The hooked coral reef continues to spawn hyperbolically

Every year after the full moons in late October and November, Australia's Great Barrier Reef begins its annual spawning – first the coral species on the coast, where the waters are warmer, and then the offshore corals, the main event. Last year, this natural spectacle coincided with the woolly dispersal of two new colonies of the Crochet Coral Reef, a long-running artisanal and scientific collaborative work of art that now inhabits the Schlossmuseum in Linz, Austria, and the Carnegie Museum of Art in Pittsburgh.

To date, nearly 25,000 reefers have created a global archipelago of more than 50 reefs – both a tribute to and a plea for these ecosystems, rainforests of the sea, threatened by climate change. The project also explores mathematical themes, as many living reef organisms biologically approximate the idiosyncratic curvature of hyperbolic geometry.

Within the domain of two dimensions, geometry is concerned with properties of points, lines, figures and surfaces: the Euclidean plane is flat and therefore exhibits no curvature. In contrast, the surface of a sphere exhibits constant positive curvature; at all points the surface curves inwards towards 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 rounded surface structure of coral maximizes nutrient absorption, and nudibranchs propel through ruffled water.

In the artworks, marine morphologies are modeled – crocheted – with a loop-like verisimilitude. Much like Monet's water lilies, the crocheted corals are abstract representations of nature, says 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 together with Margaret Wertheim, her twin sister, a science writer responsible for the scientific and mathematical components as well as management. The Wertheims, Australians who live together in Los Angeles, discovered the mother reef from their living room many moons ago, in 2005.

Crochet Coral Reef exhibitions typically consist of two main components: The Wertheims provide an anchor of sorts with works from their collection that they have crocheted over the years. They also include pieces from selected skilled international contributors. One is a 'bleached reef', reminiscent of corals under pressure from rising ocean temperatures; another, a “coral forest” made of yarn and plastic, laments the debris that pollutes reef systems.

In response to an open call, volunteers from far and wide crochet a parade of individual specimens that come together in a “satellite reef,” organized by a local curatorial team led by the Wertheims. The Wertheims liken this hive mind to a friendly version of the Borg from “Star Trek: Next Generation.” All contributors are credited.

The largest satellite reef to date was created in 2022 at the Museum Frieder Burda in Baden-Baden, Germany, with some 40,000 coral pieces from approximately 4,000 employees. The Wertheims call this the Sistine Chapel of crocheted reefs (documented in a splashy document). exhibition catalogue). But the exhibition at the Linz Schloss Museum, which is dedicated to natural sciences as well as to art and culture, is reminiscent of the work of the painter Giuseppe Arcimboldo, whose collage portraits of images of fruits, vegetables and flowers are “fantastically heterogeneous, also very funny and smart,” Ms. Wertheim said.

The Linz satellite reef unites approximately 30,000 pieces from 2,000 crocheters. The disparate sections are colorfully inspired by traditional Austrian 'craftsmanship', as the exhibition text puts it, and there is a huge, glittering coral wall that gives a nod to the artist Gustav Klimt. However, according to the Wertheims, the crochet coral project is proof that it is not always lonely geniuses who create great art, but also communities. In the art world this is a radical idea, they noted, but in science major collaborative projects and papers are being carried out with it thousands of authors are not unprecedented.

Primordial deception

Scientifically, the Linz exhibition contains special symbolism because, as the story explains, the region was previously inhabited by an “ancient primordial sea, filled with corals, the remains of which can still be found in the basins and Alps of Upper Austria.” '

The mathematical dimension of the story intersects (from a distance) with the research of applied mathematician Shankar Venkataramani and his students at the University of Arizona. They use idealized models for this study hyperbolic surfaces in nature. “It's all around us,” said Dr. Venkataramani – think of the ubiquity of kale. “The question is: why is it all around us?” The textbook evolutionary advantage is that it helps optimize processes such as circulation and nutrient absorption. His research group's studies show that there are even more advantages, such as providing a structural 'sweet spot', which makes organisms neither too rigid nor too flexible and allows them to 'move and change shape on a small energy budget'. .

When Margaret Wertheim, who studied math, physics and computer science at university, learned hyperbolic geometry, she found it “a bit mind-boggling.” She took the principles more by faith than by understanding. But by crocheting models, she said, “you really learn what hyperbolic structure is in a very profound way, and in a way that I think is very powerful pedagogically.”

Feeling the fringes

It only became clear a quarter of a century ago that the hyperbolic plane could be created with a crochet hook. Daina Taimina, a mathematician now retired from Cornell University, made this discovery while preparing a geometry course. “I had to feel it,” said Dr. Taimina. Research by the Wertheims in the early to mid-2000s planted a seed for their coral reef project (and a chapter book: “A field guide to hyperbolic space“) and for Dr.'s outreach workshops and shows. Taimina (and her own book: “Crochet adventures with hyperbolic surfaces”).

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 they do). A century later, mathematician William Thurston independently had a similar idea, using paper and tape.

Dr. Taimina came across a crumbling paper version in 1997 during 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 started looking for something more flexible and sustainable for her course. When she tried to knit, the result was too soft and cumbersome. Crocheting turned out to be the perfect medium. Dr. Taimina came up with a simple algorithm: increase the number of stitches in the constant ratio N+1. For example, say N=6: work six stitches, and on the seventh stitch increase by working two stitches into one; repeat, row after row.

“You can experiment with different proportions, but not in the same model,” she warned in an article about “The mathematical intelligence” that she and Dr. Henderson wrote. “You only get a hyperbolic surface if you always increase the number of stitches in the same proportion.”

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

The Wertheims have adopted a similar vision with their Institute for Figuring, a nonprofit organization where projects are motivated by the belief that people can play with and appreciate—and thereby gain insight into—mathematical ideas.

Because of her scientific training, Margaret's instinct had been to use Dr. To follow Taimina down to the last detail. But Christine's artistic sensibility was to break the rules and go wild. For example, crochet a few rows, increasing every third stitch, then switch to every fifth stitch, and then every second stitch. The result is not perfectly hyperbolic, because the piece does not have regular curvature.

For the Wertheims, embracing those irregular fringes was the moment their crochet reef project was born: the whimsical algorithms spawned a riotous taxonomy, a woological seascape of creatures that all the more closely mimicked the geometrically off-kilter curves of their real-life biological counterparts .

Patterns of change

Another incarnation of crocheted coral 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 Monongahela Rivers join to form the Ohio, which flows into in the Mississippi, which flows into the Gulf of Mexico, where coral spawns after the full moons of July and August. This show, organized by Alyssa Velazquez, curatorial assistant of decorative arts and design, features only a satellite reef created by 281 community crocheters.

Ms. Velazquez noted that the Wertheims' project takes inspiration from the fiber arts movement — developed by mostly women, including Sheila Hicks, Tau Lewis and Marie Watt — and then democratizes it. As (mostly) women gathered and joined loops of yarn, Ms. Velazquez observed the threads of the conversations: memories of time spent on local waterways, recycling habits, the chance to crochet something other than baby booties. In doing so, the company represents “the creative potential for environmental dialogue and new ecological behavior,” she said—invoking imaginative but concrete patterns of change.