A new puzzle turns the Earth into a Rubik’s Cube, but more complex

Another orbit around the sun and we’re back: back to where we started, but in an about-face—changed, perhaps disturbed.

Henry Segerman, a British-American mathematician and mathematical artist from Oklahoma State University, invented the puzzle for this puzzling annual event: Continental Drift, a 3D sliding puzzle that debuted earlier this year. The underlying geometric concept is holonomy: when you loop through a curved surface and return to the starting point, you arrive somewhat flipped, rotated, perhaps 180 degrees.

“Take a mathematical idea, can you make it happen? – that question, said Dr. Segerman, is what drives his inventions.

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

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

The design inspiration was a Victorian craze: the classic 15 Puzzle, in which square tiles numbered 1 through 15 are jumbled onto a 4-by-4 grid, with one square left empty; you solve the puzzle by sliding the tiles in numerical order.

In Continental Drift, a spherical version of the 15 Puzzle, it is the hexagonal tiles that are scrambled. (The pentagons are indented and stand still.) “One of the hexagons, this one in the South Pacific, is sticking out,” says Dr. Segerman on his YouTube channel. “We can then activate the San Andreas Rift and slide California south into the ocean. And we can keep going, mixing up all the continents.

Holonomy occurs when a tile travels a complete loop along the curved surface of the puzzle: slide the tile representing, say, Greenland all the way around the perimeter of a single pentagonal tile – perhaps the tile representing the North Atlantic. After a full loop, the Greenlanders return to their starting position rotated 60 degrees. If the loop encompasses two adjacent pentagons, the tile returns to the starting point with a rotation of 120 degrees. Etc.

Maker Mathematics

Dr. Segerman’s more formal research focuses on topology, the study of geometric objects without regard to lengths or angles. “All you’re left with is how things are connected – how many holes a thing has, and so on,” he said. As an old topology joke goes, “A topologist is someone who can’t tell the difference between a coffee cup and a donut.”

“Henry is a mathematician who also likes to do,” said his younger brother and occasional collaborator, Will Segerman. Mr. Segerman, who lives in Manchester, England, is a designer who loves mathematical shapes; he studied fine art and now designs and manufactures escape puzzles. Together, the creative process of the brothers is to ask everything: “But what if…? Whenever Dr. Segerman mentions a new project, he’s invariably “very, very smart,” said Mr. Segerman, who nonetheless seeks to poke holes.

A few years ago, Dr. Segerman demonstrated Expanders: a construction kit for making expander mechanisms from articulated scissor-like parts. “Not stupid enough,” said her brother, who wanted more silly things. They added an activation handle on one end and a four-pronged claw on the other. The result, which debuted in April, was the Grabber Mechanism – patent pending.

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

Ideas Collider

In a twist on the scissors theme, Dr. Segerman and former student Kyle VanDeventer 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 by scissor bonds ( like a scissor lift), and can the structure then be forced to move?

Two classes of shapes work, they proved: “boring parallelograms” and “surprising cyclic quadrilaterals”, cyclic meaning that all the vertices of a quadrilateral lie on a circle. VanDeventer, now an aerospace engineer with 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 comment on YouTube, one viewer suggested that this mechanism would serve as “one hell of a back-scratcher”.

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

A problem with some countdown dice, which are often shaped like a triangular 20-sided icosahedron, is that the number path around the shape doesn’t follow a consistent pattern, requiring you to fumble around to find the number. that you want.

The Countdown d24 overcomes this problem by being more of a sphericon, fashioned from a three-cone shape, like an awkwardly shaped soccer ball, which is then cut, twisted, and glued back together.

This invention is the result of a “collision of ideas”, like many of Dr. Segerman’s creations. He had previously collaborated in the realization of a rolling circus acrobatic device based on a spherical with two cones.

For the countdown, two cones didn’t solve this fumbling problem, but three cones did. The result shows a clear path, zigzagging up and down around the die, counting from 24 to one, making it easy to spin the die to the desired number.

And it turned out that the dice can “roll on its way,” Dr. Segerman noted. Given the right slope, gravity, and a nudge, the die wiggles along a perfect chronological countdown. “It was a surprise,” Dr. Segerman said. “Reality tends to bite back.”

fight or flight

Continental Drift isn’t the first time Dr. Segerman has circled 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 could rotate as they slid, producing the 15+4 puzzle and then the hyperbolic 29 puzzle.

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

Mr. Rubenstein met Dr. Segerman as a recreational juggler at Stanford. Dr. Segerman can juggle five balls steadily and he often takes 100 strike breaks.

“He’s actually a very no-nonsense guy with a slightly non-Euclidean sense of humor,” Rubenstein said.

Indeed, while Dr. Segerman knows his puzzles can be solved, he doesn’t care about finding the solutions.

Nevertheless, for a rough measure of Continental Drift’s complexity, he calculated that it had 7×10³¹ states, or possible configurations. (The Rubik’s Cube, with about that many moving parts, only has about 4×10¹⁹ states.) A YouTube viewer calculated that exactly half of the states in Continental Drift are feasible.

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 allows the tiles to be removed,” he said. Then he reorients himself and the tiles, and screws the puzzle back together.