In the ancient Egyptian rubbish heap of Oxyrhynchus, fragments of Euclid’s *Elements* were found on papyrus. Some of these salvaged scraps, such as one now at the University of Pennsylvania’s Penn Museum, were illustrated with diagrams expressing the influential work of the 4th-century BCE Greek mathematician. Centuries later, an Irish professor of mathematics named Oliver Byrne would transform Euclid’s mathematical proofs and propositions into one of the first multicolor printed books. His 1847 publication — *The First Six Books of The Elements of Euclid in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners* — used original illustrations to present the 300 BCE geometry treatise with diagrams … Read the rest

# Tag Archives: mathematics

# The (Imaginary) Numbers at the Edge of Reality

Have you ever sat in a math classroom and wondered, “When will I ever use this?” You might have asked yourself this question when you first encountered “imaginary” numbers, and with good reason: What could be less practical than a number described as imaginary?

But imaginary numbers, and the complex numbers they help define, turn out to be incredibly useful. They have a far-reaching impact in physics, engineering, number theory and geometry. And they are the first step into a world of strange number systems, some of which are being proposed as models of the mysterious relationships underlying our physical world. Let’s take a look at how these unfamiliar numbers are rooted in the numbers we know, but at the same time, are unlike anything … Read the rest

# A Proof About Where Symmetries Can’t Exist

Success for Robert Zimmer is defined differently these days. As the president of the University of Chicago since 2006, he’s made headlines for landing nine-figure financial gifts and writing op-eds in defense of campus free speech. But before Zimmer was a university president he was a mathematician. And long after he left serious research behind, the research plan he set in motion is finally paying off.

A year ago a trio of mathematicians solved what’s called Zimmer’s conjecture, which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries. Their proof stands as one of the biggest mathematical achievements in recent years. It settles a question that emerged for Zimmer during a period of intense intellectual activity in the … Read the rest

# What Makes the Hardest Equations in Physics So Difficult?

Physics contains equations that describe everything from the stretching of space-time to the flitter of photons. Yet only one set of equations is considered so mathematically challenging that it’s been chosen as one of seven “Millennium Prize Problems” endowed by the Clay Mathematics Institute with a $1 million reward: the Navier-Stokes equations, which describe how fluids flow.

Last month I wrote a story about an important new result related to those equations. If anything, the new work suggests that progress on the Millennium Prize will be even harder than expected. Why are these equations, which describe familiar phenomena such as water flowing through a hose, so much harder to understand mathematically than, say, Einstein’s field equations, which involve stupefying objects like black holes?

The … Read the rest