Standing at an Intersection

[Our imagination] will grow weary of conceiving things before nature tires of producing them.
                                        ~ Blaise Pascal, Pensées (1670)

Intersections are places of possibility and often of contradiction.  Last week, at the Smithsonian’s National Museum of Natural History, I stood at an intersection of science and art.  No, much more than that, a conjoining of mathematics, nature, handicraft, community, and perhaps a touch of evolution.  I visited an exhibit featuring both the Smithsonian Community Reef which was made by local handicrafters, and the Hyperbolic Crochet Coral Reef, of which the community reef is a part.  This combined project brings together hyperbolic geometry and the handicraft of crochet in a worldwide movement involving many artists in many places for the crocheting of “woolly” coral reefs.

From a distance – an unusual, huge mound of multicolored yarn.



Up close – a breathtaking work of art, and a nexus of contradictions.  The structure is the recreation in yarn and myriad other materials of a coral reef, modeled both on reefs that exist today . . . and reefs, through flights of crochet fancy, never seen before.

Here in the Smithsonian Community Reef are easily recognizable corals, such as brain, branching, or gorgonian fan coral.  Most denizens of this reef are sessile (fixed and stationary), but not all.  Floating amid the coral is a striking jellyfish with delicate purple tentacles.

In nature, a coral reef reveals manifold contradictions.  Many types of coral thrive in clear tropical water, places that are poor in nutrients.  Yet, they create structures that teem with life.  They eat other organisms, stinging and subduing prey.  Yet, for most reef building coral, vital life processes depend upon a complex symbiotic relationship with photosynthetic algae (zooxanthellae), a relationship speaking to eons of evolutionary change.  The most prosaic and visual level of this relationship means that, without the algae, the coral are white, bleached of color.  (The National Oceanic and Atmospheric Administration website provides a clear and succinct introduction to coral reefs.)

At the mathematical heart of the crochet coral reef are many hyperbolic planes crocheted from yarn, wire, and plastic.  Hyperbolic geometry, by its very contradiction of Euclidian geometry, brings us closer to an array of distinctive shapes and forms produced by nature through evolution.  As mathematician and journalist Margaret Wertheim has described in it,

[T]he natural world teems with swooping, curling, crenellated forms, from the fluted surfaces of lettuces and fungi to the frilled skirts of nudibranchs and sea slugs and anemones.  Nature just loves hyperbolic structures.                    (Article by Mick Mycoff entitled Margaret Wertheim:  Complexity, Evolution and Hyperbolic Space, appearing in the journal Evolution:  Education and Outreach in 2008, p. 531.)

Margaret Wertheim and her twin sister Christine are director and co-director of the Institute For Figuring and the moving spirits behind the Hyperbolic Crochet Coral Reef.

Let me make no pretense of actually understanding hyperbolic geometry.  I have grasped enough to recognize that it arises from a contradiction of Euclid’s fifth postulate, the parallel postulate, which most of us know through John Playfair’s axiom.  That axiom states:

Through a given point not on a given line there passes at most one line that is parallel to the given line.  (Michael Serra, Discovering Geometry:  An Inductive Approach, 1997, p. 730)

In contrast, in hyperbolic space, there are many, indeed, an infinite number of lines through a point not on a given line that are parallel to that line.  See figure below.  Margaret Wertheim notes that it is called hyperbolic because of “this abundance of parallels.”

The lines in this two dimensional Euclidian figure appear, with one exception, curved.  Margaret Wertheim asserts, “From the point of view of someone inside the hyperbolic surface, all these lines would be perfectly straight and none would meet the original line.”  (Margaret Wertheim:  Complexity, Evolution and Hyperbolic Space.)

I envision hyperbolic space as curved, fluted, twisted.  I am helped immeasurably by mathematician Daina Taimina, who in 1997, after watching her mathematician husband David Henderson nurture fragile and hard-to-make paper models of hyperbolic space, crocheted a model out of yarn.  Here’s one crocheted by my mathematician wife following Taimina’s directions in Crocheting the Hyperbolic Plane, an article Taimina and David Henderson wrote for The Mathematical Intelligencer (vol. 23, no. 2, 2001).



There is no overestimating the importance of having a model at hand, one that can be scrutinized and played with.  Reportedly, one math professor, no stranger to teaching hyperbolic geometry, commented upon seeing one of Taimina’s model, “Oh, so that’s how they look.”  (Michelle York, Professor Lets Her Fingers Do the Talking, The New York Times, July 11, 2005.)  Using a crocheted model of hyperbolic space, one can show that those apparently curved lines are straight, as Margaret Wertheim demonstrates in a video of an entertaining talk she delivered at one of the TED (Technology, Entertainment, Design) conferences in 2009.

