A New Puzzle Turns Earth Into a Rubik’s Cube, but More Complex

Another orbit around the sun and here we are again: back where we started but spun about — changed, perhaps deranged.

Henry Segerman, a British American mathematician and mathematical artist at Oklahoma State University, has invented just the puzzle for this disorienting annual event: Continental Drift, a 3-D sliding puzzle that made its debut earlier this year. The underlying geometric concept is holonomy: When you travel a loop on a curved surface and return to the starting point, you arrive somewhat turned around, rotated, perhaps by 180 degrees.

“Take a mathematical idea, can you make it real?” — this question, Dr. Segerman said, is what motivates his inventions.

He is keen on visualizing mathematics, whether with 3-D printing (he has written a book on the subject) or through non-Euclidean virtual reality experiences. But Dr. Segerman has aphantasia, an inability to construct mental pictures, or “visually hallucinate images at will,” as he puts it. This might explain his passion for making concrete pictures, especially the impressive collection he produced in 2022.

Continental Drift is Earth in miniature, mapped onto a truncated icosahedron — a soccer ball — with its regular patchwork of 12 pentagonal faces and 20 hexagonal faces.

Continental Drift puzzle.CreditCredit…Brett Deering for The New York Times

The conceptual inspiration was a Victorian craze: the classic 15 Puzzle, wherein square tiles numbered 1 to 15 are scrambled on a 4-by-4 grid, with one square left empty; you solve the puzzle by sliding tiles around into numerical order.

In Continental Drift, a spherical version of the 15 Puzzle, it’s the hexagonal tiles that are scrambled. (The pentagons are recessed and remain stationary.) “One of the hexagons, this one in the South Pacific, comes out,” Dr. Segerman explains on his YouTube channel. “We can then activate the San Andreas fault and slide California south into the ocean. And we can keep going, mixing up all of the continents.”

Holonomy happens when a tile travels a full loop along the curved surface of the puzzle: Slide the tile featuring, say, Greenland all the way around the perimeter of a single pentagonal tile — perhaps the tile featuring the North Atlantic. After a complete loop, the Greenlanders return to their starting position rotated by 60 degrees. If the loop encompasses two adjacent pentagons, then the tile returns to the starting point rotated 120 degrees. And so on.

Maker math

Dr. Segerman’s more formal investigations are in topology, the study of geometric objects without regard for lengths or angles. “All you have left is how things are connected together — how many holes a thing ha, and so on,” he said. As an old topology joke goes: “A topologist is somebody who can’t tell the difference between a coffee mug and a doughnut.”

“Henry is a mathematician who also likes making,” said his younger brother and sometime collaborator, Will Segerman. Mr. Segerman, who lives in Manchester, England, is a maker who likes mathematical shapes; he studied fine art and now designs and manufactures escape-room puzzles. Together, the brothers’ creative process is to ask of everything, “But what if…?” Whenever Dr. Segerman mentions a new project, it is invariably “very, very clever,” said Mr. Segerman, who nonetheless looks to poke holes.

Dr. Segerman’s extensor.CreditCredit…Brett Deering for The New York Times

A few years ago, Dr. Segerman demonstrated Extensors: a construction kit for making extending mechanisms from scissor-like hinged parts. “Not stupid enough,” said his brother, who wanted more silliness. They added an activator handle on one end and a four-pronged claw on the other. The result, which made its debut in April, was the Grabber Mechanism — the patent is pending.

Sabetta Matsumoto, an applied mathematician at the Georgia Institute of Technology and Dr. Segerman’s partner, gave input into the contraption’s development and came up with the name Extensor. Between them, math is “a pretty common conversation,” Dr. Matsumoto said.

Idea collider

In a variation on the scissor theme, Dr. Segerman and Kyle VanDeventer, a former student, presented Kinetic Cyclic Scissors this summer.

This invention was the answer to a problem: Given a tile pattern of “self-similar” quadrilaterals — the same shape but rotated, translated, scaled — can the tiles be replaced with scissor linkages (like a scissor lift), and can the structure then be made to move?

Two classes of shapes work, they proved: “boring parallelograms” and “surprising cyclic quadrilaterals,” cyclic meaning that all vertexes of a quadrilateral lie on a circle. Mr. VanDeventer, now an aerospace engineer at Aurora Flight Sciences in Manassas, Va., sees potential applications in the aerospace industry; for proprietary reasons, he declined to elaborate. Scissor systems have been used in architecture, space technologies and satellite panels. In a YouTube comment, a viewer suggested that this mechanism would serve as “one hell of a back-scratcher.”

Kinetic Cyclic Scissors.CreditCredit…Brett Deering for The New York Times

Also consider the Countdown d24, a 24-sided die that is the latest invention to emerge from the Dice Lab, a business partnership with Robert Fathauer, a mathematical artist and puzzle designer in Apache Junction, Ariz. The Countdown d24 is used to keep track of points, such as in the card game Magic: The Gathering.

One problem with some countdown dice, which often are the shape of an icosahedron with 20 triangular sides, is that the numerical path around the shape doesn’t follow a consistent pattern, which leaves you fumbling around to find the number you want.

The Countdown d24 overcomes this problem by instead being a sphericon, fashioned from a triple-cone shape, like an awkwardly shaped football, which is then cut up, twisted about and glued back together.

This invention resulted from a “collision of ideas,” as do many of Dr. Segerman’s creations. He had previously collaborated on making a rolling circus acrobatics apparatus based on a two-cone sphericon.

For the countdown die, two cones didn’t solve that fumbling problem, but three cones did. The result displays a clear path, zigzagging up and down around the die, counting down from 24 to one, making it a cinch to rotate the die to the number you want.

And as it turned out, the die can “roll along its path,” Dr. Segerman noted. Given the right slope, gravity and a nudge, the die wiggles along a perfect chronological countdown. “That was a surprise,” Dr. Segerman said. “Reality does tend to bite back.”

Fight or flight

Dr. Segerman demonstrating holonomy with his own arm.

Continental Drift is not Dr. Segerman’s first time around the holonomy block. Last year, he made the dodecahedral holonomy maze and more recently the Helix Cube Puzzle. His holonomy craze started with riffs on the 15 Puzzle that predated Continental Drift. He added hinges so the tiles can rotate as they slide, producing the 15+4 Puzzle and then the Hyperbolic 29 Puzzle.

“Just looking at this puzzle activates my fight-or-flight response,” a YouTube commenter wrote of the Hyperbolic 29 Puzzle. Dr. Segerman’s friend Rick Rubenstein, a former professional juggler and a semiretired software engineer in Sunnyvale, Calif., followed with: “Henry Segerman, Mad Genius.”

Mr. Rubenstein got to know Dr. Segerman as a fellow recreational juggler at Stanford. Dr. Segerman can stably juggle five balls, and he often takes 100-catch work breaks.

“He’s actually a very sensible guy with a slightly non-Euclidean sense of humor,” Mr. Rubenstein said.

Indeed, while Dr. Segerman knows that his puzzles are solvable, he doesn’t trouble himself with the task of finding the solutions.

Nonetheless, for a rough measure of Continental Drift’s complexity, he calculated that it has 7 × 10³¹ states, or possible configurations. (The Rubik’s Cube, with roughly as many moving parts, has only around 4 × 10¹⁹ states.) A YouTube viewer calculated that exactly half of Continental Drift’s states are attainable.

To Dr. Segerman’s knowledge, only one person has solved Continental Drift so far. “I solve it by unscrewing the removable part of the frame that lets you take the tiles out,” he said. Then he reorients himself and the tiles, and screws the puzzle back together.

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