Testing Gravity and Shear Strength ~ Ending the Year on a Humorous Note with a Handful of Pliocene Foram Fossils

No, not that kind of gravity.

Looking back over a decade of posts on this blog, I stumbled upon a minor leitmotif:  humor in science.  On a few occasions I’ve drawn attention to examples of refreshing and engaging humor unexpectedly interjected into serious work by otherwise quite serious scientists.  For instance, in one post, I enthusiastically pointed out that the geologist and micropaleontologist Cesare Emiliani, responsible for some quite important analysis using the chemical composition of the fossil shells of foraminifera to reveal key attributes of ancient climate, found the motivation and time (and, perhaps, the foolhardiness) to pen irreverent and funny articles and letters-to-the-editor for otherwise staid publications.  To wit, in an article that appeared in a 1993 issue of Eos:  Transactions, American Geophysical Union, Emiliani suggested that the burdens and challenges of peer reviewing scientific articles would be immensely relieved if authors peer reviewed their own articles; it was a no-brainer, he argued, because, “being the most knowledgeable person about his or her own work, [the principal investigator] can be expected to be totally objective.”

So it should not have been surprising that, despite my best intentions, I have managed to turn this current post, which is ostensibly about the Deep Sea Drilling Project (DSDP) and some Pliocene fossil foraminifera shells brought up in 1979, into one long buildup to a mildly amusing (at least I find it so) bit of scientific humor.  Still, I think it’s a fitting note on which to end this year.

First, let me be clear that I don't think my pursuit of humor in the hallowed halls of science is misguided.  Rather, such humor is singularly important.  I would point to Anthony Cooper, Earl of Shaftesbury (1671 – 1713), who in his Essay on the Freedom of Wit and Humor quoted an “ancient sage” (the Greek philosopher Gorgias Leontinus) (to be completely obsessive about it, Shaftesbury was, in fact, quoting Aristotle who had quoted Gorgias), as having said that

humor was the only test of gravity, and gravity of humor.  For a subject which would not bear raillery was suspicious; and a jest which would not bear a serious examination was certainly false wit.  (Shaftesbury, in Characteristicks [sic] of Men, Manners, Opinions, Times, 5th edition, 1732, p. 74, punctuation modernized.)

So bear with me as I begin this story.  The DSDP, begun in 1966 when the National Science Foundation and The Regents of the University of California first entered into contract, conducted coring and drilling in the floors of different oceans around the world between 1968 and 1983.  This and two subsequent drilling projects were designed to increase our understanding of the geophysical properties of the ocean floor.  Among its myriad scientific contributions, the DSDP provided strong evidence of continental drift and the constant renewal of the sea floors, crucial support for the theory of plate tectonics.  The initial reports and studies from the DSDP are available on the web.

The workhorse of the DSDP was the marine vessel Glomar Challenger, designed specifically to drill deep into the ocean floor and retrieve core samples for study; the ship was in action from 1968 to 1983.  (The picture of the vessel below is in the public domain and can be found on Wikimedia Commons,)

Each cruise of the Glomar Challenger was called a “Leg” (to my mind, a leg is one part of a whole so I guess that in its 15 year life span the Glomar Challenger was on a single voyage that consisted of 96 legs).  Well into its coring career, the Glomar Challenger began Leg 68 in Curaçao, Dutch Antilles on August 13, 1979.  The initial round of coring on this leg occupied 11 days as the ship drilled a tightly clustered series of holes (Holes 502, 502A, 502B, 502D) in the Caribbean Sea.  On the map below this cluster is approximately at the marker in the Caribbean Sea.  The Glomar Challenger then sailed through the Panama Canal and the shipboard team began coring at the first of three clustered sites in the Pacific Ocean.  These holes were Hole 503 cored from September 6 to 7, Hole 503A - from the September 7 to 11, and Hole 503B - from September 11 to 13.  For a reason that will become clear in a moment, Hole 503A is of particular interest to me and is the one marked in the Pacific Ocean on the map below.  Water depth at this position is 3,672 meters (2.3 miles) and the coring went 235 meters (771 feet) into the ocean floor.  (I drew much of the information provided here from the online reports on the coring during Leg 68.)



Among its objectives, Leg 68 was to test a new “hydraulic piston corer” which had been designed to deal with the problem of extracting cores from soft depositions such as those in the floor of the Pacific Ocean.  Conventional drilling machinery could not maintain the layering of this mushy sediment and had it been used at the 503 holes, whatever wouldn't have been lost at the outset “would most probably have been mashed into a murky paste of limited geological value.”  (Mort La Brecque, Coring Near the Mudline, Mosaic, September/October, 1981.)

La Brecque noted that the sediment in the Caribbean was a compacted mixture of mostly clay and calcium carbonate shells.  "As is usual with such sedimentary rock, it had lost its water content as it aged growing increasingly hard as material accumulated over it."  In contrast, the sediment in the Pacific Ocean floor not only had clay and calcium carbonate shells but also spiny fossils made of silica.  "These maintain their structure and act as jacks to support more porous, softer sediment column[s]."  In geological terms, the shear strength of the floor sediments differed significantly from ocean to ocean and from core depth to core depth.  As I understand it, shear strength is the amount of stress a material can sustain before failing.  The Pacific mush high in the sediment column doesn't have much.

