The previous three post have been part of a series on geometric duality. The traditional but somewhat mysterious terminology is the pole of a line and the polar of a point. As explained here, the terminology was first coined in…
The previous two posts have been about polar triangles. The first used a polar triangle to derive a theorem, and the second showed how to compute polars using linear algebra. But what exactly is a polar triangle and why is…
The previous post defined the polar triangle of a spherical triangle in terms of its sides and angles, but it doesn’t construct the triangle. It only specifies one of a family of triangles located anywhere on the sphere. How do…
I’ve written before about the law of cosines on a sphere: But that’s just one of two laws of cosines on a sphere. The equation above is the law of cosines for sides. The law of cosines for angles is…
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao 24 October, 2026 in advertising | Tags: Abhinav Nellore, Johannes Schmitt, Rachel Ward, Soledad Villar | by Terence Tao [This is a guest post by Abhinav…
An elliptic curve E over a finite field F consists of the set of points (x, y) satisfying the equation y² = x³ + ax + b with x and y in F. Typically in cryptography the field F is the integers mod p where p…
The NIST elliptic curves have been in the news twice lately. A few days ago Anthropic posted a paper on the NIST curve seeds that I wrote about here. Now the curves are in the news again, this time over…
Splitting N! into N factors is trivial. It’s defined as the product of N integers, the integers 1 through N. Can you still split N! into N factors if you put a minimum size requirement on the factors? You can’t split…
Many words can be spelled by stringing together chemical symbols. For example, sarcasm can be written S Ar Ca Sm using the symbols for sulfur, argon, calcium, and samarium. How would you write a program to determine whether a word…
NIST published a set of elliptic curves for cryptography in 2000. The seeds for the curves were supplied by the NSA, and cryptographers suspected the agency may have installed backdoors in the curves. And not without reason: the NSA did insert…
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