Another orbit around the sun and here we are again: back where we started but spun about — changed, and perhaps deranged. Henry Segerman, a British American mathematician and mathematical artist at Oklahoma State University in the US, has invented just the puzzle for this disorienting annual event: Continental Drift, a 3D sliding puzzle. 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, Segerman said, is what motivates his inventions.

He is keen on visualising mathematics, whether with 3D printing (he has written a book on the subject) or through non-Euclidean virtual reality experiences. But 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.

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 simply 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 pentagonal ones are recessed and remain stationary.) “One of the hexagons, this one in the South Pacific, comes out,” 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.

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 has, and so on,” he said.

“Henry is a mathematician who also likes making,” said his younger brother and sometime collaborator, Will Segerman. Will 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 Henry Segerman mentions a new project, it is invariably “very, very clever” said Will Segerman, who nonetheless looks to poke holes.

A few years ago, Henry Segerman demonstrated Extensors: a construction kit for making extending mechanisms from scissor-like hinged parts. They added an activator handle on one end and a four-pronged claw on the other. The result, which made its debut in April this year, was the Grabber Mechanism — the patent is still pending.

Sabetta Matsumoto, an applied mathematician at the Georgia Institute of Technology, US, and Segerman’s partner, gave input into the contraption’s development and came up with the name Extensor.

Also consider the Countdown d24, a 24-sided die that is the latest invention to emerge from the Dice Lab, a business partnership Segermanhas with Robert Fathauer, a mathematical artist and puzzle designer in Apache Junction in Arizona in the US. 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 come in 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 are looking for.

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

Continental Drift is not 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. Segerman’sfriend Rick Rubenstein, a former professional juggler and a semi-retired software engineer in Sunnyvale, California, US, followed up that comment with the pithy: “Henry Segerman, Mad Genius.”

Rubenstein got to know Segermanas a fellow recreational juggler at Stanford University in the US. Segerman can stably juggle five balls, and he is prone to taking 100-catchwork breaks.

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

NYTNS