to offer a pair of clockwise and counter-clockwise spiral patterns whose parastichy numbers fall on the Fibonacci sequence.  That is an essential attribute of the sunflower that mathematicians and others highlight when they wax enthusiastically about the deep association between mathematics and nature.  As science writer John Bohannon notes (in an article about the sunflower study described below), “Mathematical biologists love sunflowers.  The giant flowers are one of the most obvious – as well as the prettiest – demonstrations of a hidden mathematical rule shaping the patterns of life:  the Fibonacci sequence . . . .”  (Sunflowers Show Complex Fibonacci Sequences, Science, May 17, 2016.)  Mathematicians Peter Tannenbaum and Robert Arnold, in their textbook Excursions in Modern Mathematics (3rd edition, 1998), write (somewhat breathlessly) of the consistency in how Fibonacci numbers appear in some natural objects.  Of the sunflower, they observe, “[T]he seeds in the center of a sunflower spiral in 55 and 89 rows.”  (p. 305)

As to that latter assertion about sunflowers spirals, I must respond, well, that’s not always true.  Indeed, it’s often not true.  In the largest analysis to date (published last year) of parastichy numbers in sunflower blossoms, Jonathan Swinton and his colleagues found that while 74 percent of the 768 parastichy numbers (clockwise or counter-clockwise) included in the study fell precisely on the Fibonacci sequence, the remainder or 26 percent did not.  (Novel Fibonacci and Non-Fibonacci Structure in the Sunflower:  Results of a Citizen Science Experiment, Royal Society Open Science, 2016.)  Yes, 55 and 89 were among the most common Fibonacci numbers appearing in the parastichies, but the rank order for the top four such numbers was 55, 34, 89, 21.  If one were to be generous and include sequences which have “Fibonacci structure” (such as the double Fibonacci sequence, i.e., 2, 4, 6, 10, 16, . . ., or the Lucas sequence, i.e., 1, 3, 4, 7, 11, . . .), another 8 percent of the parastichies in this study might be considered Fibonacci in essence.  That leaves 18 percent of the parastichy numbers outside of the Fibonacci sequence or Fibonacci-structured sequences.  Another 8 percent of the parastichies were non-Fibonacci numbers that were very close to ones in the Fibonacci sequence, differing by only plus or minus 1.  Thus, approximately 10 percent were fully untethered to the Fibonacci sequence or Fibonacci-structured sequences, even being very generous about it.

It would appear that, at least for the sunflower, this “hidden mathematical rule shaping the patterns of life” plays out rather untidily in a significant minority of specimens.  There’s no faithful consistency here.  In light of Swinton’s study, Bohannon admits as much when he writes, “The possibility of capturing sunflower development with math just got more realistic – and more complicated.”

I was heartened that Swinton’s study is the culmination of a citizen-science project that was run by the Museum of Science and Industry (Manchester, England) as part of a celebration of the 100th anniversary of mathematician Alan Turing’s birth.  Among his myriad research interests, the brilliant and persecuted Turing toward the end of his life was pondering explanations for why and how the Fibonacci sequence appears in nature.

The photos of the coneflower from my garden that opened this post were prepared following the guidance generated by this citizen-science project.  Interestingly enough, the parastichy pair for that particular blossom (21 and 34) are, in fact, Fibonacci numbers and, better still, adjacent in the Fibonacci sequence.  Other coneflowers that I photographed and for which I generated parastichies reflected how life can get in the way of desired neatness and order.  For instance, the blossom depicted below has a very nice spiral pattern (at least, I perceive it as such) in the clockwise direction (with a parastichy number of 21 that falls on the Fibonacci sequence), but that same blossom features (again, as I perceive it) a rather complex counter-clockwise spiral pattern (third photograph below) whose order breaks down in its left half with partial spirals intersecting ones emanating from the center.  I tried to salvage the counter-clockwise parastichy count for this blossom in a couple of different ways, but I only shifted where the confusion occurred.  In his analysis of the sunflower parastichies produced in the citizen-science project, Swinton notes that in some flowers there may be overlapping or competing parastichy families, sometimes leading to “particularly awkward transitions.”  This may be a coneflower example of that.

 Yes, a bit of a mess as nature prevails over the order some would have it follow.

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