One more orbit around the sun and here we are again: back where we started but spinning—altered, perhaps deranged.
Henry Segerman, a British-American mathematician and math artist at Oklahoma State University, invented just the puzzle for this disorienting annual event: Continental Drift, a 3D sliding puzzle that debuted earlier this year. The underlying geometric concept is holonomy: when you traverse a loop over a curved surface and return to the starting point, you arrive slightly flipped, twisted, perhaps 180 degrees.
“Take a math idea, can you make it a reality?” – this question, said Dr. Segerman, is what motivates his inventions.
He loves visualizing math, either with 3D printing (he’s written a book on the subject) or through non-Euclidean virtual reality experiences. But dr. Segerman has aphantasia, an inability to construct mental images, or “visually hallucinating images at will,” as he puts it. This could explain his passion for making concrete photos, especially the impressive collection he produced in 2022.
Continental Drift is Earth in miniature, depicted on 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, in which square tiles numbered 1 to 15 are thrown onto a 4 by 4 grid, leaving one square empty; you solve the puzzle by sliding tiles around in numerical order.
In Continental Drift, a spherical version of the 15 Puzzle, it’s the hexagonal tiles that get scrambled. (The pentagons are recessed and remain stationary.) “One of the hexagons, this one in the South Pacific, is popping out,” explains Dr. Segerman on his YouTube channel. “Then we can activate the San Andreas Fault and slide California south into the ocean. And we can go on, mixing up all the continents.”
Holonomy occurs when a tile completes a full loop along the curved surface of the puzzle: slide the tile containing say Greenland all the way around the perimeter of a single pentagonal tile – perhaps the tile containing the North Atlantic Ocean. After a full loop, the Greenlanders return to their starting position turned 60 degrees. If the loop includes two adjacent pentagons, the tile will return rotated 120 degrees to the starting point. And so on.
Maker math
Dr. Segerman’s more formal research concerns 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 does a thing have, and so on,” he said. As an old topology joke goes, “A topologist is someone who can’t tell the difference between a coffee mug and a donut.”
“Henry is a mathematician who also likes to make,” says his younger brother and one-time collaborator, Will Segerman. Mr. Segerman, who lives in Manchester, England, is a maker who loves mathematical shapes; he studied fine arts and now designs and manufactures escape room puzzles. Together, the brothers’ creative process is to ask everything, “But what if…?” When dr. Segerman calls a new project, it’s invariably “very, very smart,” said Mr. Segerman, who is nevertheless out to poke holes.
Dr. Segerman demonstrated Extensors a few years ago: a kit for making extendable hinged parts and extension mechanisms. “Not stupid enough,” said his brother, wanting more silliness. They added an activation lever 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, contributed to the development of the device and came up with the name Extensor. Between them, math is “a pretty common conversation,” said Dr. Matsumoto.
Idea collider
In a variation 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 tiling pattern of “self-similar” quadrilaterals – the same shape but rotated, translated, scaled – could the tiles be replaced by scissor links (like a scissor lift), and could the structure then start moving?
They proved that two kinds of shapes work: “boring parallelograms” and “surprising cyclic quadrilaterals,” where cyclic means that all vertices of a quadrilateral lie on a circle. Mr. VanDeventer, now an aerospace engineer at Aurora Flight Sciences in Manassas, Virginia, sees potential applications in the aviation industry; for property reasons he refused 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 “a great back-scratcher”.
Also consider the Countdown d24, a 24-sided die that is the latest invention to come out of the Dice Lab, a business partnership with Robert Fathauer, a math artist and puzzle designer in Apache Junction, Ariz. The Countdown d24 is used to keep track of points, as in the card game Magic: The Gathering.
One problem with some counting dice, which are often shaped like an icosahedron with 20 triangular sides, is that the numeric path around the shape doesn’t follow a consistent pattern, leaving you fumbling to find the desired number.
The Countdown d24 solves this problem by being a sphere instead, formed from a triple cone shape, like an awkwardly shaped soccer ball, which is then cut out, spun around and glued back together.
This invention was the result of a “clash of ideas,” like many of Dr. Segerman. He had previously helped create a rolling circus acrobatics device based on a sphere with two cones.
For the countdown die, two pins didn’t solve that awkward problem, but three pins did. The result shows a clear path, zigzagging up and down around the die, counting down from 24 to one, making it a cinch to spin the die to the desired number.
And as it turned out, the die can “roll its way,” Dr. Segermann up. Given the right slope, gravity, and a nudge, the die wobbles along a perfect chronological countdown. “That was a surprise,” said Dr. Segerman. “Reality tends to bite back.”
Fight or flight
Continental Drift isn’t the first time Dr. Segerman goes around the holonomy block. Last year he made the Dodecahedral Holonomy Maze and more recently the Helix Cube Puzzle. His holonomy craze started with riffing on the 15 Puzzle that predated Continental Drift. He added hinges to allow the tiles to rotate as they slide, creating the 15 + 4 puzzle and then the hyperbolic 29 puzzle.
“Just looking at this puzzle triggers my fight-or-flight response,” one YouTube commenter wrote of the Hyperbolic 29 Puzzle. Dr. Segerman’s friend Rick Rubenstein, a former professional juggler and semi-retired software engineer in Sunnyvale, California, followed up with, “Henry Segerman, Mad Genius.”
Mr. Rubenstein taught Dr. Segerman as a fellow recreational juggler at Stanford. Dr. Segerman can juggle five balls stably and he often takes work breaks of 100 catches.
“He’s actually a very sensible man with a somewhat non-Euclidean sense of humour,” Mr. Rubenstein said.
Although dr. Segerman knows that his puzzles are solvable, he is not concerned with the task of finding the solutions.
Nevertheless, 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 about the same number of moving parts, only has about 4 × 10¹⁹ states.) A YouTube viewer calculated that exactly half of Continental Drift’s states are feasible.
As far as Dr. Segerman knows, only one person has so far solved Continental Drift. “I solve it by unscrewing the removable part of the frame that allows you to take out the tiles,” he said. Then he reorients himself and the tiles, and screws the puzzle back together.
