Over the last two years, mathematicians have identified the best versions of a child’s playroom’s worth of shapes. These results occupy a quirky corner of math and, fittingly, have been produced by unlikely collaborations, involving a mathematician practicing origami with his wife and a professor teaching her undergraduates to play with paper.

The work takes place within the study of “optimal” shapes, which involves understanding which version of a shape best achieves a goal given some constraints. Bees understand this implicitly: They build honeycombs with hexagonal cells because hexagons provide the most storage capacity using the fewest resources.

At least in lore, the first person to search for such a shape was Dido, the founding queen of Carthage. After she landed on what is today the shore of Tunisia, she struck a deal with the Berber king, Iarbas. He agreed to give her whatever land she could enclose in a single ox hide. Rather than lay the meager hide flat, as Iarbas had anticipated, Dido cut it into thin strips, which she used to encircle and claim an entire hill. The ascendant queen’s insight was that given a fixed amount of material, the optimal area-enclosing shape, which defined the city limits of Carthage, is the circle.

“They usually have this flavor. There’s a family of objects, and you want to know which one maximizes this or minimizes that,” said Richard Schwartz of Brown University, who posted three results about optimal shapes in quick succession starting this past August, including one with his wife, Brienne Elisabeth Brown.

All of the recent results are about minimizing the amount of paper, rope or string used to make a particular shape. Schwartz’s recent run started with the Möbius strip, which is formed by taking a strip of paper, giving it a twist, and joining the ends. It has the bizarre feature of being a surface that only has one side, which means you can trace its entire surface without ever lifting your finger.

As far back as the 1930s, mathematicians have tried to find the stubbiest possible rectangle that can be twisted into a Möbius strip. It seems intuitively clear that it’s easy to twist a long, skinny rectangle into a one-sided strip, but that doing so with a square is impossible. But where exactly is the boundary?

Optimal shapes arise when we try to minimize or maximize some value, like, in this case, the ratio of the width of a strip to its length. In crucial mathematical ways, they are the most extreme version of a shape. The study of optimal shapes is a bridge between geometry, in which length matters, and topology, a branch of math that deals with idealized objects that are endlessly stretchable and compressible. In topology, Möbius strips of different sizes are interchangeable, since a small strip can be stretched into a big one, a wide one squished into a skinny one, and so forth. Similarly, rectangular strips of any size are all, topologically, the same.

However, the operation of twisting a strip and joining the ends changes things. To reckon with optimal shapes is to reckon with the limits of topology. Yes, you can squeeze one Möbius strip into another. But how much can you squeeze before it becomes impossible to go any further?

“One question is, what is the least length and the other is, is there a way to attain that least length and what does it look like,” said Elizabeth Denne of Washington and Lee University.

Altogether, there have been at least five results in recent years that have identified new best values for different shapes, including the Möbius strip (with one twist), the three-twist Möbius strip and the simple knot. Some of these results identify the best known value for a shape; others go a step further and prove that no better value is possible.

The Optimal Möbius Strip

To formalize how close to a square a rectangle is, mathematicians use a number called the aspect ratio. It is simply the length divided by the width. A square has an aspect ratio of 1, while a long, skinny, ribbonlike rectangle has a much larger aspect ratio. That ribbon has plenty of slack, allowing the rectangle’s ends to be twisted and attached to one another. But as the strip gets shorter, and the aspect ratio gets closer to 1 — a square  — it gets harder. At a certain point it’s not possible anymore.

In 1977, two mathematicians conjectured that to be twisted in a Möbius strip, a rectangle of width 1 must be longer than $latex sqrt{3}$, as in the strip on the lower right. In August 2023, Schwartz proved that they were correct: Any closer to a square than that, and there’s no way to twist the rectangle into a Möbius strip.

You might be tempted to find a clever workaround. If you fold a square up like an accordion, creating a thin strip of paper, you can then twist it into a Möbius strip. But that doesn’t count, because the folds are sharp, not smooth. (Smoothness has a particular mathematical meaning that aligns with the plain English meaning.)

One central tool in figuring out what optimal shapes look like is called a “limiting shape.” Limiting shapes are different in crucial respects from the shapes being optimized but also share some of their properties. By rough analogy, think about how if you stretch out a rectangle out to make it longer and skinnier it begins to look like a line, or how polygons with more and more sides begin to resemble a circle.

In this case, Schwartz creates a limiting shape for the Möbius strip. Start with a flat piece of paper that is one unit wide and $latex sqrt{3}$ units long. Begin by folding it following the instructions below. This will create sharp creases much like those of the accordion, but in a moment we’ll make those creases smooth by relaxing the paper just a little bit.

Fold down from the upper left corner and up from the bottom right corner, creating a diamond. Then fold up across the midline of the diamond and tape together the two edges, shown by blue and yellow dotted lines, that meet in the interior of the diamond. Now add just the tiniest bit of slack by making the strip just a little bit longer, or a little bit narrower, so that you can tug the triangles apart. This is your Möbius strip. An infinitesimally small ant who traveled on the surface of the triangle, following the folds, would go all the way around — it has only one side.