Nature creates hyperbolic structures for a good reason; hyperbolic structures greatly increase the surface area exposed per amount of volume, a boon to sessile filter feeding animals and plants.  The process of crocheting hyperbolic figures, according to the Wertheims, mimics evolution.  Crocheters have deviated from the “mathematical perfection” that Taimina used in creating her models of hyperbolic space, constantly tweaking the underlying patterns and discovering, in the process, that small changes in the algorithm may have large consequences for the final form.  New shapes emerge, some find favor and end up being selected to inhabit the Hyperbolic Crochet Coral Reef or one of the satellite communities.  On the Hyperbolic Crochet Coral Reef website, the Wertheims invoke the word “species” in their discussion of the evolution of new shapes.  Yes, in the first instance, they place the word in quotation marks, but, as they go on, they become more exuberant in drawing connections to evolution.

Just as the teeming variety of living species on earth result from different versions of the DNA-based genetic code, so too a huge range of crochet hyperbolic species have been brought into being through minor modifications to the underlying code.  As time progresses the models have “evolved” from the simple purity of Dr. Taimina’s mathematically precise algorithms to more complex aberrations that invoke ever more naturalistic forms. 

They conclude, “The Crochet Reef thus serves to engage audiences on the subject of evolution and to demonstrate playfully how evolution works.”

I agree that the shapes and creatures are evolving, but at some remove from what is meant in nature by evolution.  It is intellectually stimulating to cast the process of creating crochet coral in terms that speak of evolution, but, to my mind, it is at some cost to the substance of the concept and process in nature.  Still, that is a minor quibble in the face of the amazing Hyperbolic Crochet Coral Reef.

As I stood at that intersection in the National Museum of Natural History last week, I was reminded of the late Benoit Mandelbrot, who saw yet another kind of geometry in aspects of nature (and in so doing, robbed me of sleep many years ago as I wrote and tweaked computer programs to generate the strange Mandelbrot set).  He observed

I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid – a term used in this work to denote all standard geometry – Nature exhibits not simply a higher degree but an altogether different level of complexity.
                  ~ Benoit B. Mandelbrot, The Fractal Geometry of Nature (1983)

Glaciers, Evolution, and Reefs ~ Darwin and the Agassizs

I’d been waiting for some personal event or insight of my own to hang this story on, then paleontologist Alton Dooley’s son had a class assignment to write about a famous scientist’s contributions to science, an assignment which prompted Dooley (père) to post “Darwin’s Other Theory” (his work, not his son’s) on his blog Updates from the Vertebrate Paleontology Lab. (Alton Dooley is Associate Curator of Paleontology at the Virginia Museum of Natural History.) Since it was a father-son thing and very relevant to the story, I thought it would be an appropriate moment to write this post.

The people and debate surrounding this “other” theory have been percolating in my mental filing cabinet for some time because the theory is the focal point for the third signal encounter in the 19th Century between, on the one side, naturalist Charles Darwin, and, on the other, either Swiss-born naturalist Louis Agassiz or his son Alexander, also a naturalist. (In the late 19th Century, Alexander Agassiz was one of the wealthiest men in the United States as a result of investing in cooper mines which he then managed and made productive.) These scientists were in some degree of conflict for much of their adult lives. If they’d been playing chess, there would be three matches to study:

• one ended almost before it started, though the loser, Darwin, took a couple of decades to admit it with any finality;
• another, the one that mattered most, saw Darwin devastating Louis Agassiz; and
• the last consumed Agassiz fils, who devoted much time and treasure to a battle settled only long after all were dead.

The irony was that Alexander Agassiz and Charles Darwin liked each other.

Since I’ve already covered two of the three battles between the Agassizs and Darwin (a little on each below), this post is principally about the third.

Match One – Glaciers

The first of these encounters centered on the explanation of the parallel “roads” or terraces along either side of the valley of Glen Roy in Scotland. (I’ve written about this previously.) In his first formal scientific paper (1839), Darwin argued that the land at Glen Roy had risen in stages from below or at sea level, at each stage, a shoreline was carved on either side of the valley as the water level in the valley fell. (He’d seen the action of geological uplift first hand during the earthquake he experienced in Chile in 1835.)

Louis Agassiz (pictured here) countered swiftly, arguing that glaciers plugged the egress from the valley, flooding it. At different times, the ice plugs shrank, letting out some of the water, and, so, parallel traces of shorelines were etched on the valley walls. Agassiz’s view prevailed in scientific circles, though Darwin fought on, until, at last, in 1861, he agreed he had to “give up the ghost.”








