An anonymous reader quotes a report from The New York Times: OpenAIsaid on Tuesday that its newest artificial intelligence technology had solved one of the "Millennium Problems," a collection of important unanswered math questions meant to push the world's leading mathematicians to new heights. The announcement is an another clear sign that A.I. is fundamentally changing the upper reaches of mathematics, which have long been viewed as a pinnacle of human achievement. The change has excited some mathematicians, while stirring concern among others. Over the past year, A.I. systems successfully solved a wide range of problems that have bedeviled mathematicians for decades. But these problems were not as complex, nor as closely watched, as the one that OpenAI's technology has solved over the past several days. The Millennium Problems are among the most heavily researched in the field. "This is a spectacular culmination of the arc we have seen over the past twelve months," OpenAI researcher Sebastien Bubeck said of the company's new solution. The company announced that one of its latest models, which has not yet been released to the public, needed just 88 hours to solve what mathematicians call "the Navier-Stokes existence and smoothness problem." This problem involves a series of equations that are often used to predict the weather. The equations describe the movement of water and other fluids. The Navier-Stokes problem, which has no clear practical value, asks whether these equations completely break down in certain situations. OpenAI's proof claims to have defined just such a situation. This would imply, at least theoretically, that the laws of physics themselves would break down under certain conditions: that, for example, water could be made to spontaneously explode. But mathematicians and physicists do not believe that this mathematical breakdown could really lead to such an outcome in the physical world. The Navier-Stokes problem was one of seven "Millennium Problems" selected by the Clay Mathematics Institute in the year 2000 as a way of tracking the progress of mathematics in the new millennium. The institute, founded by an American businessman named Landon T. Clay, offered a million dollars for the first correct solution to each problem. Before OpenAI's announcement, only one of the problems had been solved. OpenAI says it managed to solve the Navier-Stokes problem by coordinating as many as 10,000 AI agents, an extremely expensive process that may have cost millions of dollars in computing power. For those curious about the findings, the company has released a paper (PDF) and formal Lean proof.

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Are computers in mathematics rewriting the definitions of stupid and smart?

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Dark energy might be getting weaker. Scientists are wondering if interactions with dark matter in a “dark dimension” may be responsible.

Symmetry (noun, “SIM-meh-tree”)

A shape with symmetry has parts that match each other in some way.

Imagine cutting a heart from a piece of paper. To ensure the two halves match perfectly, you might fold your paper in half. Then, you can cut both sides of the heart at once. When you unfold the paper, the two sides are identical. The heart is symmetrical.

The line down the center of the heart, left over from folding your paper, is called the line of symmetry. Folding a shape along a line of symmetry matches up the parts. This kind of symmetry, across one line down the middle, is called mirror symmetry. That’s because the two sides of the heart are shaped like mirror reflections of one another.

But symmetry can come in different forms. Rotational symmetry is another example. This describes a shape that you can spin into a new position that looks identical to the first one. A propeller often displays this kind of symmetry.

Symmetry appears in nature. The elegantly branching crystal shards of a snowflake often have rotational symmetry. Leaves of a tree may display mirror symmetry. The body shape of a cat or dog or person also shows this kind of symmetry. When referring to living things, biologists often use the word bilateral symmetry. It means the same thing as mirror symmetry.

Radial symmetry is another example of symmetry in body shapes. This describes a body shape where you can draw several different lines of symmetry that cross the center. Jellyfish and starfish are examples of creatures with radial symmetry.

Not all living things are symmetrical. A marine sponge usually lacks matching parts. So, this type of body plan is described as asymmetrical.

In a sentence

Floral designs on ancient pottery suggest people understood symmetry 8,000 years ago.

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