Mathematicians have long known that such a triangle is a limiting shape for Möbius strips. Schwartz showed that other limiting shapes that would allow for a stubbier strip do not exist. To do this, he used the “T” formed by the triangle’s folds, as seen in the rightmost triangle above.

Schwartz combined arguments from topology and geometry. He used topology to show that on every paper Möbius strip it’s possible to draw intersecting lines that form a T in a particular way. Then, using some basic geometry — the Pythagorean theorem and the triangle inequality — he showed that if such a T exists (which it must), the aspect ratio of the strip has to be greater than $latex sqrt {3}$.

The Optimal Twisted Paper Cylinder

After Schwartz identified the optimal Möbius strip, people asked him: What would happen with more twists? Any odd number of twists produces a Möbius strip, because the resulting shape still has only one side. On the other hand, an even number of twists yields a two-sided structure called a twisted cylinder (shown below left). Unlike an ordinary cylinder, it does not have a well-defined inside and outside.

After his paper on the Möbius strip, Schwartz proved in late September that the limiting shape of the twisted cylinder can be made by folding a 1-by-2 rectangle formed from four stacked right isosceles triangles (as shown above right). To begin, fold triangle B behind triangle A, and triangle D above triangle C. (Dotted-line arrows indicate folds going backward, and solid arrows indicate folds in the forward direction.) Then fold the resulting triangle in two by putting the bottom half behind the top half. Then tape together the dotted blue and yellow lines (which were originally the top and bottom of the rectangle). Finally, make the starting rectangle just a smidge longer, so that you have enough slack to pull the flat shape up into a squished-up twisted cylinder. “The basic idea is to build the limiting shape first and then relax the shape a bit and round out the folds,” wrote Schwartz. “I think of this as a bit like making the thing and then soaking it overnight in water.” As you can see in the figure (above right), the stacked triangle shape is twice as long as it is wide, so the optimal aspect ratio of the twisted cylinder is 2.

The Optimal Three-Twist Möbius Strip

Schwartz then turned his attention to the three-twist Möbius strip. Like the one-twist strip, this is a one-sided figure, but because of the two extra twists, its boundary is more complicated. Schwartz thought that its limiting shape was going to be the hexaflexagon, a bewildering shape popularized by Martin Gardner in a 1956 column in Scientific American. Hexaflexagons are made by folding a strip of equilateral triangles and gluing the ends together. A flattened hexaflexagon looks like a hexagon split into six triangles. But it can be “flexed” by pinching adjacent sides together, as in the children’s game MASH. When it’s opened up again, a different set of triangles is facing outward. “This thing is like if a fortuneteller and a Möbius band had a baby,” Schwartz said.

But Schwartz’s wife, Brienne Elisabeth Brown, started playing around with paper herself and revealed the hexaflexagon to be “a bit of a red herring,” Schwartz said. Brown found a construction she calls the “crisscross” (shown below) that is a limiting shape of a three-twist Möbius strip and is three times as long as it is wide. First you fold along the diagonal line in the middle of the strip, taking the bottom part in front of the top part. Then you fold the top right triangle in front of the triangle below and to its left. You now have the shape shown in step 2: a slanted parallelogram with a square jutting out to the right. Bring the square behind the parallelogram, and the triangle at top in front of the square that is now below it. This makes a new square, shown in step 3.

What were originally the top and bottom edges (shown by dotted blue and yellow lines) are now both on the left edge of the square; tape them together and you have created a limiting shape for a three-twist Möbius strip. As in the case of the one-twist strip, this flat shape is not itself a Möbius strip, but if it’s given just a little extra length so it can relax into three dimensions without sharp bends, it will form a three-twist strip.

Brown and Schwartz also found a completely different limiting shape for the three-twist cylinder, which they call the cup. Unlike the crisscross, the cup cannot be made to lie flat. However, like the crisscross, it is three times as long as it is wide. In a paper posted on October 16, Brown and Schwartz explain why they think that the optimal three-twist strip has an aspect ratio of 3. But they haven’t yet been able to prove it, in part because the existence of the cup, which can’t be flattened, means that the kinds of arguments Schwartz made in the one- and two-twist cases can’t be extended to the three-twist case.

Optimal Trefoil Knots

Not all optimal shapes are variants on the Möbius strip. Mathematicians also ponder how much material you need to make different kinds of knots. In 2020, Denne and two of her undergraduate students — John Carr Haden and Troy Larsen — were studying knots that can be drawn on the surface of a torus, or doughnut.

The simplest torus knot — indeed, the simplest nontrivial knot, period — is called a trefoil. It’s like the one many people use in the first step of tying their shoelaces by making a loop in a piece of rope and pulling one end through, if instead of then tying a bow, they just glued the tips of the shoelaces together to form an overhand knot with the two loose ends connected.

The usual way of tying the trefoil is equivalent to wrapping a piece of string around the torus as shown here: