Every year after the full moons in late October and November, Australia’s Great Barrier Reef begins its annual spawning — first the coral species inshore, where waters are warmer, then the offshore corals, the main event. Last year, this natural spectacle coincided with the woolly propagation of two new colonies of the Crochet Coral Reef, a long-running craft-science collaborative artwork now inhabiting the Schlossmuseum in Linz, Austria, and the Carnegie Museum of Art in Pittsburgh.

To date, nearly 25,000 crocheters (“reefers”) have created a worldwide archipelago of more than 50 reefs — both a paean to and a plea for these ecosystems, rainforests of the sea, which are threatened by climate change. The project also explores mathematical themes, since many living reef organisms biologically approximate the quirky curvature of hyperbolic geometry.

Within the realm of two dimensions, geometry deals with properties of points, lines, figures, surfaces: The Euclidean plane is flat and therefore displays zero curvature. By contrast, the surface of a sphere displays constant positive curvature; at all points, the surface bends inward toward itself. And a hyperbolic plane exhibits constant negative curvature; at all points, the surface curves away from itself. Reef life thrives on hyperbolism, so to speak; the curvy surface structure of coral maximizes nutrient intake, and nudibranchs propel through water with frilly flanges.

In the artworks, marine morphologies are modeled — crocheted — with loopy verisimilitude. A bit like Monet’s water lilies, the crochet corals are abstract representations of nature, said Christine Wertheim, an artist and writer now retired from the California Institute of the Arts. Dr. Wertheim is the driving artistic force behind the project, which she created with Margaret Wertheim, her twin sister, a science writer who is in charge of scientific and mathematical components as well as management. The Wertheims, Australians who live together in Los Angeles, spun out the mother reef from their living room many moons ago, in 2005.

Crochet Coral Reef exhibitions typically have two main components: The Wertheims provide an anchor, of sorts, with works from their collection that they have crocheted over the years. They also incorporate pieces by select skilled international contributors. One is a “bleached reef,” evoking corals stressed by increases in ocean temperature; another, a “coral forest” made from yarn and plastic, laments the debris that pollutes reef systems.

Then in response to an open call, volunteers far and wide crochet a pageant of individual specimens that agglomerate in a “satellite reef,” staged by a local curatorial team with guidance from the Wertheims. The Wertheims liken this hive mind to a friendly iteration of the Borg from “Star Trek: Next Generation.” All contributors are credited.

The largest satellite reef thus far coalesced in 2022 at the Museum Frieder Burda in Baden-Baden, Germany, with some 40,000 coral pieces by about 4,000 contributors. The Wertheims call this the Sistine Chapel of crochet reefs (documented in a splashy exhibition catalog). But the show at the Linz Schlossmuseum, which is dedicated to natural science as well as art and culture, is reminiscent of the work of the painter Giuseppe Arcimboldo, whose collage portraits from depictions of fruits, vegetables and flowers are “fantastically heterogeneous, also very funny and clever,” Ms. Wertheim said.

The Linz satellite reef unites some 30,000 pieces by 2,000 crocheters. The disparate parts take colorful inspiration from traditional Austrian “craftswomanship,” as the exhibit text puts it, and there is a vast, glittery coral wall that gives a nod to the artist Gustav Klimt. In the Wertheims’ view, however, the crochet coral project is proof that it is not always lone geniuses who create great art, but also communities. In the art world, that is a radical idea, they noted, yet in science big collaborative projects and papers with thousands of authors are not unprecedented.

Scientifically, the Linz exhibition holds special symbolism since, as the narrative explains, the region was previously occupied by an “ancient primordial sea, filled with corals whose remains can still be found in the basins and Alps of Upper Austria.”

The mathematical dimension of the story intersects (from afar) with research by the applied mathematician Shankar Venkataramani and his students at the University of Arizona. They use idealized models to study hyperbolic surfaces in nature. “It’s all around us,” Dr. Venkataramani said — consider the ubiquity of curly kale. “The question is, Why is it all around us?” The textbook evolutionary benefit, he said, is that it helps optimize processes like circulation and nutrient absorption. His research group’s studies show additional advantages, such as affording a structural “sweet spot,” making organisms neither too rigid nor too flexible and allowing them “to move and change shape with a small energy budget.”

