John D. Cook
Applied Mathematics Consulting
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Latest posts
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Making peace with the projective plane
In my newsletter I give an introduction to the latest few posts and occasionally give some additional commentary. In my latest newsletter I wrote Projective space and homogeneous coordinates seemed like arbitrary abstractions when I was a student because they were presented…
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A unified view of poles and polars
I wrote several posts on poles and polars a few days ago. First, I wrote about the dual of a spherical triangle, called the polar triangle. Then I wrote about poles and polars in plane geometry and briefly mentioned the extension to projective geometry. Poles and polars are a…
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The Unbalanced Group Theorem
The classification of finite simple groups (CFSG) has been called the biggest theorem in mathematics. It is big, not necessarily in terms of its importance, but in the sense that its proof developed over the course of decades and thousands of pages of journal articles. Two days…
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The Exponential Time Hypothesis is False
If you have an expression in n Boolean variables, how long does it take to find out whether some assignment of values to the variables makes the expression true? There is an obvious upper bound: you can find out in 2n steps by exhaustive search, trying all possible ways of…
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Inversion in an ellipse
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 context of spherical geometry where the term…
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Where else to find me
This blog is the main place I write, but I also use other platforms, either to write new content or to announce blog posts. X (née Twitter) I have a several X accounts, one personal account (@JohnDCook) and several specialized accounts (@CompSciFact, @AlgebraFact, @DataSciFact,…
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Poles and polars
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 it called that? Poles and polars on a sphere Let’s start…
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Computing polar triangles with linear algebra
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 we know that polar triangles exist? There is a…
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Duality in spherical trigonometry
I’ve written before about the law of cosines on a sphere: cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C). 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 typographically very similar,…
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Working around a strange Python problem
A Python script that worked a few days ago wouldn’t work today. I suspect it might have something to do with my updating everything I could think of in response to recent security breaches. I got an error message that said error: The script contains multiple PEP 723 metadata…
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How the eMudrha certificate debacle might have happened
You may have seen references to a password being the number 4. There are countless allusions to this online, but they don’t give the context. As I mentioned in an earlier post, eMudrha was using the numbers 4, 56, 115, 155, 156, 230, and 237 as secret keys. I haven’t seen…
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The number of points on an elliptic curve
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 is a very large prime number. The number of points on E is roughly the…
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Backdoors, Incompetence, and Elliptic Curves
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 broken certificates. My post a few days ago was about the possibility that…
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Deliberate imperfection
I was watching a video by John Higgins this morning about the poem in Ecclesiastes 3, one of the most famous passages in the Bible. There is a time for everything, and a season for every activity under the heavens: a time to be born and a time to die, a time to plant and […] The…
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Refactoring factorials
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 3! into three factors unless one of the factors is 1, but you…
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A frivolous problem with a deeper lesson
About 14% of English 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 can be written by stringing…
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Cloud of suspicion lifted from NIST curves
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 a backdoor in the Dual_EC_DRBG…
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Monoliths and Lagrange Points
This post began as a thread on X last night. I’ve edited the thread to turn it into a blog post, making some additions and corrections. I’m watching 2010: The Year We Make Contact. The movie opens saying the monolith is located at the Lagrange point between Jupiter and Io. I got…
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LaTeX package for Pascal-like triangles
When I wrote about the harmonic triangle yesterday, I displayed the triangle left-aligned using fixed-width text because that was easier than producing the format you usually see for Pascal’s triangle and related triangles. 1 1/2 1/2 1/3 1/6 1/3 1/4 1/12 1/12 1/4 1/5 1/20 1/30…
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A generalization of the Pythagorean theorem
G. D. Chakerian [1] discovered the following generalization of the Pythagorean theorem. Take any triangle and attach squares to each side. From each triangle vertex, extend a line perpendicular to the opposite side and extend the line through the square, as shown in the figure…