Match Two – Evolution

The second Agassiz vs. Darwin (pictured below) match (described in a prior posting) was precipitated by Darwin’s publication of On the Origin of Species By Means Of Natural Selection.



Agassiz, in 1859, held sway in the United States, probably the most well-known scientist in the country in both scientific and popular circles. He’d worked hard to create renowned scientific establishments in the U.S. to rival those in Europe. While teaching at Harvard, he’d trained many of the leading American biologists and paleontologists of his time. But, he could not embrace evolution because it so thoroughly undermined his belief in a God-driven plan for the natural world, in which species were independent, coming into being or going extinct through repeated catastrophes, all in keeping with that divine blueprint. Besides, by then, Agassiz had stopped doing science. He launched a campaign in the U.S. in opposition to the theory, an effort he carried out in popular journals and in lectures. He had some success with the general public but was thoroughly and embarrassingly routed in the scientific arena. He remained in opposition to the theory until his death in 1873.

Initially, Alexander took his father’s side (as a dutiful son should), though, by 1869, he was admitting privately to a belief in the theory (he confided it once to Darwin), a position he never made public. He visited Darwin a couple of times, visits both men enjoyed. After Darwin’s death, Agassiz would express disappointment when evidence emerged that Darwin had promoted, and even taken pleasure in, the attacks that the senior Agassiz had endured for his anti-evolution stance.

Match Three – Reefs

The third Darwin-Agassiz debate involved coral reef formation. As I understand it, any theory about coral formation has to resolve the problem that coral reefs are often found mid-ocean in areas of deep water, far exceeding the maximum depth (roughly 200 feet or so) at which living coral can survive. Darwin described the problem and his solution as follows:

The facts then stand as follows:—there are many large spaces of ocean, without any high land, interspersed with reefs and islets formed by the growth of those kinds of corals, which cannot live at great depths; and the existence of these reefs and low islets in such numbers and at such distant points, is inexplicable, excepting on the theory that their rocky bases slowly and successively sank beneath the level of the sea, whilst the corals continued to grow upwards. No positive facts are opposed to this view, and some direct evidence, as well as general considerations, render it probable.
~ Darwin, The Structure and Distribution of Coral Reefs (3rd edition, 1897, of the volume originally published in 1842), p. 132..

He posited that the sinking (or subsidence) of an island accounted for the main variations in the structures of coral reefs. He theorized that, first, coral would form a reef around the perimeter of an ocean island (a fringing reef). As the island sank, the coral reef would grow upwards in an effort to keep the living coral within the depth at which it can survive; a lagoon would now separate the sinking island from the reef (a barrier reef). When the island sank below the waves, the coral would seek to maintain its presence, continuously adding to a ring circling the spot where the island once stood (an atoll). The subsidence of the ocean floor that drew down the islands was, in Darwin’s view, the natural counterpart to the uplift he’d witnessed in his travels in Chile and elsewhere in South America. Dooley’s posting on Darwin’s reef formation theory is much better than the description I’ve just written (link here). His is nicely done, very succinct and accessible, and enhanced by some great pictures and graphics.

Frankly, I think what particularly galled Alexander Agassiz about Darwin’s reef theory was the way he had conceived of it. Darwin’s inspiration for the theory came from studying the charts of the Pacific route the Beagle would travel on its way home from South America. As he wrote in his Autobiography,

No other work of mine was begun in so deductive a spirit as this, for the whole theory was thought out on the west coast of S. America before I had seen a true coral reef. (p. 34)

The deliberate, time-consuming process of gathering the mountains of evidence needed to support a theory was one that he, Agassiz (pictured here), thought any self-respecting scientist had to undertake before advancing a theory. This was the kind of process Agassiz undertook in his decades-long effort to dethrone Darwin’s reef theory. (This was also the process Darwin followed with evolution.)

Agassiz’s rejection of Darwin’s method in this instance can be read in the sarcastic remarks he made in the opening to his study of The Islands and Coral Reefs of Fiji (Bulletin of the Museum of Comparative Zoölogy, May, 1899). Agassiz wrote (I’ve added emphasis to the particularly biting comments which I venture to guess were directed at Darwin):

On looking over the literature on coral reefs, one cannot fail to be struck with the amount of irrelevant matter which has been passed down from writer to writer. Statements made on hearsay have gradually become facts. The observations of inexperienced persons receive general recognition. Special cases are discussed without reference to their limited or exceptional application. The whole question is often threshed out de novo, so that it is difficult to separate the new from the old. And, finally, information gathered from charts is substituted for observation in dealing with questions which the latter alone can settle. (p. 5)

Agassiz was also frustrated by the unwillingness of Darwin’s supporters in this battle to recognize what Agassiz considered to be the incontrovertible evidence against the coral reef theory that had accumulated during the last quarter of the 19th Century.