That the new corer succeeded meant that another objective of this leg could be worked toward – dating of the closing of the Isthmus of Panama and analysis of its impact.  The cores that were brought up with the new corer on either side of Panama contained material that ranged in age from the present to some eight million years ago.  One of the initial studies of the planktic (living in the water column) foraminifera shells that were present in the cores drilled from the 502 and 503 cluster of holes showed that the diversity of the planktic foraminifera assemblages in the Pacific and the western Atlantic Oceans remained generally analogous from the eight million year mark until around roughly the beginning of the Pliocene Epoch (this epoch ran from 5.3 to 2.6 million years ago).  (L.D. Keigwin, Jr., Neogene Planktonic Foraminifers from Deep Sea Drilling Project Sites 502 and 503, appearing in Initial Reports of the Deep Sea Drilling Project, Volume LXVIII, 1982)  According to Keigwin, differences in the chronological ranges of selected foraminifera species on either side of Panama in the Pliocene (e.g., Globorotalia pertenuis was found in the Pacific cores until about 3.2 million year ago, but it survived until approximately 2.5 million years ago in the Atlantic) offered “further evidence that the emergent Panama Isthmus was an effective barrier to the migration of planktonic organisms by 3.0 to 3.2 MA [million years ago].”  (p. 276)

The fossil foram shells extracted from the Leg 68 (and all other) cores were carefully picked, sorted and identified, and then mounted on slides.  I know from first hand experience working with similar material from a different drilling project that this is a challenging task.  An example of one of the DSPS foram slides, in this case from core 33 of Hole 503A, is shown below.  The fossils on this slide are from the early Pliocene.  The initial photo shows the entire slide (it’s evident that some of the shells have come loose and now rattle around the slide well).  The photo of cell number 2 shows forams identified as Globorotalia menardii while that of cell 23 shows Globigerinella aequilateralis (as it was identified by the DSDP but is now known as Globigerinella siphonifera).  One of the G. aequilateralis specimens was dyed green at some point, presumably to increase the visibility of some minute attribute for identification or for photographing.

This particular slide is in my personal collection, acquired from a seller on eBay who (if I remember correctly) described its provenance shortly and sweetly as:  “It came from an estate sale.”  One wonders if it should have gone rogue like that given that the DSPS was a federally funded project.  A rock one should not turn over.

Returning for a moment to the problem that the hydraulic piston corer was designed to solve, it is clear that the shear strength of the material being cored is a key attribute and a critical element in the process, presumably dictating how easy it is to drill into the material and likelihood of being able to extract an intact core.  Not surprisingly, shear strength in the material being cored is likely to increase with depth because the deeper material is usually more compacted, but that trend is not always uniform.  Even in my ignorance, I can see how important this attribute of the sea floor at different depths is for the success or failure of any coring effort.

As part of the analysis performed during the coring process, the DSPS team on Leg 68 took what are called “vane shear measurements.”  Based on some online videos and a bit of reading, I’ve come to understand (misunderstand?) that a vane shear test is conducted with an instrument which has several blades at one end.  The extent and ease with which those blades can be turned through material at different depths yield vane shear measurements.

The formal report by the “shipboard scientific party” on Site 503:  Eastern Equatorial Pacific (Initial Reports of the Deep Sea Drilling Project, Volume LXVIII, 1982), provided and discussed shear strength measurements from the coring.  For instance, very low shear strength measurements were found high in the cores, only increasing at about 15 meters (49 feet) where it was about 400 g/cm².  The maximum shear strength for Hole 503 was found to be 1,686 g/cm² reached at 210 meters (689 feet).

And here’s the prize for sticking with me so far.  Supposedly in an effort to be helpful to readers of their report interested in shear strength, the authors penned the following paragraph which repaid me immeasurably for having to wade through otherwise dense, soporific, data-ladened text:

In order to place the vane shear measurements in the proper perspective, shear strengths were determined on several calibration samples.  Each sample was run ten times with the utmost of care.  The shear strength of day-old Jewish rye bread dough (without seeds) was found to be 47.53 g/cm².  Cream cheese proved to have a strength of 66.13 g/cm².  Ginger cookie dough had values of 70.26 g/cm².  A value could not be determined for lime jello, probably because of the large pineapple inclusions interspersed throughout the host material.  Several attempts were made to measure the shear strength of chocolate chip cookie dough, but in each case the cookies were eaten before a measurement could be made.  (p. 171)

On that note, I bid farewell to 2017.

A Colonial Clay Pipe Stem Fragment

The opening scene of the first episode of actor and author Mackenzie Crook’s comedy TV series Detectorists is set in a plowed field somewhere in Essex, England.  Our heroes Andy and Lance are working the field with metal detectors, rhythmically swinging them back and forth while listening through headphones for telltale pings signaling metal in the ground.  Lance suddenly stops, drops to the ground, digs out an object which he inspects with a jeweler’s loupe.  Andy asks what he’s found.

Lance:  Ring pull.  ’83.  Tizer.

Lance carefully puts the ring pull into a plastic baggie.  (Some context here:  What he’s found is typically called here in the States a “pop tab” or “pull tab” or, as Jimmy Buffett styled it, “Stepped on a pop top.  Cut my heel.  Had to cruise on back home.”  Tizer is a British red-colored, citrus soda.)

Andy:  What do you do with ’em?

Lance:  Bag ’em up, stick ’em on eBay.  People buy this shit.

Andy:  Sad tits.

Lance:  You said it.

That exchange captures the gently mocking, almost self-deprecating humor of this superb series.

For a couple of reasons, the Detectorists came to mind this past summer while I was on vacation at the North Fork of Long Island, New York.  Not much of a vacation actually as I was trying to cope with nasty back issues that kept me from looking much above eye-level without excruciating pain.  A friend, who is a birding authority, was visiting, so my wife and I ventured out with her to several of the nature preserves that dot the North Fork.  They vary greatly in quality but the impulse behind them is praise worthy.  Armed with binoculars, which I couldn’t raise to look high in tree canopies, I went birding.

The first preserve we visited was alive with birds even at midday.  Despite my clear limitations, in time I saw 14 different bird species including an Eastern Wood-Pewee, a Great Crested Flycatcher, and a Least Tern.  Not too shabby given how limited my field of vision was.

I was compelled for much of this outing to keep my eyes focused on the ground (lifting them only when someone else spotted a bird), but, as an instance of how contingent life is, at one point I spied a small, off-white object pressed into the trail.  It looked for all the world like a piece of heavy duty electrical cable, but once I pried it free, I realized with delight that it was more likely to be a fragment of a clay pipe stem.

That’s what it was; I have since given it to the group that manages this preserve.

