“If I have seen further than others, it is by standing upon the shoulders of giants.” This famous quote by Isaac Newton points to an axiom that lies at the heart of The Sciences — knowledge precedes ...

What if the deepest part of a black hole is not defined only by crushed matter and broken spacetime, but by the logic of prime numbers? That question has moved from mathematical curiosity to a serious ...

On Monday, I looked at the possibility of artificial intelligence through the perspective of history: the history of the field and of expert conjecture. That analysis revealed that, up until now, ...

Water acts familiarly at familiar temperatures. It freezes into ice at 32 degrees Fahrenheit, which corresponds to 0 degrees Celsius and 272 Kelvin. But "supercooled," water can remain liquid even at ...

Before it became the province of futurists, the word singularity had significance in both mathematics and the physical sciences. A mathematical singularity is a point at which a function is not “well ...

Envision energy trajectories that curve and cross each other. Now envision energy trajectories that are straight lines and do not cross. Now see an explosion of energy at the points where the curved ...

Watching this soap film stretch, rupture and reform in slow motion is mesmerising – but mathematicians see more than a trippy image. The point where the bubble breaks explosively is a mathematical ...