Theoretical computer scientists deal with complicated ideas. But whenever possible, they’d prefer to work with simpler ones. A 2009 tool known as the regularity lemma gives them a great way to do this. It effectively lets them break a given computational problem or function into simpler pieces. For computational complexity theorists, who study the relative hardness of different problems...
The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction. The first proof of this...
The universe seems like it should be unfathomably complex. How then is science able to crack fundamental questions about nature and life? Scientists and philosophers alike have often commented on the “unreasonable” success of mathematics at describing the universe. That success has helped science probe some profound mysteries — but as the physicist Nigel Goldenfeld points out, it also helps that...
Each summer, like clockwork, millions of beech trees throughout Europe sync up, tuning their reproductive physiology to one another. Within a matter of days, the trees produce all the seeds they’ll make for the year, then release their fruit onto the forest floor to create a new generation and feed the surrounding ecosystem. It’s a reproductive spectacle known as masting that’s common to many tree...
We learn in grade school that water freezes at zero degrees Celsius, but that’s seldom true. In clouds, scientists have found supercooled water droplets as chilly as minus 40 C, and in a lab in 2014, they cooled water to a staggering minus 46 C before it froze. You can supercool water at home: Throw a bottle of distilled water in your freezer, and it’s unlikely to crystallize until you shake it.
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