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# Quantized Columns

## Latest Articles

### The Uselessness of Useful Knowledge

Today’s powerful but little-understood artificial intelligence breakthroughs echo past examples of unexpected scientific progress.

### The Simple Math Behind the Mighty Roots of Unity

Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.

### The Journey to Define Dimension

The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.

### How Steven Weinberg Transformed Physics and Physicists

When Steven Weinberg died last month, the world lost one of its most profound thinkers.

### How to Find Rational Points Like Your Job Depends on It

Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others.

### How ‘Long COVID’ Keeps Us Sick

Other diseases with long-term symptoms can help us understand how COVID can affect us long after the virus itself is gone.

### What Makes Quantum Computing So Hard to Explain?

To understand what quantum computers can do — and what they can’t — avoid falling for overly simple explanations.

### Can Machines Control Our Brains?

Advances in brain-computer interface technology are impressive, but we’re not close to anything resembling mind control.

### How to Solve Equations That Are Stubborn as a Goat

Math teachers have stymied students for hundreds of years by sticking goats in strangely shaped fields. Learn why one grazing goat problem has stumped mathematicians for more than a century.