When Margaret Wertheim, who studied math, physics and computer science at university, learned hyperbolic geometry, she found it “a bit bamboozling.” She took the principles more on faith than understanding. Yet through crocheting models, she said, “you really do learn in a very deep way what a hyperbolic structure is, and in a way that I think is very powerfully pedagogical.”

That the hyperbolic plane could be looped into existence with a crochet hook became apparent only a quarter-century ago. Daina Taimina, a mathematician now retired from Cornell University, made this discovery while preparing a geometry course. “I needed to feel it,” Dr. Taimina said. Investigations with the Wertheims in the early to mid-2000s planted a seed for their coral-reef project (and a chap book, “A Field Guide to Hyperbolic Space”) and for Dr. Taimina’s outreach workshops and shows (and a book of her own, “Crocheting Adventures With Hyperbolic Planes”).

Further back, in 1868, the Italian mathematician Eugenio Beltrami constructed a parchment model of the hyperbolic plane — and he rolled it into a negatively curved surface called a pseudosphere (as one does). A century later the mathematician William Thurston independently had a similar idea, using paper and tape.

Dr. Taimina encountered a crumbling paper version in 1997 at a workshop by David Henderson, a Cornell mathematician and her partner. Dr. Henderson had learned the model-making technique from Dr. Thurston. On the spot, Dr. Taimina set out to construct something more pliable and durable for her course. When she tried knitting, the result was too floppy, unwieldy. Crochet proved the perfect medium. Dr. Taimina devised a simple algorithm: Increase the number of stitches in the constant ratio N+1. For instance, say N=6: crochet six stitches, and on the seventh stitch, increase by crocheting two stitches into one; repeat, row upon row.

“You can experiment with different ratios, but not in the same model,” she cautioned in an article for “The Mathematical Intelligencer” that she wrote with Dr. Henderson. “You will get a hyperbolic plane only if you increase the number of stitches in the same ratio all the time.”

Dr. Taimina also joined Dr. Henderson, who died in 2018, as a co-author for revised editions of his book “Experiencing Geometry,” wherein he described his belief “that mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone.”

The Wertheims adopted a similar vision with their Institute for Figuring, a nonprofit where projects are motivated by the belief that people can play with and aesthetically appreciate — and thereby acquire an understanding of — mathematical ideas.

With her science training, Margaret’s instinct had been to follow Dr. Taimina’s algorithm to the stitch. But Christine’s artistic sensibility was to break the rules and go wild. For instance, crochet a few rows, increasing every third stitch, and then switch to every fifth stitch, and then to every second stitch — the result is not perfectly hyperbolic, because the piece does not have regular curvature.

For the Wertheims, embracing that irregular frilliness was the moment their crochet reef project was born: The erratic algorithms begot a riotous taxonomy, a woological seascape of creatures that all the more closely emulated the geometrically aberrant curvatures of their real-life biological counterparts.

Another crochet-coral incarnation recently emerged from a pond of creativity organized by the Carnegie Museum of Art in Pittsburgh, a city known for its three rivers: The Allegheny and the Monongahela Rivers converge to form the Ohio, which empties into the Mississippi, which empties into the Gulf of Mexico, where coral spawns after July and August full moons. This show, organized by Alyssa Velazquez, a curatorial assistant of decorative arts and design, features only a satellite reef made by 281 community crocheters.

Ms. Velazquez noted that the Wertheims’ project takes inspiration from the fiber-art movement — advanced by mostly women, among them Sheila Hicks, Tau Lewis and Marie Watt — and then democratizes it. As (mostly) women gathered and interlocked loops of yarn, Ms. Velazquez observed the lines of conversations: memories of time spent at local waterways, recycling habits, the chance to crochet something other than baby bootees. At that, the enterprise represents “the creative potential for environmental dialogue and new ecological behaviors,” she said — invoking imaginative yet concrete patterns of change.

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