This serendipitous find opened a door to the fascinating world of the archaeological study of English colonial clay pipes, particularly stem fragments.  Lance’s Tizer ring pull is analogous, in a very small way, to clay pipe stem fragments.  Were archaeologists on a dig in that same Essex field as depicted in the Detectorists some 200 years hence to come upon several examples of that same ring pull, they might well date the layer of their excavation which give them up to 1983, give or take.  So it is with pipe stem fragments; they have become one important means of dating archaeological sites of colonial America.  And of course their range is much broader than the decade, or decade and a half, during which pop tops were torn from soda cans and discarded like so many cigarette butts.

Though pipe bowls are also used in the dating process, pipe stem fragments are apparently go-to artifacts partly because these pieces are often found in staggering numbers (stem fragments were so plentiful in colonial times they were sometimes used as ballast in ships; one path in Williamsburg was “paved” with some 12,000 stem fragments) and complete bowls are relatively rare.

It was the analysis of the diameters of the bore holes in pipe stem fragments by archaeologist Jean Carl Harrington that apparently launched the widespread use of such fragments in the dating process.  He reported,

In working with the Jamestown pipe collection I had observed that the early pipes have relatively large holes through the stems, while the holes in later specimens are much smaller.  If this represented a definite and consistent trend, then it might possibly be useful as a dating criterion.  (J.C. Harrington, Dating Stem Fragments of Seventeenth and Eighteenth Century Clay Tobacco Pipes, Quarterly Bulletin of the Archeological Society of Virginia, 1954.)

His work showed that there was in fact a “definite and consistent” trend in how the bore hole diameters clustered in five broad time ranges.  Within different ranges, different hole sizes accounted for the majority of stem fragments.  The key element in his article was a graph (below) that showed the clustering across each time period, a graph that could easily be used in the dating process if one had the average bore hole sizes for a sample of stem fragments from an undated site.  The Y-axis for each time period measures the percentage of the stem fragments in that period featuring a specific bore hole measurement.  The X-axis presents the range of bore hole measurements in increments of 64ths of an inch.

The graph can be read as follows:  in the 1650 – 1680 time range (second from the bottom), Harrington found that 57% of the holes were 7/64ths of an inch wide, another 25% were 8/64ths of an inch wide, and the remaining 18% were 6/64ths of an inch wide.  Were the average bore hole size of a cluster of fragments from an undated site to fall somewhere within these sizes, it was a good bet the site was probably from the 1650 – 1680 time period.

One of the most fascinating aspects of Harrington’s work is its unit of measurement – 64ths of an inch.  It’s such a curious standard but one easily explained by the tool that he used to measure the bore hole diameters of the fragments he analyzed – the metal bits for drills.

Pictured above is one of my sets of drill bits.  Note that they are measured in 64ths of an inch (or some factor of 64 – e.g., 1/16ths is 4/64ths).  Harrington, I have to believe, decided early on that calipers wouldn’t do the trick to measure the hole diameters, nor would trying to rest a ruler across the uneven surface of either end of a fragment.  Wondering what to use, he cast his eyes about his lab (workshop?) and spotted his drill bits.  As he wrote,

In making use of this dating device [his graph], the first requirement is a 39-cent set of drills; the second is common sense.  (Dating Stem Fragments.)

The cost of drill bits has risen markedly since the early 1950s.  Common sense?  In short supply today in many areas.  Harrington cautions against using his graph for a single specimen unless one is using common sense and making only rough estimates about the age of the specimen.  Certainly, it would be foolhardy to use a single specimen to date a site.

So what does Harrington’s graph say about the age of the pipe stem I found?  Well, I cannot duplicate his work precisely with that stem fragment because I give it to the preserve administrators before I’d learned enough about his dating process to turn to my drill bits and see which one fit.  So I worked from the crude measurements I had taken.  The 3.3 mm hole diameter shown in the picture above is slightly less than 0.13 inches.  I make the not unreasonable leap that a drill bit of 8/64ths inch size (or 1/8ths inch) would have fit nicely into that bore hole but not any other drill bit sizes (the rest too big or too little).  The numbers bear me out:  3.3 mm is slightly less than 0.13 inches; 8/64ths inch is 0.125 inches.

Given that, I turned to Harrington’s graph and saw that the time period in which the majority (59%) of bore holes are 8/64ths inch in diameter is 1620 – 1650.  In the next youngest time period (1650 – 1680), only 25% are 8/64ths of an inch.  What can I say with confidence?  Only that this pipe stem probably dates from the 1600s, and possibly from the first half of that century.  (These formulas regarding bore hole diameters apply principally to English clay pipes, not Dutch pipes, whose bore hole diameters are more variable.  Distinguishing between these white clay pipes isn’t always easy, though Dutch pipes are reportedly much softer than English pipes – my fragment is quite hard.)  The possible 17th century dates for this fragment square quite nicely with what I know about the history of the particular preserve where I found the stem.

Harrington’s article presented a very useful piece of analysis with a practical application which, in time, apparently prompted archaeologists working colonial sites where pipe stem fragments were present to do field work or lab work armed with cheap sets of drill bits.

There has been a great more work done by archaeologists trying to render data on the hole diameters in pipe stem fragments into a more mathematically precise dating tool.  Archaeologist Lauren K. McMillan turned her masters thesis into a succinct and accessible article that nicely summarizes the work done to date.  An Evaluation of Tobacco Pipe Stem Dating Formulas (McMillan, Northeast Historical Archaeology, 2016) offers strong evidence for her choice of the best of the analytical methods using these hole diameters to date sites.  She argues in favor of the formula derived by Robert F. Heighton and Kathleen A. Deagan (A New Formula for Dating Kaolin Clay Pipestems, The Conference on Historic Site Archaeology Papers 1971, Volume 6, Part 1, Stanley South, Editor, June, 1972).

