Mathematicians created an ‘impossible’ shape that shouldn’t exist

Scientists have designed a new kind of paradoxical shape

By Clara Moskowitz & Robert Ghrist edited by Jen Christiansen & Clara Moskowitz

A rectangle is filled with lines of cubes creating paths at right angles and ladders through the space. — Robert Ghrist

An impossible object is something that looks realistic when drawn but can’t exist in real life. Dutch artist M. C. Escher is famous for depicting, for instance, staircases and waterfalls that are impossible to build in three dimensions. Many of Escher’s works are based on constructions by British mathematicians Roger and Lionel Penrose, such as the Penrose triangle and Penrose stairs, which they published in the 1950s.

Three impossible figures are shown. The first two are variations on what is generally known as a Penrose triangle: a three‑sided geometric shape made of rectangular beams that appear to connect in a continuous loop. The third is a Penrose staircase: a drawing of a continuous loop of steps that appear to ascend or descend forever. All are optical illusions that defy real-world geometry. — Robert Ghrist

Now mathematicians Robert Ghrist of the University of Pennsylvania and Zoe Cooperband of the U.S. Naval Research Laboratory have created a mathematical classification system for visual paradoxes. These objects, they explain, are locally but not globally consistent. A ladybug walking along a Penrose staircase, for example, will feel like it has climbed a full set of stairs, yet it will have returned to the same height it was at when it started. “The essence of a paradox is: you walk around a loop, and something has changed,” Ghrist says. “It’s a mismatch between where you are and where you thought you were.”

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Ghrist and Cooperband used their framework to invent an impossible object that breaks reality in novel ways. It starts with a variant of the Penrose staircase. A bug walking around the blue path in the graphic, for instance, will feel like it is traveling a level course, but if it takes the ladder connecting two opposite sides, it will feel as if it has climbed to a new height. Both courses are locally consistent but globally inconsistent.

A rectangular blue path is created by a series of blue cubes, placed as if they were stepping stones, and settled into apparent three-dimensional space. A tower of pink cubes connects two of the rectangle's sides. — Robert Ghrist

The researchers then imagined rearranging this rectangular path into a line and pasting it onto a cylinder so that the left-hand side connected to the right-hand side. In that case, a bug that walked to the right from its starting point would find itself exactly back where it started.

Within a rectangular plane, a blue path is created by a series of blue cubes, placed as if they were stepping stones, and settled into apparent three-dimensional space. A staircase of pink cubes connects two blue segments, creating an illusion of a height change. Next to that schematic, a cylinder is shown with a ladybug making a full circuit and ending up where it started. — Robert Ghrist

The scientists further imagined winding the path like a Möbius strip—a form one can make by twisting a strip of paper and attaching the two ends. A bug that traveled to the right from its starting point would find that after it completed the loop, what it once considered right side up had changed.

The schematic setup is very similar to the preceding image. Next to that schematic, a mobius strip is shown with a ladybug making a full circuit and ending upside down, underneath is its original position. — Robert Ghrist

This path forms the basis of the new impossible shape, which is a continuous multilevel staircase modeled on a shape called a Klein bottle, invented by German mathematician Felix Klein in 1882.

A rectangle is filled with lines of cubes creating paths at right angles and ladders through the space. Paths touch—and apparently extend off—all four sides of the rectangle. — Robert Ghrist

In the impossible Klein ladder, a bug’s orientation flips when it crosses a vertical edge, just as it does in the Möbius strip. A ladybug can make a horizontal loop from its starting point by moving up a ladder, across a pathway, up another ladder and across a vertical edge. When the bug has finished the loop, it is upside down relative to the way it started (a).

The schematic in the previous image now hosts a ladybug. Arrows show its path as described in the text. — Robert Ghrist

When the ladybug makes a vertical loop and crosses a horizontal edge, however, its orientation stays the same, just as on the cylinder. To make this kind of loop, the bug will start again at the same spot, move up one ladder, then head left to cross over the horizontal edge, completing the loop without having made any flips (b).

The base image is repeated, this time with arrows showing a different path for the ladybug, as described in the text. — Robert Ghrist

The grid below represents the “unwrapped” perceptual space that our ladybug experiences; flips are factored into the tiling by reflections. If the ladybug is in the center column, it’s not flipped. If it moves horizontally into the leftmost or rightmost column, it has reflected and become flipped—the meaning of “up” becomes reversed. The black cubes all mark the “same” starting point with an unknowable absolute height and orientation.

The rectangle of paths shown in the previous three images is now tiled into a grid of three by three. The central column is stacked. The paths continue seamlessly from tile to tile. The left and right columns mirror the central column. The central column is labelled "up is up." The left and right columns are labelled "down is up." — Robert Ghrist

Consider a ladybug that makes both a horizontal and a vertical loop in this space. The order of those loops is important. In scenario 1 (yellow), the ladybug does a horizontal (reflecting) loop first (a), then a vertical loop (b). The result: it climbs up two ladders, then its orientation flips and it climbs “up” a third from its perspective. But from the outside it seems it has really climbed down. In scenario 2 (green), the ladybug does a vertical loop first (b), then a horizontal (reflecting) loop (a). The result: it climbs up three ladders to get back to what it perceives to be the same spot it started in.

The previous tiled image now hosts a ladybug. Arrows show two path scenarios, as described in the text. — Robert Ghrist

This new shape is the first impossible object for which such ordering produces different outcomes—a property called nonabelian. “We deal with nonabelian things all the time in math,” Ghrist says, “but it’s never been seen in a visual paradox before.”

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