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)