Based on their analysis of English pipe stem fragments from an array of sites, the Heighton/Deagan formula reads as follows:

X=(-logY+1.04435)/0.05324
Date=1600+22X

In this formula, Y is the bore hole measurement (the mean of the hole diameters in fragments from a site).  Reflecting the measurement standard set by Harrington, that bore hole measurement is in 64ths of inch.

So, assuming my fragment’s bore hole would be 8/64ths of an inch, the Heighton/Deagan formula gives a date of 1658, a date much too precise to be assigned to this single fragment, but certainly within the range that the Harrington graph suggested.

Later, I came upon a couple of clay pipe bowls in an antique store on the North Fork which I purchased despite their very vague provenience:  “Oh, they were found somewhere around here.”  One of these is pictured below with an image of its bore hole.  A 4/64ths inch drill bit fits perfectly in that hole suggesting this bowl dates from the latter half of the 18th century (using Harrington’s graph) or about 1782 (using Heighton/Deagan).

All of this begs a fundamental question:  Why did the bore holes trend smaller as time went on?  The commonly proffered answer is that stems grew longer and thinner, and so the bore holes had to become smaller.  I don’t find that altogether satisfying.  Why were stems longer?  Other aspects of the pipes changed over time, particularly the size of the bowls which grew larger as tobacco became cheaper.  Might there be something going on with trying to keep the amount of smoke inhaled with each breath roughly constant over time?  A puzzlement.

I turn back to the Detectorists.  During Season 2, Andy, who has completed an archaeological certificate, interviews for a job on a dig that will occur in Botswana (Episode 4).  Seated in a hall outside the room where the interviews are being conducted he notes that some of the other candidates have better beards than his and dirty finger nails to boot.  He momentarily leaves the building to find some dirt to rub into his hands.  While crouched down and rooting in a flower bed he comes upon some objects which he pockets.  At that moment, one of the leaders of the Botswana dig comes by and spots him.  Andy looks up sheepishly, though he has no idea who the man is.  Needless to say he discovers the man’s identity during the interview which is a disaster.  Later, while Andy is standing outside bemoaning the interview experience, the dig leader passes.

Andy:  Sorry, I just wanted to say, I wasn’t picking up cigarette butts.

Dr. Tendai:  Excuse me?

Andy:  Earlier when you saw me, I wasn’t picking up cigarette butts.

Dr. Tendai:  Oh.

Andy:  Clay pipes.

Dr. Tendai:  Pardon?

Andy:  I . . . I saw some bits of clay pipe in the flower bed. . . .  I . . . .

Andy hands the fragments to Dr. Tendai.

Dr. Tendai:  What are they?

Andy:  Broken bits of pipe, you know, that people used to . . . smoke.

Dr. Tendai:  How old are they?

Andy:  Uh . . . These ones are Victorian.  But that one’s early 18th, maybe late 17th century.

Dr. Tendai:  Hmmm.  How can you tell?

Andy:  The older ones are thicker and they had a much smaller bulb because tobacco was so expensive.

Dr. Tendai:  Okay.  And you found these just here?

Andy:  Yeah, just . . . .  Yeah.

Dr. Tendai:  Can I keep these?

Andy:  Yeah.

Ah, the myriad contingencies of life.  I wasn’t picking up cigarette butts, I was bird watching.

An Abundance of Forams and a Few Thoughts on D'Arcy Thompson

The sand my wife brought back from St Andrews, Scotland, in a small vial was filled to abundance with foraminifera (those tiny, single-celled protists, many of which secrete graceful calcium carbonate shells).  The hook for me was the nexus of forams and St Andrews because it was at the University of St Andrews that the biologist and classical scholar D’Arcy Wentworth Thompson (1860 – 1948) taught for over 30 years, and where he wrote the second edition of his magnum opus, On Growth and Form (the first edition was published in 1917).

In this hefty tome, Thompson espoused the singular importance of physical laws (those of engineering and physics) and mathematical principles in the shaping of organic morphology, laws and principles which in his view largely trumped evolution through natural selection.  A centerpiece for his argument was the minute foraminifera, and it’s not hard to see why.  The shapes of many foraminifera shells shout quite loudly “mathematics” and these shells have been used by Thompson and others to illustrate how mathematical spirals are found in nature, evidence, he would argue, of the preeminence of those governing forces.  Thompson wrote:

It is obvious enough that the spiral shells of the Foraminifera closely resemble true logarithmic spirals.  Indeed so precisely do the minute shells of many Foraminifera repeat or simulate the spiral shells of Nautilus [the quintessential exemplar of a logarithmic spiral in nature] and its allies that to the naturalists of the early nineteenth century they were known as Céphalopodes microscopiques [microscopic cephalopods], . . . .  (On Growth and Form, 1917 edition, p. 591)

To get some flavor of how Thompson shaped his argument with these microscopic shells, here is part of the opening paragraph of Chapter XII:  The Spiral Shells of the Foraminifera:

We have already dealt in a few simple cases with the shells of the Foraminifera; and we have seen that wherever the shell is but a single unit or single chamber, its form may be explained in general by the laws of surface tension: the assumption being that the little mass of protoplasm which makes the simple shell behaves as a fluid drop, the form of which is perpetuated when the protoplasm acquires its solid covering. . . .  When the foraminiferal shell becomes multilocular [having more than one chamber], the same general principles continue to hold; the growing protoplasm increases drop by drop, and each successive drop has its particular phenomena of surface energy, manifested at its fluid surface, and tending to confer upon it a certain place in the system and a certain shape of its own. (p. 587)

Indeed, as evolutionary biologist Stephen Jay Gould observed in a lengthy and devastatingly critical dissection of On Growth and Form, it’s at this microscopic level, the level of the foraminifera, that Thompson’s contention that physical laws are paramount in creating the shapes of living creatures may have its greatest (though certainly not complete) validity.

