How do you prove something is true? For mathematicians, the answer is simple: Start with some basic assumptions and proceed, step by step, to the conclusion. QED, proof complete. If there’s a mistake anywhere, an expert who reads the proof carefully should be able to spot it. Otherwise, the proof must be valid. Mathematicians have been following this...

Data is almost always incomplete. Patients drop out of clinical trials and survey respondents skip questions; schools fail to report scores, and governments ignore elements of their economies. When data goes missing, standard statistical tools, like taking averages, are no longer useful. “We cannot calculate with missing data, just as we can’t divide...

Imagine, for a moment, that you’re a honeybee. In many ways, your world is small. Your four delicate wings, each less than a centimeter long, transport your half-gram body through looming landscapes full of giant animals and plants. In other ways, your world is expansive, even grand. Your five eyes see colors and patterns that humans can’t, and your...

If instruments do someday detect evidence of life beyond Earth, whether it’s in this solar system or in the farther reaches of space, astrobiologists want to be ready. One of the best ways to learn how alien life might function can be to study the organisms called extremophiles, which live in incredibly challenging environments on or in the Earth. In...

In the fall of 2022, a Princeton University graduate student named Carolina Figueiredo stumbled onto a massive coincidence. She calculated that collisions involving three different types of subatomic particles would all produce the same wreckage. It was like laying a grid over maps of London, Tokyo and New York and seeing that all three cities had train...

Anyone who has seen the classic rabbit-duck optical illusion knows the magic and confusion of duality. One might see a duck facing to the left with its bill hanging slightly open or a rabbit with its nose pointing to the right and its ears extending behind it. Both perspectives are valid. In physics, duality is when a single physical system can be described...