Jan 7, 2024 08:52 PM
https://www.quantamagazine.org/mathemati...-20240105/
INTRO: 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... (MORE - details)
INTRO: 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... (MORE - details)