D’Arcy Thompson continued to maintain – and he may well have been right in some cases – that good matches between simple organic conformations (primarily the outward forms of unicellular creatures) and geometric shapes of well know mathematical definition and easily accomplished mechanical construction probably illustrate his favored principle of direct imposition by physical forces.  But he had to admit that he could not apply this line of reasoning to the basic form of a horse or a tuna.  (The Structure of Evolutionary Theory, 2002, p. 1197)

Frankly, I come away from On Growth and Form wondering whether its message for me is a truism:  evolution through natural selection cannot fashion a living shape that defies the basic laws of physics and certain mathematical principles.  In other words, there are some givens that have to be accounted for.  Science writer Philip Ball puts a positive spin on this, noting that Thompson was arguing against the propensity at the time for Darwinians to posit that every feature of an organism was an example of adaptation.  “Thompson’s insistence that biological form had to make sense in engineering terms was a necessary reminder.”  (In Retrospect:  On Growth and Form, Nature, February 2013.)

I would acknowledge that Ball seems to call me out in his article when he notes that “On Growth and Form is a book more often name-checked than read.”  Yes, I admit I’ve only dipped into it, enjoying some of its literary flourishes (which admittedly can go over the top) and appreciating some aspects of his argument, but never aspiring to read it cover to cover (the 1942 edition is over 1,000 pages long).

Ball rightly identifies the logarithmic spiral as the “central motif” of the book (which he notes “appears on the plaque commemorating Thompson’s former residence in St Andrews”).  Here is a representation of the logarithmic spiral.

(This image is in the public domain and was downloaded from Wikimedia Commons [].)

Earlier this year, my wife during her stay in Thompson’s city climbed down some steps to the shoreline of St Andrews Bay at low tide and scraped some sand into a vial.  She came away with a rich mixture of quartz, mica, bits of mollusc shell, fragments of ostracode tests, broken sponge spicules, and a wealth of foraminifera shells.  [Later edit:  I should note that I assume all of the foram shells in my sample of St Andrews sand are from organisms that recently died (hopefully not in the vial of sand as it was transported back to the States).  So, if I'm correct, they are not fossils.]

 Here’s the same image of this smidgeon of St Andrews sand with some of the prominent forams marked.

The most abundant species in this sand are from at least two genera:  Elphidium and Ammonia, shown below in that order.

For these identifications I have relied on the very nice taxonomic appendix in Quantifying Holocene Sea Level Change Using Intertidal Foraminifera:  Lessons from the British Isles by Benjamin P. Horton and Robin J. Edwards (University of Pennsylvania Scholarly Commons, 5-1-2006; originally published as Cushman Foundation for Foraminiferal Research, Special Publication, Volume 40, 2006).

I’ll close by suggesting that in the following passage from On Growth and Form is a hint that foraminifera held some intimate, personal significance for Thompson (or perhaps he was simply waxing eloquent as was his want):

But in days gone by I used to see the beach of a little Connemara bay bestrewn with millions upon millions of foraminiferal shells, simple Lagenae, less simple Nodosariae, more complex Rotaliae: all drifted by wave and gentle current from their sea-cradle to their sandy grave: all lying bleached and dead: one more delicate than another, but all (or vast multitudes of them) perfect and unbroken.  (p. 609)

Evolution and Historical Contingency ~ A Review of Improbable Destinies

Stephen Jay Gould threw down the gauntlet when he argued in Wonderful Life:  The Burgess Shale and the Nature of History (1989) that, were we able to replay the tape of life – that is, erase the metaphorical tape back to some point in the distant past – then “any [subsequent] replay of the tape would lead evolution down a pathway radically different from the road actually taken.”  (Wonderful Life, p. 51)  Evolution, in Gould’s view, is clearly not deterministic, rather, historical contingency holds sway.  By historical contingency, Gould meant that any particular state (of history, of the evolutionary paths of species) was “dependent, or contingent, upon everything that came before – the unerasable and determining signature of history.” (Wonderful Life, p. 283)

Thirty years later debate over the validity of Gould’s proposition continues.  Are the various constraints on evolution through natural selection such that, regardless of where we re-start the tape, the results shown in the replay eventually would be remarkably similar to what we see around us today, or does contingency play such a powerful role that ending up in the same place is not only improbable but perhaps out of the realm of possibility, dependent as it is upon myriad events occurring in just the right order?

This is the context for evolutionary ecologist and herpetologist Jonathan B. Losos’ new book Improbable Destinies:  Fate, Chance, and the Future of Evolution, a riveting and remarkably accessible exploration of convergent evolution and its import for Gould’s thought experiment.

Convergent evolution is the phenomenon in which different species evolve similar features, that is, they converge on very similar morphological features, say, wings in insects and wings in birds among other vertebrates.  The list of examples of convergent evolution in the natural world is rich and long and growing.  As Improbable Destinies makes clear, anole lizards, an early example and the centerpiece of Losos’ research, have been joined by a host of other instances of convergence involving many different kinds of organisms, ranging from snails to guppies, from bats to stickleback fish.

The wealth of such examples has led some scientists, paleontologist Conway Morris prominent among them, to conclude that (as Losos writes), “evolution is deterministic, predictable, following the same course time after time.  The reason, they argue, is that there are only so many ways to make a living in the world,” and so the tape actually would end up in the same place every time it is replayed.  (p. 5)

At the same time, Losos shows that the list of “evolutionary idiosyncrasies” isn’t short and often features creatures, such as the kiwi or the solendon, that have evolved on isolated land masses such as New Zealand, Australia, and Madagascar.  There’s a diverse richness here.  Losos notes, “Conway Morris and his colleagues have made long lists of examples of convergence, but it would be just as easy to make comparable catalogs of species without counterparts.”  (p. 87)  He argues that these one-offs may appear because “natural selection is either not as predictable or as powerful as some make it out to be.  That is, even when species experience identical environments, they might not evolve in the same way.”  (p. 88)

Losos describes many research projects relevant to our understanding of convergent evolution.  These include observation of convergence in the wild, particularly in settings showcasing natural experiments (principally islands), deliberate experiments in the wild as scientists transfer organisms to different locations and analyze their subsequent evolution (yes, as he makes abundantly clear, evolution need not proceed at the snail’s pace that Darwin would have had it), and finally experiments in the laboratory, often in petri dishes or vials involving microorganisms, such as E. coli.

The highlight of the book for me (and here it’s a page-turning pleasure) is Losos’ account of his research on the Anolis lizard (commonly referred to as the anole lizard) and his field work in the Greater Antilles (Cuba, Hispaniola, Jamaica, and Puerto Rico).  Perhaps their most spectacular feature (reserved in most species to males) is the dewlap, “a flap of skin under their throat,” usually hidden from sight.

But when the lizard has an announcement to make – ‘Get lost, buddy, this is my territory” or “Hey, ladies, come check me out.  I’d make a good baby daddy’ – out comes the dewlap, arching downward from the jaw, forming a semicircle so large that the lizard often has to straighten its legs, pushing its body off the ground, to provide clearance.  (p. 58)

A nice example of the prose that graces the book.  Losos is able to make what is certainly tedious and physically difficult research on these lizards into an exciting narrative, a narrative that shows why the anole lizards are robust examples of convergent evolution.

He found that, although each of the four islands in the Greater Antilles has a different array of anole species, those species generally occupy similar ecological niches (bush, or different parts of trees) across the islands.  Those in each niche have evolved to be so similar from island to island that each of these “habitat specialists” can be mistaken for each other.  Yet all of the species on any of these islands are more closely related to each other than they are to any of the similar habitat specialists across the islands.

There is a great deal to absorb from Losos’ walk through the relevant research from the wild to the laboratory, much of it, particularly those natural and deliberate experiments in the wild, thoroughly engaging.  Yet two aspects of the research covered here are, to me, disquieting.

First, I appreciate what can be learned from projects that involve researchers moving organisms in the wild from one place to another (e.g., guppies from pools with predators to those without, or anoles to islands with no lizards), but, given how complex the array of contingencies is in the natural world, what are the real consequences of this manipulation?  I am reminded of how off putting I found ecologist E.O. Wilson’s research that involved fumigating entire mangrove islands in the Florida Keys to kill off all the arthropods.  Yes, he gained insight into how populations recover and how quickly, but at what cost?  Certainly I’m being overly sensitive about this and I’m sure my concerns would be discounted by those involved.  Still the deliberate interference bothers me.

Second, though much of Losos’ explication of the relevant laboratory research is extremely well done and informative, I had a nagging unease about this research.  It didn’t help that I often lost track of what the various projects were trying to prove and what they did prove.  Perhaps the more substantive part of my discomfort with the laboratory portion of the book has to do with my sense that test tube evolution with microbes is somewhat analogous to running computer programs that seek to replicate evolution:  a limited set of conditions are programmed in, with some random function added to mimic the mutations that natural selection has to work with.  The results can be instructive, yet they are, ultimately, artificial.  The full force of contingency that occurs in the natural world is largely missing in either the lab or the computer program.

None of this is to gainsay the importance of tracking myriad generations of microorganisms in controlled environments in the laboratory, not only to see if identical populations always end up in the same place, but also because, as at least one project was structured, researchers could, literally, replay the tape, taking a population to some previous generation and letting it reproduce from there. The message from such experiments, as I understand it, is that, yes, largely the tape replays exactly as it has run before.  But, and it’s a big but, not always.

Indeed, though most of the research from the field and from the laboratory would appear to support the evolutionary determinist camp – that the replay ends up in the same place – closer examination of the results, often over a longer range of time, shows that isn’t always true.  It’s most likely to be true when the populations begun with are closely related; such relatives are likely to evolve similarly when placed in new environments.  Not so when they are more distantly related.

But I didn’t think that Gould’s thought experiment envisioned the tape replaying exactly as it had before and the only thing in play being how evolution would be affected by random mutations in organisms.  Losos, citing philosopher of science John Beatty, acknowledges as much, positing that Gould actually embraced two concepts of contingency.  The first being that replaying of the tape with nothing changed to see if evolution would or could take a different course.  The second being the replaying of the tape from a specific point with the full panoply of historical contingencies allowed to be at work affecting living organisms.  Though these contingencies may be small events with important or unimportant consequences, they could, as well, be more dramatic things such as a storm wiping out the sole representatives of a species or an asteroid hitting the planet and decimating a host of species.  Each of these specific contingencies cannot be anticipated when we replay the tape – they may or may not occur – and they are likely to make all the difference in the world and we are likely not to end up in the same place with any replay.

In fact, and this needs no spoiler alert given the title of the book, I think Losos comes down largely in the Gould camp though his stance is somewhat nuanced.  As he writes,

So, can we predict evolution?  In the short-term, yes, to some extent.  But the longer the passage of time and the more different the ancestors or conditions, the less likely we are to prognosticate successfully.  (p. 336)

He adds, regarding our species,

If any of a countless number of events had occurred differently in the past, Homo sapiens  wouldn’t have evolved.  We were far from inevitable and are lucky to be here, fortunate that events happened just as they did.  Asteroids, of course, but what other events critically tipped evolution’s path in our favor?  Who knows how slight a difference in the past – a tree falling on great-great-great-to-the-millionth-degree-grandpa Ernie, a forest fire, a mutation – might have snuffed out our future existence?  (p. 335)

As he notes, we weren’t fated to evolve though possibly something like us might have.  Ultimately, no matter the outcomes one might anticipate from evolution, it is highly improbable that any specific one will occur.

In closing, I have to explain why I felt compelled to buy and read Improbable Destinies.  I had no choice after reading his interview in Current Biology (August 21, 2017).  He had me from the moment he answers the question:  How did you get started in biology?

Sure, he mentions dinosaurs as a gateway drug, but he goes wonderfully so much further, offering credit to:

a particular episode of Leave it to Beaver, the one in which the Beaver purchases a mail-order baby alligator.