To Agassiz, All Reefs Are Local
According to David Dobb’s superb account, Reef Madness: Charles Darwin, Alexander Agassiz, and the Meaning of Coral (2005), the strongest, clearest public statement by Alexander Agassiz rejecting Darwin’s reef theory and advancing his own came in the Fiji study. Agassiz had undertaken a voyage to the Fiji Islands in 1897, in part, because they had been touted by geologist James Dwight Dana as offering the clearest examples of subsidence at work in the creation of coral reef structures. (Despite never embracing evolution, Dana had played a role in dethroning Louis in the 1860s. Further, he had written substantive analyses of the reef question, coming down on the side of Darwin’s theory.) In contrast, Agassiz saw only evidence of uplift and then erosion leading to the formation of coral reefs, nothing of sinking islands.

The islands of the whole group have been elevated, and since their elevation have, . . . , remained nearly stationary, and exposed to a great and prolonged process of denudation and of aerial and submarine erosion, which has reduced them to their present height. . . . These atolls and islands, surrounded in part of wholly by encircling and barrier reefs, have not been built (as is claimed by Dana and Darwin) by the subsidence of the islands they enclose. They are not situated in an area of subsidence, but on the contrary in an area of elevation. The theory of Darwin and Dana is therefore not applicable to the Fiji Islands. (p. 135)

His ultimate conclusion was that Darwin’s theory had little applicability to the many coral reefs that scientists had studied worldwide, and that “all reefs are local” (to misquote former Speaker of the U.S. House of Representatives Tip O’Neill):

My observations in Fiji only emphasize what has been said so often, that there is no general theory of the formation of coral reefs, either of barrier reefs or atolls, applicable to all districts, and that each district must be examined by itself. (p. 144)

Agassiz died in 1910, never having published the summary account he said he was working on, the one that would tie all of his reef analyses together and bury Darwin. In fact, little trace of any serious work on it has been found.

The Evidence Comes In [Spoiler Alert]

It wasn’t until the middle of the 20th Century that definitive evidence was first gathered resolving the coral reef issue. This involved drilling through reefs. If the cores brought up by the drilling showed only extensive coral surmounting the island base, subsidence would have been at work. Cores that showed shallow ranges of coral on top of non-coral limestone sediments before reaching the island "basement" would support uplift and erosion.

In 1950, on Eniwetok, an atoll in the Marshall Islands, deep cores were extracted through drilling in the reefs as part of an analysis of the local environment prior to upcoming atomic bomb testing (a sad fate). This drilling technology had been unavailable to the previous combatants in the fight. Dobbs dramatically recounts how this initial evidence came in (evidence that was supported repeatedly over the years by drilling elsewhere):

The first cores were clearly reef rock, as expected. As the drill passed the first few hundred feet and out of coral reef depth, the cores changed little. They still appeared to be reef rock. . . . So it went as the drills cut deeper – 500 feet, 1,000, 2,000, 3,000, 4,000. Finally, at 4,200 feet, the drills hit what was unequivocally basement, a greenish basalt, the volcanic mountain on which the reef had originated. . . . For more than thirty million years this reef had been growing – an inch a millennium – on a sinking volcano, thickening as the lava beneath it subsided. Darwin was right, Agassiz was wrong. (p. 254-5)

Closing Comment and Sources

I’m not sure why I find the Agassizs so interesting. I clearly don’t agree with Louis’ stance on evolution and don’t find much about his character very appealing. Perhaps, it’s easier to explain for Alexander, who seems quite quixotic, devoting a vast fortune to ocean voyages to study coral reefs around the world and then never being able to bring it all together. He is a tragic figure, unable to avenge his father’s scientific demise, and losing both his father and wife in a ten day period in 1873. Ultimately, though, I think it’s because the two men are fortunate to have been the subjects of fine books by skilled writers – Edward Lurie’s Louis Agassiz: A Life in Science (1988) and Dobbs’ Reef Madness (cited earlier).

I must give credit where credit’s due; though I read other material, my thoughts on the reef issue were shaped by Dobbs’ book. (All errors are my own.)

The photographs included in this posting are from the Smithsonian Institution. That of Louis Agassiz can be found here; that of Darwin can be found here; and that of Alexander Agassiz can be found here. They were located through the Collections Search Center.