PALENVILLE, NY – While on a winding mushroom hunt north-south of the lake in New York’s Catskill Mountains, Jessica Rosenkrantz spotted a favorite mushroom: the hexagonal-pore polypore. Ms. Rosenkrantz loves life forms different from humans (and mammals in general), although two of her favorite humans joined the ride: her husband Jesse Louis-Rosenberg and their toddler, Xyla, who gave the your. Ms. Rosenkrantz likes mushrooms, lichens and corals because, she says, “they are quite strange, compared to us”. From the top, the hexagonal polypore looks like any boring brown mushroom (albeit sometimes with an orange glow), but flip it over and there’s a perfect array of six-sided polygons decorating the underside of the cap.

Ms. Rosenkrantz and Mr. Louis-Rosenberg are algorithmic artists who make laser-cut wooden puzzles—among other curiosities—in their design studio, Nervous System, in Palenville, NY. Inspired by the way shapes and forms emerge in the wild, they write bespoke software to “grow” intertwined Puzzle Pieces. Their signature puzzle cuts go by names like dendrite, amoeba, labyrinth, and wave.

Beyond the natural and algorithmic domains, the couple draws its creativity from many points around the compass: science, mathematics, art and the gray areas in between. Chris Yates, an artist who makes hand-cut wooden puzzles (and a collaborator), described their puzzle-making as “not just pushing boundaries – they’re tearing them up and starting over.”

On the day of the hike, Ms. Rosenkrantz and Mr. Louis-Rosenberg’s new puzzle came out hot from the laser cutter. This creation combined the age-old craft of paper marbling with a time-tested invention of the nervous system: the infinity puzzle. Having no fixed form or defined boundary, an infinite puzzle can be assembled and reassembled in many, seemingly endless ways.

Nervous System kicked off this concept design with the “Infinite Galaxy Puzzle”, featuring a photograph of the Milky Way on both sides. “You can only see half of the picture at a time,” Mr. Louis-Rosenberg said. “And each time you do the puzzle, you theoretically see a different part of the picture.” Mathematically, he explained, the design is inspired by the “stunning” topology of a Klein bottle: a “closed, non-orientable surface,” with no interior, exterior, top or bottom. “Everything is continuous,” he says. The puzzle goes on and on, winding up and down, side to side. With a trick: the puzzle “tiles with a flip”, which means that any piece on the right side connects to the left side, but only after the piece has been flipped.

Ms Rosenkrantz recalled that the beginnings of the infinity puzzle prompted some to philosophize on social media: “’A puzzle that never ends? What does it mean? Is it even a puzzle if it doesn’t end?’ There were also questions about the motives of his brains. “What evil, crazy, maniacal people would ever create such a dastardly puzzle that you could never complete?” she says.

A “convoluted” process

Ms. Rosenkrantz and Mr. Louis-Rosenberg were trained at the Massachusetts Institute of Technology. She obtained two degrees, biology and architecture; he dropped out after three years of math. They call their creative process “convoluted” – they are captivated by the seed of an idea, then seek out its telos.

Nearly a decade ago, they began researching paper marbling: drops of ink – swirled, distorted, stretched in water and then transferred to paper – capture patterns similar to those found in the rock that has turned into marble. “It’s like an art form that’s also a science experiment,” Ms. Rosenkrantz said.

In 2021, duo Nervous System began a collaboration with Amanda Ghassaei, an artist and engineer who had built an interactive physics-based paper marbling simulator powered by fluid dynamics and mathematics. (She refined her approach over time.) Ms. Ghassaei created the turbulent flows of psychedelic colors that plunge through the wavy puzzle pieces. Ms. Rosenkrantz and Mr. Louis-Rosenberg created the wave cut especially for the Marbling Infinity Puzzle, which is available in different sizes and colors.

“There’s so much more to explore when you’re not constrained by the physical realities of working with a water pan,” Ms. Ghassaei said. By relying on classic marbling patterns such as the bouquet and the bird’s wing, the simulator allowed for more free results: she could combine the Japanese style of blowing ink, using a puff or fan, with the European style of pushing the ink in different directions. using combs. And it could modify the physical properties of the system to get the most out of each technique: with combing, the fluid must be more viscous; blowing requires lower viscosity and faster flow.