According to the IMDb synopsis of this particular episode (frankly, I don’t remember this one which explains, I guess, why I didn’t become a biologist), Wally and the Beaver acquire a Florida alligator sub rosa and the laughs (such as they are) stem from their efforts to raise an alligator in the house without their parents discovering.  In contrast, Losos, smitten by the idea, was above board in his campaign to be allowed to acquire this kind of reptile.

. . . I knew that local pet stores sold baby caimans, the neotropical relative of alligators. The question was: how to convince my mother to allow a crocodilian in the house. Fortunately, my mom — not liking to say ‘no’ — passed the buck to the local zoo director, a family friend, expecting him to put the kibosh on the idea. To her dismay and my delight, however, he said that having an alligator was how he got his start in herpetology, and the next thing you know, I had a caiman in a kiddie’s wading pool in the basement.

In closing, I must say that my encountering this interview (which in turn prompted me to read and review the book) was contingent on a long, long chain of events that included spotting a turtle in my backyard several years ago and deciding to write a blog post on the turtle shell.  Current Biology had a relevant article and I actually paid for access to it (shows how desperate I was) which got me snared in the publisher's email web which meant the index to this August’s issue reached me.  Then again, even if I hadn’t seen the interview in Current Biology, given what I read I still would have seen reviews of the book and I might very well have ended up in the same place.

Where Worlds Meet or Perhaps Collide

Several years ago on a whim, I purchased a packet of 100 worldwide stamps that mostly feature dinosaurs.  The Mystic Stamp Company originally assembled and sold this packet.  For reasons not relevant to this post, my philatelic interest from my early teen years has robustly revived and that dinosaur packet (found under a bed after a dusty search) now sits on my desk, the object of some study, offering a sense of two worlds – paleontology and philately – meeting.

Here are a few examples of these stamps.

I’ve concluded that this collection, regardless of how it was brought together, actually constitutes a fairly representative sample of how dinosaurs, and by extension, things paleontological, have been treated on postage stamps.

There are several sites on the web that allow me to make this kind of generalization beyond just my small sample of 100 stamps.  For instance, I consulted with Stamps2Go, a great marketplace for folks selling and those buying postage stamps, which currently has 750 stamps for sale that are nestled under the topic “Animals:  Extinct:  Dinosaurs.”  Admittedly, not all of them are dinosaurs, but most are.  (Later edit:  To be sure, among the 750 stamps are duplicates of the same issue being offered by different sellers.)

Then there’s another website that proves once again that if you can imagine it, it’s probably already on the web.  The Paleophilatelie site is the brainchild of Paleophilatelist in Munich, Germany, who married his interest in fossils with his stamp collecting, creating in the process a beautiful virtual collection of worldwide postage stamps (and related postal items such as first day covers and cancellations) with some relationship to paleontology.  It’s a source of endless fascination (though perhaps that may be true for me just because I’ve been sucked into the black holes of these two interests).  Anyway, I have found it great fun to go through his collection of stamps; one can either browse the full gallery or select stamps from specific countries.

So, based on my sample and what I see at sites like the two just described, I’ve reached the two following conclusions:

• The artwork and details in these stamps are mostly second rate.  No other way to say it (unless third rate is more appropriate).  Details often seem wrong.  Among the offending aspects are the proportions of various body parts of the animals, the structure of appendages, the animals’ posture, and their general environment.  Even if the details are right, the artwork mostly fails to bring these creatures to life.  Sad stuff.

• Fossils are missing from the vast majority of these stamps.  In general, postage stamps don’t depict the fossils that underlie our understanding of how extinct ancient animals (and plants) looked and lived.  In my sample of 100 stamps, only one shows a fossil skeleton of a dinosaur.  (I certainly won’t extrapolate from that and suggest that only one percent of postage stamps with dinosaurs or other things paleontological shows fossils.)  The one in my collection was the lowest denomination issue that was part of a five-stamp set released in 1991 to honor that nasty, ill-tempered British paleontologist Sir Richard Owen, a doyen of paleontology in the mid 19th century who coined the word dinosaur.  The stamps feature somewhat stylized portions of skeletons of various dinosaurs, including Iguanodon, the only one of the dinosaurs depicted on these stamps whose fossils Owen actually knew.  (The discussion about these stamps on the Paleophilatelie site is helpful.)

Although some countries do quite nicely with fossils on their stamps, such as Germany, the question remains why fossils are the general exception.  Are fossils harder to illustrate?  Are we (the general public, postage stamp users, or collectors) assumed to be more attracted to depictions of the living creatures or, perhaps, considered likely to be put off by fossilized bones on our stamps?  Maybe fossils are thought to be too static, failing to convey action very well.  Frankly, I don’t think that’s true of fossils, and the inferior artwork used for many dinosaur stamps certainly puts a lie to the notion that illustrating the living animals is necessarily the avenue to attractive, action-filled stamps.

How do U.S. stamps fare in this kind of discussion?  Most of the stamps in my dinosaur packet come from African and Asian nations.  None come from the U.S. though the U.S. has featured illustrations of living dinosaurs on a number of occasions.  For instance, here is a stamp issued in 1970 titled The Age of Reptiles.  (It is in the public domain and downloaded from Wikimedia Commons.)

The artwork on the U.S. stamps I’ve looked at is certainly passable, if generally not memorable.

What of fossils on U.S. stamps?  My search of Arago database of all U.S. stamps on the Smithsonian’s National Postal Museum website turned up exactly one stamp with fossils, featuring a fairly abstract illustration of a trilobite and some ferns.  It was issued in conjunction with the Knoxville World’s Fair in 1982, and bears the title Fossil fuels, one of four stamps in a block with an energy theme (another of the stamps was titled Breeder reactor).  It's telling that that's how fossils came to be on a stamp.  But is that it?  It’s what I could find though I’d be happy to be corrected.

