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Banach-Tarski and the Paradox of Infinite Cloning
...a mathematical theorem involving infinity that makes it possible, at least in principle, to turn one apple into two. That argument is called the Banach-Tarski paradox, after the mathematicians Stefan...
How Tadayuki Watanabe Disproved a Major Conjecture About Spheres
...symmetries. The Mystery Continues Watanabe’s result closed one front of mathematical research. It also opened another. After his work, mathematicians knew the Smale conjecture was false in all dimensions four...
Machine Learning Gets a Quantum Speedup
...than a classical computer. And getting quantum computers to outlearn traditional machines means finding AI problems that boil down to mathematical operations congruous with quantum physics. “Rather than forcefully trying...
Computer Scientists Prove Why Bigger Neural Networks Do Better
...performing humanlike tasks. As they’ve gotten bigger, they have come to grasp more. This has been a surprise to onlookers. Fundamental mathematical results had suggested that networks should only need...
Father-Son Team Solves Geometry Problem With Infinite Folds
...of Technology. Erik Demaine learned basic math and the visual arts from his father, but eventually taught Martin advanced math and computer science. “Now we’re both artists and both mathematicians/computer...
New Proof Illuminates the Hidden Structure of Common Equations
...help solve lots of other equations in our mathematical system, like x2 – 20 = 0. (Note that here, our mathematical system is limited to polynomials and rational numbers.) But...
How Complex Is a Knot? New Proof Reveals Ranking System That Works.
“Ribbon concordance” will let mathematicians compare knots by linking them across four-dimensional space. Back in 1981, Cameron Gordon introduced a new way to relate two knots — mathematical constructs modeled...
Surfaces Beyond Imagination Are Discovered After Decades-Long Search
Unimaginable Surfaces Discovered After Decades-Long Search Using ideas borrowed from graph theory, two mathematicians have shown that extremely complex surfaces are easy to traverse. Last July, two mathematicians from Durham...
How to Weigh Truth With a Balance Scale
How to Weigh Truth With a Balance Scale and Math In recreational mathematics, the balance scale is an endless source of puzzles that require precise and elaborate logic and teach...