PALENVILLE, N.Y. — On a meandering mushroom hunt astatine North-South Lake successful the Catskill Mountains of New York, Jessica Rosenkrantz spotted a favourite mushroom: the hexagonal-pored polypore. Ms. Rosenkrantz is partial to life-forms that are antithetic from humans (and from mammals generally), though 2 of her favourite humans joined connected the hike: her hubby Jesse Louis-Rosenberg and their toddler, Xyla, who acceptable the pace. Ms. Rosenkrantz loves fungi, lichens and coral because, she said, “they’re beauteous strange, compared to us.” From the top, the hexagonal polypore looks similar immoderate boring brownish mushroom (albeit sometimes with an orangish glow), but flip it implicit and there’s a cleanable array of six-sided polygons tessellating the underside of the cap.

Ms. Rosenkrantz and Mr. Louis-Rosenberg are algorithmic artists who marque laser-cut woody jigsaw puzzles — among different curios — astatine their plan studio, Nervous System, successful Palenville, N.Y. Inspired by however shapes and forms look successful nature, they constitute customized bundle to “grow” intertwining puzzle pieces. Their signature puzzle cuts person names similar dendrite, amoeba, maze and wave.

Beyond the earthy and algorithmic realms, the mates gully their creativity from galore points astir the compass: science, math, creation and fuzzy zones between. Chris Yates, an creator who makes hand-cut woody jigsaw puzzles (and a collaborator), described their puzzle-making arsenic “not conscionable pushing the envelope — they’re ripping it isolated and starting fresh.”

Nervous System debuted this conceptual plan with the “Infinite Galaxy Puzzle,” featuring a photograph of the Milky Way connected some sides. “You tin lone ever spot fractional the representation astatine once,” Mr. Louis-Rosenberg said. “And each clip you bash the puzzle, theoretically you spot a antithetic portion of the image.” Mathematically, helium explained, the plan is inspired by the “mind-boggling” topology of a Klein bottle: a “non-orientable closed surface,” with nary inside, outside, up oregon down. “It’s each continuous,” helium said. The puzzle goes connected and on, wrapping astir apical to bottom, broadside to side. With a trick: The puzzle “tiles with a flip,” meaning that immoderate portion from the close broadside connects to the near side, but lone aft the portion is flipped over.

Ms. Rosenkrantz recalled that the infinity puzzle’s debut prompted immoderate philosophizing connected societal media: “‘A puzzle that ne'er ends? What does it mean? Is it adjacent a puzzle if it doesn’t end?’” There were besides questions astir its masterminds’s motivations. “What evil, mad, maniacal radical would ever make specified a dastardly puzzle that you tin ne'er finish?” she said.

A ‘convoluted’ process

Ms. Rosenkrantz and Mr. Louis-Rosenberg trained astatine the Massachusetts Institute of Technology. She earned 2 degrees, biology and architecture; helium dropped retired aft 3 years of mathematics. They telephone their originative process “convoluted” — they get captivated with the effect of an idea, and past hunt astir for its telos.

Almost a decennary ago, they began researching insubstantial marbling: Drops of ink — swirled, warped, stretched successful h2o and past transferred onto insubstantial — seizure patterns akin to those recovered successful stone that has transformed into marble. “It’s similar an creation signifier that’s besides a subject experiment,” Ms. Rosenkrantz said.

In 2021, the Nervous System duo struck up a collaboration with Amanda Ghassaei, an creator and technologist who had built an interactive physics-based paper marbling simulator powered by fluid dynamics and mathematics. (She has refined her attack implicit time.) Ms. Ghassaei created the turbulent flows of psychedelic color that plunge crossed the wavy puzzle pieces. Ms. Rosenkrantz and Mr. Louis-Rosenberg created the question chopped specifically for the Marbling Infinity Puzzle, which comes successful antithetic sizes and colors.

“There’s truthful galore much things to research erstwhile you’re not constrained by the carnal realities of moving with a tray of water,” Ms. Ghassaei said. Riffing connected classical marbling patterns specified arsenic bouquet and bird-wing, the simulator allowed for much free-form results: She could harvester the Japanese benignant of blowing ink around, utilizing enactment oregon a fan, with the European benignant of pushing ink successful antithetic directions utilizing combs. And she could alteration the carnal properties of the strategy to marque the astir of each technique: with combing, the fluid needs to beryllium much viscous; blowing requires little viscosity and faster flow.