[Well, in this later edit (that itself has been edited further), I will correct myself.  I've found at least two other instances in which fossils appear on U.S. stamps.  In 1955, the U.S. Postal Service issued a stamp commemorating Charles Willson Peale (1741 - 1827) and his museum.  The gifted Peale was, among other things, an artist, politician, and naturalist, and he turned his massive collection of natural history specimens into a museum.  He painted a portrait of himself lifting the curtain on a view of his museum and this is what the 1955 stamp depicts.  At his feet (on the right side of the stamp) are mastodon fossils.

It was the Paleophilatelist on his Paleophilatelie site in his Milestones Paleontology Related Philatelic Items who led me to this stamp which was already in my collection.  Also please see his comment below on this blog post.

And, as I discovered in conversation with a stamp collector, the Postal Service issued stamps in 1974 celebrating the abundance of minerals in the United States.  One of those minerals is, in fact, a fossil - petrified wood.  Here is that stamp:

(This image in the public domain and downloaded from Wikimedia Commons.)]

One final note which may relate to a place where the worlds of paleontology and philately do collide, at least in this country.  As I looked at many hundreds of paleontologically oriented postage stamps from across the globe, it was fairly easy to note when the bicentennial of Charles Darwin’s birth occurred (2009) because at roughly that point there was an explosion of Darwin-related stamps from many countries.  The Darwin OnLine website offers a selection of worldwide stamps featuring the great naturalist.  Conspicuously, though not unexpectedly, missing, is the U.S. where I conclude that, even though the published criteria for selection of individuals to be honored on U.S. stamps pose no particular barrier to the British Charles Darwin, the U.S. Postal Service appears to have shied away from offending the religious right.

Buying a Fossil for its Label

There are myriad reasons for acquiring a fossil from a dealer, but it was a new one for me when the impulse to buy a small gastropod fossil came not from the specimen itself but, rather, from the typewritten paper label that was nestled in the box with the fossil.

The label is 6.3 cm long.

I should make clear that my use of the term “label” in this post is not, I suspect, strictly paleontologically kosher.  For example, the American Museum of Natural History, in its Collections Management site on its Paleontology Portal, defines a label as the numbering that is applied to a fossil specimen itself, not the separate cataloguing record, which is much closer to what in this post I am referring to as a “label.”  That said, the two – numbering on the specimen and the catalogue record – are inextricably (we hope) linked.

Labels (as I am defining them here) are the lifeblood of a fossil collection, accomplishing the essential task of linking specimens to the specific places they were found.  Two pieces of information in the label accomplish that:  description of the location and a specimen number.  Yes, more information can be, and usually is, assigned to a specimen through a label, including, among other data, the scientific name of the fossil (the organism from which it came), the name of collector, date of collection, formation where collected, and age of the specimen.  Though in many ways the label shown above is a prime example of a useful label, it has some limitations.

The location description leaves a bit to be desired.  I suspect that this label’s identification of place – “E. of Labelle, Hendry Co., Florida” – meant a very specific location to the collector R.J. Bland, Jr.  Nevertheless, as it comes to me, this probably would not be sufficient for me to find and explore the spot where the fossil was found.

And, of course, absent the same specimen number written on both the fossil and the label, I cannot be absolutely certain that this label was prepared for this specific fossil, but the likelihood is, I think, overwhelming.

Here then is the specimen to which this label was associated – a fairly well preserved example of a fossil Apple Murex which is 4.9 cm long.

Things have changed since this fossil was found and its label prepared.  For instance, Chicoreus pomum is now known as Phyllonotus pomum.  The formation (as written on the label, ‘“Glades” (Unit A)’) is currently designated the Bermont Formation and its age is considered to be Middle Pleistocene, some 1.1 to 1.8 million years ago.  (Primary sources for this information are the World Register of Marine Species (entry for Phyllonotus pomum), and Molluscs:  Bermont Formation (Middle Pleistocene) by Roger W. Portell and B. Alex Kittle, Part 13 of Florida Fossil Invertebrates, a publication of the Florida Paleontological Society, December 2010.)

What hasn’t changed is the name of the collector R. J.  Bland, Jr.  And therein lies one of the most attractive aspects for me of many fossil labels – the link between specimen and collector.  I have, for example, seen labels for specimens in the Smithsonian’s National Museum of Natural History where the collector is identified as John Bell Hatcher, one of my paleontology heroes (see post).  When someone of Hatcher’s stature is directly connected to the specimen before me the feeling is electric.  Though it wasn’t quite the same experience when I spotted the name of the collector of the Apple Murex, there was a spark.  It was a name I knew.

In a post a couple of years ago on state geological surveys and the rocks and minerals (and sometimes fossils) that they put up for sale to the general public, I described the rocks and minerals that the Virginia Department of Mines, Minerals, and Energy (the state’s geological survey) sent me. They were small samples affixed to the inside of a box.  The label described these samples as coming “from the collection of Rudolph J. Bland, Jr.”  No further identification of Bland was provided and I could uncover little additional information on the web.  Yet, through some good fortune, today here in my hands is a gastropod fossil collected in Florida by R.J. Bland, Jr., who, I feel, has to be the same Rudolph J. Bland, Jr., whose rock and mineral collection the Virginia state geological survey is busy dispersing, small specimen by small specimen.  (How the state geological survey came to be doing that is yet another tale waiting to be told.)

Finally, how this fossil shell found its way into the dealer’s collection of wares that he had on sale at a particular gem and mineral show that I attended is a story with a touch of serendipity.  This dealer had very, very few shells for sale amid the myriad fossilized remains of dinosaurs, sharks, marine reptiles, and the like.  Indeed, he admitted that shells are of little interest to him because he felt they’re of little interest to collectors.  This specific specimen with its label came into his possession because of some horse trading he had entered into several years ago with a woman who was trying to clear her basement of an unwanted fossil collection.  He offered her a price for the good stuff (that is, the teeth and bones) and she countered by asking that he name a price that would cover the entire collection, shells included.  He goosed his offer a small bit and both went away happy.

There’s a moral in here somewhere about the fate of collections and collectors, and I wonder if that moral might be stronger if I knew how the woman came in possession of this Apple Murex with its little label in the first place.

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.