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Quantized Academy

The Simple Math Behind the Mighty Roots of Unity

By Patrick Honner
September 23, 2021
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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.

Quantized Academy

How to Find Rational Points Like Your Job Depends on It

By Patrick Honner
July 22, 2021
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Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others.

3D illustration of Gardam's crystallographic group
group theory

Mathematician Disproves 80-Year-Old Algebra Conjecture

By Erica Klarreich
April 12, 2021
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Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

Illustration of mathematicians descending through a cavern whose walls are lined with polynomials.
polynomials

Mathematicians Resurrect Hilbert’s 13th Problem

By Stephen Ornes
January 14, 2021
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Long considered solved, David Hilbert’s question about seventh-degree polynomials is leading researchers to a new web of mathematical connections.

An abstract illustration showing broken tools, cubes, numbers and other abstract representations of impossible math
Quantized Columns

When Math Gets Impossibly Hard

By David S. Richeson
September 14, 2020
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Mathematicians have long grappled with the reality that some problems just don’t have solutions.

Diagram showing a colorful visualization of the simple Lie group
Abstractions blog

The ‘Useless’ Perspective That Transformed Mathematics

By Kevin Hartnett
June 9, 2020
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Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Multimedia

The Map of Mathematics

By Kevin Hartnett
February 13, 2020
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Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

Animation showing two sets of tangrams cycling between identical squares and different shapes.
geometry

Mathematicians Cut Apart Shapes to Find Pieces of Equations

By Kevin Hartnett
October 31, 2019
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New work on the problem of “scissors congruence” explains when it’s possible to slice up one shape and reassemble it as another.

Quantized Academy

The (Imaginary) Numbers at the Edge of Reality

By Patrick Honner
October 25, 2018
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Odd enough to potentially model the strangeness of the physical world, complex numbers with “imaginary” components are rooted in the familiar.


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