There was a fine line, however, between psychedelic adornment and “letting the color stretch and distort too far,” Ms. Ghassaei said. “That’s where the undo button came in handy.”

Cultivate algorithms

Trial and error is the methodology of the nervous system. Ms. Rosenkrantz and Mr. Louis-Rosenberg started making jewelry in 2007 (a current line uses their Floraform design system), followed by a 3D printed sculpture (Growing Objects) and a Kinematics dress that resides in the MoMA collection. The journal Science presented their research on 3D printed organs with Jordan Miller, a bioengineer at Rice University. They also create software for New Balance – deployed for data-driven midsoles and other aspects of sneaker styling. The same code was reused, in collaboration with fashion designer Asher Levine, to create a bodysuit inspired by dragonfly wings for musician Grimes.

The path from one project to another is marked by mathematical concepts such as Laplacian growth, Voronoi structures and the Turing model. These concepts, which basically govern how shapes emerge and evolve in nature, “cultivate the algorithms,” Rosenkrantz wrote. The same algorithms can be applied to very different media, from winding maze pieces to complex components of 3D printed organs. And the algorithms also solve practical manufacturing problems.

A project that came to fruition this year, the Puzzle Cell Lamp, is based on research into how to cut curved surfaces so that puzzle pieces can be effectively flattened, making it easier to manufacture and ship.

“When you’re trying to build a curved object out of flat material, there’s always a fundamental tension,” said Keenan Crane, surveyor and computer science professor at Carnegie Mellon University. “The more cuts you make, the easier it is to flatten but the harder it is to put together.” Dr. Crane and Nicholas Sharp, Principal Investigator at NVIDIA, a 3D technology company, have devised an algorithm that attempts to find an optimal solution to this problem.

Using this algorithm, Ms. Rosenkrantz and Mr. Louis-Rosenberg delineated 18 flat puzzle pieces that are shipped in what looks like a large pizza box. “By putting the curvy shapes together,” the Nervous System blog explains, “you’ll create a spherical lampshade.”

From Dr. Crane’s perspective, the work of Nervous System adopts a philosophy similar to that of great artists like da Vinci and Dalí: an appreciation of scientific thought as “something that should be incorporated into art, rather than a category of opposing thought”. (He noted that Dalí described himself as a fish swimming between “the cold water of art and the warm water of science.”) Ms. Rosenkrantz and Mr. Louis-Rosenberg have dedicated their careers to finding deep connections between the worlds of creativity and the world of mathematics and science.

“It’s something that people imagine happening more than it actually does,” Dr Crane said. “The reality is that it takes someone who is willing to do the very, very dirty job of translating between worlds.”

Recreate the Earth

The Puzzle Cell Lamp takes its name from the interlocking puzzle cells found in many sheets, but this lamp is not a puzzle per se – it comes with instructions. Again, one could ignore the instructions and organically design an assembly strategy.

According to Mr. Louis-Rosenberg, this is what makes a good puzzle. “You want the puzzle to be an experiment in strategy — recognizing certain patterns and then turning that into a methodology for solving the puzzle,” he said. The psychedelic swirls of marbled infinity puzzles can seem daunting, he added, but there are areas of color that lead the way, piece by piece.

The most difficult infinity puzzle of the nervous system is a map of the Earth. It has the topology of a sphere, but it’s a sphere unfolded flat by an icosahedral map projection, preserving geographic area (unlike some map projections which distort area) and giving every inch of the planet equal billing .

“I’ve had some serious headache complaints about the difficulty,” Ms. Rosenkrantz said. The puzzle pieces have a more complex behavior; rather than tiling with a flip, they rotate 60 degrees and “zip the seams of the card,” she explained. Rosenkrantz finds the infinity factor particularly significant in this context. “You can create your own map of the Earth,” she said, “by centering it on what interests you – making all the oceans continuous, or making South Africa the center, or whatever it is you want to see in a prime position.” In other words, she advised on the blog, “Start anywhere and see where your journey takes you.”