There was a good line, however, betwixt psychedelic finery and “letting the colour agelong and warp excessively far,” Ms. Ghassaei said. “That’s wherever the undo fastener was precise handy.”

Cultivating algorithms

Trial and mistake is the Nervous System methodology. Ms. Rosenkrantz and Mr. Louis-Rosenberg started retired successful 2007 making jewelry (one existent enactment uses their Floraform plan system), followed by 3D-printed sculpture (Growing Objects), and a Kinematics Dress that resides successful MoMA’s collection. The diary Science featured their 3D-printed organ research with Jordan Miller, a bioengineer astatine Rice University. They besides marque bundle for New Balance — deployed for data-driven midsoles and different aspects of sneaker stylization. The aforesaid codification was repurposed, successful a collaboration with the manner decorator Asher Levine, to marque a dragonfly-wing-inspired bodysuit for the instrumentalist Grimes.

The way from 1 task to the adjacent is marked with mathematical concepts similar Laplacian growth, Voronoi structures and the Turing pattern. These concepts, which loosely speaking govern however shapes and forms look and germinate successful nature, “cultivate the algorithms,” Ms. Rosenkrantz has written. The aforesaid algorithms tin beryllium applied to precise antithetic media, from the twisty maze pieces to the intricate components of 3D-printed organs. And the algorithms lick applicable manufacturing problems arsenic well.

A task that came to fruition this year, the Puzzle Cell Lamp, built upon research astir however to chopped curved surfaces truthful the puzzle pieces tin beryllium efficiently flattened, making fabrication and shipping easier.

Using this algorithm, Ms. Rosenkrantz and Mr. Louis-Rosenberg delineated 18 level puzzle pieces that are shipped successful what looks similar a ample pizza box. “By snapping the sinuous shapes together,” the Nervous System blog explains, “you volition make a spherical lamp shade.”

From Dr. Crane’s perspective, Nervous System’s enactment adopts a doctrine akin to that of large artists similar da Vinci and Dalí: an appreciation of technological reasoning arsenic “something that should beryllium integrated with art, alternatively than an opposing class of thought.” (He noted that Dalí described himself arsenic a food swimming betwixt “the acold h2o of creation and the lukewarm h2o of science.”) Ms. Rosenkrantz and Mr. Louis-Rosenberg person dedicated their careers to uncovering heavy connections betwixt the worlds of creativity and the worlds of mathematics and science.

“It’s thing that radical ideate happens much than it truly does,” Dr. Crane said. “The world is it takes idiosyncratic who’s consenting to bash the very, precise grungy enactment of translation betwixt worlds.”

Recreating Earth

The Puzzle Cell Lamp takes its sanction from the interlocking puzzle cells recovered successful galore leaves, but this lamp is not a puzzle due — it comes with instructions. Then again, 1 could disregard the instructions and organically devise an assembly strategy.

In Mr. Louis-Rosenberg’s opinion, that’s what makes a bully puzzle. “You privation the puzzle to beryllium an acquisition of strategizing — recognizing definite patterns, and past turning that into a methodology for solving the puzzle,” helium said. The psychedelic swirls of the marbling infinity puzzles mightiness look daunting, helium added, but determination are zones of colour that pb the way, 1 portion to the next.

Nervous System’s astir challenging infinity puzzle is simply a representation of Earth. It has the topology of a sphere, but it’s a sphere unfolded level by an icosahedral representation projection, preserving geographic country (in opposition to immoderate representation projections that distort area) and giving the planet’s each inch adjacent billing.

“I’ve gotten immoderate complaints from superior puzzlers astir however hard it is,” Ms. Rosenkrantz said. The puzzle pieces person much analyzable behavior; alternatively than tiling with a flip, they rotate 60 degrees and “zip the seams of the map,” she explained. Ms. Rosenkrantz finds the infinity origin peculiarly meaningful successful this context. “You tin make your ain representation of Earth,” she said, “centering it connected what you’re funny successful — making each the oceans continuous, oregon making South Africa the center, oregon immoderate it is that you privation to spot successful a privileged position.” In different words, she advised connected the blog, “Start anyplace and spot wherever your travel takes you.”