One more orbit around the Sun and here we are again: back where we started but spun around – changed, maybe disturbed.
Henry Segerman, a British-American mathematician and mathematical artist at Oklahoma State University, invented just the puzzle for this bewildering annual event: Continental Drift, a 3-D sliding puzzle that made its debut earlier this year. The underlying geometric concept is holonomy: if you loop around a curved surface and come back to where you started, you arrive slightly upside down, rotated, maybe 180 degrees.
“Take a mathematical idea, can you realize it?” – This question, said Dr. Segerman, motivates his inventions.
He is interested in visualizing mathematics, whether with 3D printing (he has 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 hallucinate images at will,” as he puts it. This could explain his passion for concrete images, especially the impressive collection he produced in 2022.
Continental Drift is the Earth in miniature, depicted on a truncated icosahedron – a soccer ball – with its regular patchwork of 12 pentagonal and 20 hexagonal faces.
The conceptual inspiration was a Victorian craze: the classic 15 puzzle, where square tiles numbered 1 through 15 are jumbled up on a 4 by 4 grid, leaving one square blank; You solve the puzzle by moving tiles in numerical order.
In Continental Drift, a spherical version of the 15 puzzle, it’s the hexagonal tiles that get messed up. (The pentagons are recessed and remain stationary.) “One of the hexes, this one in the South Pacific, comes out,” explains Dr. Segerman on his YouTube channel. “We can then activate the San Andreas Fault and slide California south into the ocean. And we can go ahead and mess up all the continents.”
Holonomy happens when a tile makes a full loop along the curved surface of the puzzle: slide the tile with, say, Greenland all the way around the perimeter of a single pentagonal tile — perhaps the North Atlantic tile. After a complete loop, the Greenlanders return to their starting position rotated 60 degrees. If the loop includes two adjacent pentagons, the tile will return to the starting point rotated 120 degrees. Etc.
dr Segerman’s more formal investigations concern topology, the study of geometric objects without regard to length or angle. “All you have left is how things are connected — how many holes a thing has and stuff like that,” he said. As the old topology joke says, “A topologist is someone who can’t tell the difference between a coffee cup and a donut.”
“Henry is a mathematician who also enjoys tinkering,” said his younger brother and occasional collaborator, Will Segerman. Mr. Segerman, who lives in Manchester, England, is a maker who enjoys mathematical forms; He studied fine arts and now designs and makes escape room puzzles. The brothers’ creative process involves asking everything, “But what if…?” Segerman mentions a new project, it’s invariably “very, very clever,” said Mr. Segerman, who nonetheless tries to poke holes.
A few years ago, Dr. Segerman extensors: a kit for producing stretch mechanisms from scissor-like joint parts. “Not stupid enough,” said his brother, wanting more silliness. They added an activator 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, contributed to the development of the apparatus and came up with the name Extensor. Between them, math is “a pretty common conversation,” said Dr. Matsumoto.
In a variation on the scissors theme, Dr. Segerman and Kyle VanDeventer, a former student, did Kinetic Cyclic Scissors this summer.
This invention was the answer to a problem: given a tiling pattern of “self-similar” squares—same shape but rotated, translated, scaled—can the tiles be replaced with scissor linkages (like a scissor lift), and then can the structure be moved?
They proved that two classes of shapes work: “boring parallelograms” and “amazing 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 aerospace industry; For copyright reasons, he declined to elaborate further. Scissor systems have been used in architecture, aerospace engineering, and satellite panels. In a YouTube comment, a viewer suggested that this mechanism would serve as “a back-scratcher from hell.”
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 mathematical artist and puzzle designer in Apache Junction, Arizona. The countdown d24 is used to keep track of points, as in the card game Magic: The Gathering.
A problem with some countdown dice, which are often in the shape of a 20-triangular-sided icosahedron, is that the numeric path around the shape doesn’t follow a consistent pattern, requiring you to fiddle around to find the number you want.
The countdown d24 overcomes this problem by instead being a sphere icon formed from a triple cone shape, like an awkwardly shaped soccer ball, which is then cut up, turned around and glued back together.
This invention resulted from a “collision of ideas,” as many of Dr. Segerman’s creations. Previously, he had helped create a rolling circus acrobatic contraption based on a two-cone sphericon.
For the countdown die, two pins didn’t solve this fiddling problem, but three pins did. The result shows a clear path that zigzags up and down around the cube, counting down from 24 to one, making rotating the cube to the desired number a breeze.
And as it turns out, the cube can “roll on its orbit,” noted Dr. Segerman. With the right tilt, gravity and a nudge, the cube wobbles along a perfect chronological countdown. “It was a surprise,” said Dr. Segerman. “Reality tends to bite back.”
fight or flight
Continental Drift is not Dr. Segerman’s first encounter with the holonomy block. Last year he did the dodecahedral holonomy maze and more recently the helix cube puzzle. His holonomy craze began with riffs on the 15 Puzzle that predated Continental Drift. He added hinges to allow the tiles to rotate while sliding, creating 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, California, followed with: “Henry Segerman, Mad Genius.”
Mr. Rubenstein met Dr. Segerman as a recreational juggler at Stanford. dr Segerman can stably juggle five balls and often takes 100-catch work breaks.
“He’s actually a very reasonable guy with a slightly non-Euclidean sense of humor,” said Mr. Rubenstein.
Although dr Segerman knows his puzzles are solvable if he doesn’t bother finding the solutions.
Nonetheless, as a rough measure of the complexity of continental drift, he calculated that it has 7 × 10³¹ states or possible configurations. (The Rubik’s Cube, with about as many moving parts, only has about 4 × 10¹⁹ states.) One YouTube viewer has calculated that exactly half the states of continental drift are achievable.
after dr Segerman’s knowledge has so far only one person solved the continental drift. “I solve it by unscrewing the removable part of the frame that you use to take out the tiles,” he said. He then realigns himself and the tiles and puts the puzzle back together.