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: Once you journey a loop on a curved floor and return to the start line, you arrive considerably rotated, rotated, maybe by 180 levels.

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

He’s eager on visualizing arithmetic, whether or not with 3-D printing (he has written a guide on the topic) or via non-Euclidean digital actuality experiences. However Dr. Segerman has aphantasia, an lack of ability to assemble psychological footage, or “visually hallucinate pictures at will,” as he places it. This may clarify his ardour for making concrete footage, 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 traditional 15 Puzzle, whereby sq. tiles numbered 1 to fifteen are scrambled on a 4-by-4 grid, with one sq. left empty; you remedy 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 hold going, mixing up all the 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 an entire 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 start line 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 could have left is how issues are linked collectively — what number of holes a factor has, and so forth,” he stated. As an previous topology joke goes: “A topologist is any individual who can’t inform the distinction between a espresso mug and a doughnut.”

“Henry is a mathematician who additionally likes making,” stated his youthful brother and someday collaborator, Will Segerman. Mr. Segerman, who lives in Manchester, England, is a maker who likes mathematical shapes; he studied positive artwork and now designs and manufactures escape-room puzzles. Collectively, the brothers’ artistic course of is to ask of every little thing, “However what if…?” At any time when Dr. Segerman mentions a brand new undertaking, it’s invariably “very, very intelligent,” stated 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 components. “Not silly sufficient,” stated his brother, who needed extra silliness. They added an activator deal with on one finish and a four-pronged claw on the opposite. The end result, 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 stated.

Thought collider

In a variation on the scissor theme, Dr. Segerman and Kyle VanDeventer, a former scholar, introduced 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 raise), and might the construction then be made to maneuver?

Two courses of shapes work, they proved: “boring parallelograms” and “shocking cyclic quadrilaterals,” cyclic that 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 functions within the aerospace trade; for proprietary causes, he declined to elaborate. Scissor techniques have been utilized in structure, house applied sciences and satellite tv for pc panels. In a YouTube remark, a viewer advised 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 recreation Magic: The Gathering.

One downside with some countdown cube, which frequently are the form of an icosahedron with 20 triangular sides, is that the numerical path across the form doesn’t comply with 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, customary from a triple-cone form, like an awkwardly formed soccer, which is then reduce 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 primarily based on a two-cone sphericon.

For the countdown die, two cones didn’t remedy that fumbling downside, however three cones did. The end result shows a transparent path, zigzagging up and down across the die, counting down from 24 to at least one, 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 stated. “Actuality does are inclined to chunk again.”

Battle or flight

Continental Drift is just not Dr. Segerman’s first time across the holonomy block. Final 12 months, he made the dodecahedral holonomy maze and extra lately 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 received to know Dr. Segerman as a fellow leisure juggler at Stanford. Dr. Segerman can stably juggle 5 balls, and he typically takes 100-catch work breaks.

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

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 doable configurations. (The Rubik’s Dice, with roughly as many transferring components, 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 thus far. “I remedy it by unscrewing the detachable a part of the body that allows you to take the tiles out,” he stated. Then he reorients himself and the tiles, and screws the puzzle again collectively.