Sep 22, 2024 06:49 AM
Mathematicians discover new class of shape seen throughout nature
https://www.nature.com/articles/d41586-024-03099-6
INTRO: Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
The work considers the mathematical concept of ‘tiling’: how shapes tessellate on a surface. The problem of filling a plane with identical tiles has been so thoroughly explored since antiquity that it’s tempting to suppose that there is nothing left to be discovered about it. But the researchers deduced the principles of tilings with a new set of geometric building blocks that have rounded corners, which they term ‘soft cells’.
“Simply, no one has done this before”, says Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics in New York City, who was not involved in the work. “It’s really amazing how many basic things there are to consider.”
It has been known for millennia that only certain types of polygonal tile, such as squares or hexagons, can be packed together to fill 2D space with no gaps. Tilings that fill space without a regularly repeating arrangement, such as Penrose tilings, have attracted interest since the discovery of non-periodic structures called quasicrystals in the 1980s. Last year, the first quasiperiodic tiling, lacking any true periodicity, that uses just a single tile shape was announced by Goodman-Strauss and his colleagues... (MORE - details)
Mathematicians discover new shapes to solve decades-old geometry prroblem
https://www.quantamagazine.org/mathemati...-20240920/
INTRO: In 1986, after the space shuttle Challenger exploded 73 seconds into its flight, the eminent physicist Richard Feynman was called in to find out what had gone wrong. He later demonstrated that the “O-ring” seals, which were meant to join sections of the shuttle’s solid rocket boosters, had failed due to cold temperatures, with catastrophic results. But he also discovered more than a few other missteps.
Among them was the way NASA had calculated the O-rings’ shape. During preflight testing, the agency’s engineers had repeatedly measured the width of the seals to verify that they had not become distorted. They reasoned that if an O-ring had been slightly squashed — had become, say, an oval, instead of maintaining its circular shape — then it would no longer have the same diameter all the way around.
These measurements, Feynman later wrote, were useless. Even if the engineers had taken an infinite number of measurements and found the diameter to be exactly the same each time, there are many “bodies of constant width,” as these shapes are called. Only one is a circle.
Arguably the best known noncircular body of constant width is the Reuleaux triangle, which you can construct by taking the central region of overlap in a three-circle Venn diagram. For a given width in two dimensions, a Reuleaux triangle is the constant-width shape with the smallest possible area. A circle has the largest.
In three dimensions, the largest body of constant width is a ball. In higher dimensions, it’s simply a higher-dimensional ball — the shape swept out if you hold a needle at a point and let it rotate freely in every direction.
But mathematicians have long wondered if it’s always possible to find smaller constant-width shapes in higher dimensions. Such shapes exist in three dimensions: Though these Reuleaux-like blobs might look a bit pointy, sandwich them between two parallel planes and they will roll smoothly, like a ball. But it’s much harder to tell whether this is true in general. It could be that in higher dimensions, the ball is optimal. And so in 1988, Oded Schramm, then a graduate student at Princeton University, asked a simple-sounding question: Can you construct a constant-width body in any dimension that is exponentially smaller than the ball?
Now, in a paper posted online (opens a new tab) in May, five researchers — four of whom grew up in Ukraine and have known each other since their high school or college days — have reported that the answer is yes.
The result not only solves a decades-old problem, but gives mathematicians their first glimpse into what these mysterious higher-dimensional shapes might look like. Although these shapes are easy to define, they’re surprisingly mysterious, said Shiri Artstein (opens a new tab), a mathematician at Tel Aviv University who wasn’t involved in the work. “Any new thing we learn about them, any new construction or computation, is at this point interesting.” Now researchers can finally access a corner of the geometric universe that was once completely unapproachable... (MORE - details)
https://www.nature.com/articles/d41586-024-03099-6
INTRO: Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
The work considers the mathematical concept of ‘tiling’: how shapes tessellate on a surface. The problem of filling a plane with identical tiles has been so thoroughly explored since antiquity that it’s tempting to suppose that there is nothing left to be discovered about it. But the researchers deduced the principles of tilings with a new set of geometric building blocks that have rounded corners, which they term ‘soft cells’.
“Simply, no one has done this before”, says Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics in New York City, who was not involved in the work. “It’s really amazing how many basic things there are to consider.”
It has been known for millennia that only certain types of polygonal tile, such as squares or hexagons, can be packed together to fill 2D space with no gaps. Tilings that fill space without a regularly repeating arrangement, such as Penrose tilings, have attracted interest since the discovery of non-periodic structures called quasicrystals in the 1980s. Last year, the first quasiperiodic tiling, lacking any true periodicity, that uses just a single tile shape was announced by Goodman-Strauss and his colleagues... (MORE - details)
Mathematicians discover new shapes to solve decades-old geometry prroblem
https://www.quantamagazine.org/mathemati...-20240920/
INTRO: In 1986, after the space shuttle Challenger exploded 73 seconds into its flight, the eminent physicist Richard Feynman was called in to find out what had gone wrong. He later demonstrated that the “O-ring” seals, which were meant to join sections of the shuttle’s solid rocket boosters, had failed due to cold temperatures, with catastrophic results. But he also discovered more than a few other missteps.
Among them was the way NASA had calculated the O-rings’ shape. During preflight testing, the agency’s engineers had repeatedly measured the width of the seals to verify that they had not become distorted. They reasoned that if an O-ring had been slightly squashed — had become, say, an oval, instead of maintaining its circular shape — then it would no longer have the same diameter all the way around.
These measurements, Feynman later wrote, were useless. Even if the engineers had taken an infinite number of measurements and found the diameter to be exactly the same each time, there are many “bodies of constant width,” as these shapes are called. Only one is a circle.
Arguably the best known noncircular body of constant width is the Reuleaux triangle, which you can construct by taking the central region of overlap in a three-circle Venn diagram. For a given width in two dimensions, a Reuleaux triangle is the constant-width shape with the smallest possible area. A circle has the largest.
In three dimensions, the largest body of constant width is a ball. In higher dimensions, it’s simply a higher-dimensional ball — the shape swept out if you hold a needle at a point and let it rotate freely in every direction.
But mathematicians have long wondered if it’s always possible to find smaller constant-width shapes in higher dimensions. Such shapes exist in three dimensions: Though these Reuleaux-like blobs might look a bit pointy, sandwich them between two parallel planes and they will roll smoothly, like a ball. But it’s much harder to tell whether this is true in general. It could be that in higher dimensions, the ball is optimal. And so in 1988, Oded Schramm, then a graduate student at Princeton University, asked a simple-sounding question: Can you construct a constant-width body in any dimension that is exponentially smaller than the ball?
Now, in a paper posted online (opens a new tab) in May, five researchers — four of whom grew up in Ukraine and have known each other since their high school or college days — have reported that the answer is yes.
The result not only solves a decades-old problem, but gives mathematicians their first glimpse into what these mysterious higher-dimensional shapes might look like. Although these shapes are easy to define, they’re surprisingly mysterious, said Shiri Artstein (opens a new tab), a mathematician at Tel Aviv University who wasn’t involved in the work. “Any new thing we learn about them, any new construction or computation, is at this point interesting.” Now researchers can finally access a corner of the geometric universe that was once completely unapproachable... (MORE - details)