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

One other orbit across the solar and right here we’re once more: again the place we began however spun about — modified, maybe deranged.

Henry Segerman, a British American mathematician and mathematical artist at Oklahoma State College, has invented simply the puzzle for this disorienting annual occasion: Continental Drift, a 3-D sliding puzzle that made its debut earlier this 12 months. The underlying geometric idea is holonomy: While you journey a loop on a curved floor and return to the place to begin, you arrive considerably rotated, rotated, maybe by 180 levels.

“Take a mathematical thought, are you able to make it actual?” — this query, Dr. Segerman mentioned, is what motivates his innovations.

He’s eager on visualizing arithmetic, whether or not with 3-D printing (he has written a e-book on the topic) or via non-Euclidean digital actuality experiences. However Dr. Segerman has aphantasia, an lack of ability to assemble psychological photos, or “visually hallucinate photographs at will,” as he places it. This may clarify his ardour for making concrete photos, particularly the spectacular assortment he produced in 2022.

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

The conceptual inspiration was a Victorian craze: the basic 15 Puzzle, whereby sq. tiles numbered 1 to fifteen are scrambled on a 4-by-4 grid, with one sq. left empty; you resolve the puzzle by sliding tiles round into numerical order.

In Continental Drift, a spherical model of the 15 Puzzle, it’s the hexagonal tiles which are scrambled. (The pentagons are recessed and stay stationary.) “One of many hexagons, this one within the South Pacific, comes out,” Dr. Segerman explains on his YouTube channel. “We are able to then activate the San Andreas fault and slide California south into the ocean. And we will maintain going, mixing up the entire continents.”

Holonomy occurs when a tile travels a full loop alongside the curved floor of the puzzle: Slide the tile that includes, say, Greenland all the way in which across the perimeter of a single pentagonal tile — maybe the tile that includes the North Atlantic. After a whole loop, the Greenlanders return to their beginning place rotated by 60 levels. If the loop encompasses two adjoining pentagons, then the tile returns to the place to begin rotated 120 levels. And so forth.

Maker math

Dr. Segerman’s extra formal investigations are in topology, the research of geometric objects with out regard for lengths or angles. “All you may have left is how issues are linked collectively — what number of holes a factor ha, and so forth,” he mentioned. As an previous topology joke goes: “A topologist is any person who can’t inform the distinction between a espresso mug and a doughnut.”

“Henry is a mathematician who additionally likes making,” mentioned his youthful brother and someday collaborator, Will Segerman. Mr. Segerman, who lives in Manchester, England, is a maker who likes mathematical shapes; he studied nice artwork and now designs and manufactures escape-room puzzles. Collectively, the brothers’ artistic course of is to ask of every thing, “However what if…?” Every time Dr. Segerman mentions a brand new undertaking, it’s invariably “very, very intelligent,” mentioned Mr. Segerman, who nonetheless appears to be like to poke holes.

A couple of years in the past, Dr. Segerman demonstrated Extensors: a development equipment for making extending mechanisms from scissor-like hinged elements. “Not silly sufficient,” mentioned his brother, who wished extra silliness. They added an activator deal with on one finish and a four-pronged claw on the opposite. The outcome, which made its debut in April, was the Grabber Mechanism — the patent is pending.

Sabetta Matsumoto, an utilized mathematician on the Georgia Institute of Expertise and Dr. Segerman’s associate, gave enter into the contraption’s improvement and got here up with the title Extensor. Between them, math is “a fairly frequent dialog,” Dr. Matsumoto mentioned.

Concept collider

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

This invention was the reply to an issue: Given a tile sample of “self-similar” quadrilaterals — the identical form however rotated, translated, scaled — can the tiles get replaced with scissor linkages (like a scissor elevate), and might the construction then be made to maneuver?

Two courses of shapes work, they proved: “boring parallelograms” and “shocking cyclic quadrilaterals,” cyclic which means that each one vertexes of a quadrilateral lie on a circle. Mr. VanDeventer, now an aerospace engineer at Aurora Flight Sciences in Manassas, Va., sees potential purposes within the aerospace business; for proprietary causes, he declined to elaborate. Scissor programs have been utilized in structure, house applied sciences and satellite tv for pc panels. In a YouTube remark, a viewer steered that this mechanism would function “one hell of a back-scratcher.”

Additionally contemplate the Countdown d24, a 24-sided die that’s the newest invention to emerge from the Cube Lab, a enterprise partnership with Robert Fathauer, a mathematical artist and puzzle designer in Apache Junction, Ariz. The Countdown d24 is used to maintain observe of factors, corresponding to within the card sport Magic: The Gathering.

One downside with some countdown cube, which regularly are the form of an icosahedron with 20 triangular sides, is that the numerical path across the form doesn’t observe a constant sample, which leaves you fumbling round to search out the quantity you need.

The Countdown d24 overcomes this downside by as an alternative being a sphericon, long-established from a triple-cone form, like an awkwardly formed soccer, which is then lower up, twisted about and glued again collectively.

This invention resulted from a “collision of concepts,” as do lots of Dr. Segerman’s creations. He had beforehand collaborated on making a rolling circus acrobatics equipment based mostly on a two-cone sphericon.

For the countdown die, two cones didn’t resolve that fumbling downside, however three cones did. The outcome shows a transparent path, zigzagging up and down across the die, counting down from 24 to 1, making it a cinch to rotate the die to the quantity you need.

And because it turned out, the die can “roll alongside its path,” Dr. Segerman famous. Given the precise slope, gravity and a nudge, the die wiggles alongside an ideal chronological countdown. “That was a shock,” Dr. Segerman mentioned. “Actuality does are inclined to chew again.”

Combat or flight

Continental Drift isn’t Dr. Segerman’s first time across the holonomy block. Final 12 months, he made the dodecahedral holonomy maze and extra not too long ago the Helix Dice Puzzle. His holonomy craze began 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 after which the Hyperbolic 29 Puzzle.

“Simply taking a look at this puzzle prompts my fight-or-flight response,” a YouTube commenter wrote of the Hyperbolic 29 Puzzle. Dr. Segerman’s good friend Rick Rubenstein, a former skilled juggler and a semiretired software program engineer in Sunnyvale, Calif., adopted with: “Henry Segerman, Mad Genius.”

Mr. Rubenstein bought to know Dr. Segerman as a fellow leisure juggler at Stanford. Dr. Segerman can stably juggle 5 balls, and he usually takes 100-catch work breaks.

“He’s truly a really wise man with a barely non-Euclidean humorousness,” Mr. Rubenstein mentioned.

Certainly, whereas Dr. Segerman is aware of that his puzzles are solvable, he doesn’t bother himself with the duty of discovering the options.

Nonetheless, for a tough measure of Continental Drift’s complexity, he calculated that it has 7 × 10³¹ states, or attainable configurations. (The Rubik’s Dice, with roughly as many shifting elements, has solely round 4 × 10¹⁹ states.) A YouTube viewer calculated that precisely half of Continental Drift’s states are attainable.

To Dr. Segerman’s information, just one individual has solved Continental Drift up to now. “I resolve it by unscrewing the detachable a part of the body that permits you to take the tiles out,” he mentioned. Then he reorients himself and the tiles, and screws the puzzle again collectively.