Quanta Magazine

Given a central sphere, how many identical spheres can “kiss” it without overlapping? Arrange six pennies around a central penny, and you’ll find they fit snugly into a daisylike pattern. But in higher dimensions, the problem gets much harder. www.quantamagazine.org/mathematicia...

The Polyglot Neuroscientist Resolving How the Brain Parses Language | Quanta Magazine Is language core to thought, or a separate process? For 15 years, the neuroscientist Ev Fedorenko has gathered evidence of a language network in the human brain — and has found some similarities to LL...

In brain scans of around 1,400 people, the cognitive scientist Ev Fedorenko has identified a sort of digestive system for language. Fedorenko spoke with Quanta about the system’s workings and how they might be compared to an LLM or the digestive system. www.quantamagazine.org/the-polyglot...

Why are protons always paired with neutrons? Why does the energy of an atom come in discrete quantities? A powerful mathematical object called a Lie group has helped scientists answer these questions.
www.quantamagazine.org/what-are-lie...

‘Reverse Mathematics’ Illuminates Why Hard Problems Are Hard | Quanta Magazine Researchers have used metamathematical techniques to show that certain theorems that look superficially distinct are in fact logically equivalent.

Most math works like this: Start with the axioms that you know, then use them to prove a theorem. A recent complexity theory paper flipped this formula upside down.

🗺️ 📍Read about the groundbreaking 2020 result: www.quantamagazine.org/computer-sci...

No one has improved on the results since 2020. But the traveling salesperson problem has applications ranging from DNA sequencing to ride-sharing logistics. Over the decades, it has inspired many of the most fundamental advances in computer science.

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“[The traveling salesperson problem] isn’t a problem, it’s an addiction.” — Christos Papadimitriou, computational complexity expert

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It's hard to find the best tree to use. The updated algorithm uses a random process to generate a tree in which cities with an odd number of connections tend to have nearby partners. Then it moves on to step 2 from the old method.

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Nicos Christofides' algorithm starts by selecting the shortest possible tree (a network with no closed loops) connecting the cities. Then it adds connections until every city has an even number of connections, producing a closed route. In this case, that requires connecting the two farthest cities, resulting in a long round trip.

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An algorithm devised in 2020 broke a 44-year record for finding the best approximate solutions to the traveling salesperson problem, which seeks the shortest round-trip route for any set of cities.

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But the new solution was only able to subtract 0.2 billionth of a trillionth of a trillionth of a percent.

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In 2020, a team of computer scientists finally found a better way to find approximate solutions to the traveling salesperson problem.

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At the time, computer scientists expected that someone would soon improve on Christofides’ simple algorithm and come closer to the true solution. But the anticipated progress did not arrive.

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In 1976, the mathematician Nicos Christofides came up with an algorithm that efficiently finds approximate solutions — round trips through each city that are at most 50% longer than the ideal round trip.

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Most computer scientists believe that there is no algorithm that can efficiently find the best solutions for all possible combinations of cities. But, so far, nobody knows how to prove this hunch.

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The problem asks solvers to find the shortest round-trip route that passes through every city on a map exactly once. All known methods for solving the problem are painfully slow on maps with many cities.

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The traveling salesperson problem is one of the most notorious, longstanding problems in theoretical computer science. 🧵

Why Computer Scientists Consult Oracles | Quanta Magazine Hypothetical devices that can quickly and accurately answer questions have become a powerful tool in computational complexity theory.

P = NP? It’s a question that computational complexity theorists have been asking for over 50 years. Oracles have helped them better understand what they’re working with.

Lie groups underlie some of the most fundamental laws of physics. www.quantamagazine.org/what-are-lie...

Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal | Quanta Magazine Three high schoolers and their mentor revisited a century-old theorem to prove that all knots can be found in a fractal called the Menger sponge.

A large part of doing mathematics involves struggling with the failure of a promising avenue of attack. “Math has no mercy.” —Malors Espinosa, mathematician

The famously intimidating field of metamathematics analyzes math proofs. For example: Why are some problems hard to solve, while others are straightforward? A recent proof shows that three distinct theorems are logically equivalent. www.quantamagazine.org/reverse-math...

What Are Lie Groups? | Quanta Magazine By combining the language of groups with that of geometry and linear algebra, Marius Sophus Lie created one of math’s most powerful tools.

Leila Sloman explains how Lie groups blend the language of group theory, geometry and linear algebra to help mathematicians understand important symmetries: www.quantamagazine.org/what-are-lie...

Hi there. When a page is at 200% zoom, our layout moves the date stamp to the bottom of the page. We are creating a ticket to change this so that the date stamp remains at the top.

An estimated 1 to 4% of people have aphantasia, a condition where they don’t experience mental imagery or a ‘mind’s eye.’ Neuroscience research into people with aphantasia is now revealing how… Aphantasia: Why Some People Can't 'See' Mental Images

How do you measure someone else’s inner reality?

A Cell So Minimal That It Challenges Definitions of Life | Quanta Magazine The newly described microbe represents a world of parasitic, intercellular biodiversity only beginning to be revealed by genome sequencing.

How can a cell that’s unable to process nutrients or grow on its own be alive?

At the deepest lake in America, scientists are trying to understand how climate change breaks the fundamental physics of deep freshwater systems.

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The Geometric Tool That Solved Einstein’s Relativity Problem | Quanta Magazine Tensors are used all over math and science to reveal hidden geometric truths. What are they?

Tensors are instrumental in physics, machine learning and even biology. Einstein once begged a friend to help him understand them, fearing he was going mad. Joseph Howlett explains how they work:

‘Reverse Mathematics’ Illuminates Why Hard Problems Are Hard | Quanta Magazine Researchers have used metamathematical techniques to show that certain theorems that look superficially distinct are in fact logically equivalent.

A recent paper shows that many seemingly distinct theorems in computational complexity are actually equivalent. @benbenbrubaker.bsky.social reports:
www.quantamagazine.org/reverse-math...

What Is Entropy? A Measure of Just How Little We Really Know. | Quanta Magazine Exactly 200 years ago, a French engineer introduced an idea that would quantify the universe’s inexorable slide into decay. But entropy, as it’s currently understood, is less a fact about the world…

To truly wrap one’s head around entropy requires taking some deeply uncomfortable philosophical leaps. What is the purpose of science if physical qualities only make sense in relation to the observer?

A Cell So Minimal That It Challenges Definitions of Life | Quanta Magazine The newly described microbe represents a world of parasitic, intercellular biodiversity only beginning to be revealed by genome sequencing.

A newly discovered archaeal cell has a tiny genome and can’t metabolize biomolecules. It’s upending biologists’ definition of a living thing. “These types of organisms have been found before, but not as extreme as this,” said microbiologist Thijs Ettema.