On April 10, 2019, the Event Horizon Telescope (EHT) collaboration released the first image of a black hole: a glowing orange ring surrounding a dark central shadow. The object was the supermassive black hole at the center of the galaxy Messier 87 (M87), located 55 million light-years from Earth. It…

By the late 19th century, mathematics had grown so vast that no single person could master all of it. The era of the universal mathematician, the scholar who could make original contributions to every branch of the discipline, was ending. The last person widely acknowledged to have achieved this was…

The history of physics can be told through a handful of books. Not textbooks or popular summaries, but the original publications in which physicists first presented the ideas that changed everything. These are the works that introduced new frameworks for understanding the universe, frameworks so powerful that they replaced everything…

Around 300 BCE, the Greek mathematician Euclid compiled the Elements, a systematic presentation of geometry built on a small number of definitions, common notions, and five postulates (axioms). The first four postulates are simple and self-evident: you can draw a straight line between any two points; you can extend a…

Every student of mathematics encounters, at some point, the statement that the square root of negative one is called i, and that numbers involving i are called “complex” or “imaginary.” The terminology is unfortunate. It suggests that these numbers are artificial, mysterious, or somehow less real than ordinary numbers. Mathematicians…

On July 20, 1969, the Apollo 11 Lunar Module Eagle was descending toward the Sea of Tranquility. Neil Armstrong and Buzz Aldrin were three minutes from becoming the first humans to land on the Moon. Mission control in Houston was monitoring every parameter. Everything was nominal. Then the alarms started.…

In the winter of 1961, the meteorologist Edward Lorenz at MIT was running a simple computer simulation of weather patterns. The model used twelve equations to simulate atmospheric convection, and Lorenz had been running it to study long-term weather behavior. One day, wanting to re-examine a particular sequence, he restarted…

The Philosophiæ Naturalis Principia Mathematica, published in 1687, is widely regarded as the most important scientific book ever written. It contains Newton’s laws of motion, his law of universal gravitation, his derivation of Kepler’s laws of planetary motion, his theory of tides, his calculation of the speed of sound, and…

In the year 2000, the Clay Mathematics Institute in Cambridge, Massachusetts, announced seven “Millennium Prize Problems,” the most important unsolved problems in mathematics. Each carried a prize of one million dollars. As of today, only one has been solved (the Poincaré Conjecture, by Grigori Perelman in 2003, who declined the…

Nikola Tesla’s completed inventions changed the world. The alternating current motor, the polyphase power system, the Tesla coil, and the rotating magnetic field are foundations of modern electrical technology. But Tesla’s ambitions extended far beyond what he managed to build. Throughout his career, he announced, designed, and sometimes partially constructed…