John D. Cook

A few years ago I wrote about a curious result: a/bis usually greater thanc/dwhena+dis greater thanb+c. And usually" means with probability 11/12. As stated in the post, you can make this statement rigorous by choosing integers up to sizeN at random and taking the limit asN goes to infinity. We can also choosea,b,c, andd uniformly [...]The post Fraction comparison trick revisited first appeared on John D. Cook.

Spherical geometry is very tangible because we live on a sphere. Projective geometry seems much more exotic, and yet they're closely related. To show how they're connected, let's go back to the origins of non-Euclidean geometry. Euclidean geometry Here are Euclid's five postulates, statements about what geometric objects can be constructred. To draw astraight linefrom [...]The post Euclidean, Spherical, and Projective Geometry first appeared on John D. Cook.

This post will show how a graphic programmer's hack could lead to a rigorous definition of the projective plane. As mentioned in the previous post, we start by adding some points at infinity" to Euclidean plane to take care of exceptional situations. For example, instead of saying most pairs of lines intersect at a single [...]The post How a programming might reinvent the projective plane first appeared on John D. Cook.

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 without any motivation. Much later I came to realize that they're natural, almost [...]The post Making peace with the projective plane first appeared on John D. Cook.

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 little awkward in the plane because [...]The post A unified view of poles and polars first appeared on John D. Cook.

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 ago Axiom announced it had formalized the proof [...]The post The Unbalanced Group Theorem first appeared on John D. Cook.

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 setting each variable to true [...]The post The Exponential Time Hypothesis is False first appeared on John D. Cook.

The previous three post have been part of a series on geometric duality. The traditional but somewhat mysterious terminology is thepole of a line and thepolar of a point. As explained here, the terminology was first coined in the context of spherical geometry where the term pole" makes sense. It was then extended to the [...]The post Inversion in an ellipse first appeared on John D. Cook.

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 (nee Twitter) I have a several X accounts, one personal account (@JohnDCook) and several specialized accounts (@CompSciFact, @AlgebraFact, @DataSciFact, etc.).I announce new blog posts on the most relevant account. RSS [...]The post Where else to find me first appeared on John D. Cook.

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 exactlyis a polar triangle and why is it called that? Poles and polars on a sphere Let's start with a sphereS and a [...]The post Poles and polars first appeared on John D. Cook.

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 geometric construction that starts with a [...]The post Computing polar triangles with linear algebra first appeared on John D. Cook.

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 cosinesfor sides. The law of cosines for angles is typographically very similar, just inverting the case of [...]The post Duality in spherical trigonometry first appeared on John D. Cook.

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 blocks I searched on what that [...]The post Working around a strange Python problem first appeared on John D. Cook.

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 details of how this [...]The post How the eMudrha certificate debacle might have happened first appeared on John D. Cook.

An elliptic curve E over a finite field F consists of the set of points (x, y) satisfying the equation y^2 = x^3 + ax + b withx andy inF. Typically in cryptography the fieldF is the integers modp wherep is a very large prime number. The number of points onE is roughly the number [...]The post The number of points on an elliptic curve first appeared on John D. Cook.

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 the NSA [...]The post Backdoors, Incompetence, and Elliptic Curves first appeared on John D. Cook.

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 post Deliberate imperfection first appeared on John D. Cook.

SplittingN! intoN factors is trivial. It'sdefined as the product ofN integers, the integers 1 throughN. Can you still splitN! intoN 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 can split 4! into factors >= 2: 4! [...]The post Refactoring factorials first appeared on John D. Cook.

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 together chemical element abbreviations ignoring case? Obviously [...]The post A frivolous problem with a deeper lesson first appeared on John D. Cook.

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 random number generator. Two days ago researchers at Anthropic [...]The post Cloud of suspicion lifted from NIST curves first appeared on John D. Cook.

I mentioned in the previous post that I watched 2010: The Year We Make Contact last night and got nerd-sniped by the mention of Lagrange points at the beginning. I got nerd-sniped again in a scene describing a password. The instructions are Put in nine nines, take the square root and press the integer. I [...]The post Movie password first appeared on John D. Cook.

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 nerd sniped and [...]The post Monoliths and Lagrange Points first appeared on John D. Cook.

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 1/20 1/5 ... I asked Claude to write a [...]The post LaTeX package for Pascal-like triangles first appeared on John D. Cook.

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 below. Then the areas of the rectangles with the same [...]The post A generalization of the Pythagorean theorem first appeared on John D. Cook.

What's a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel? Poe's story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function [...]The post Poe, Bessel, and Drones first appeared on John D. Cook.

Pascal's triangle is the most famous triangular arrangement of numbers. I've written about other numerical triangles, such as Bell's triangle and the trinomial triangle. There's another well-known triangle that somehow I never heard of until recently: the harmonic triangle, first explored by Leibniz around 1700. To create the harmonic triangle, start with the harmonic series [...]The post Harmonic triangle first appeared on John D. Cook.

A few days ago I wrote about topological models of modal logic. If you find general topology more familiar than modal logic, this correspondence could be helpful. If you think the motivation for general topology is evident but the motivation for modal logic is suspect, the correspondence between the two fields might relieve some of [...]The post Categorical models of logics first appeared on John D. Cook.

If you've followed this blog for a while, you may know that I do a lot of work with data privacy and that I have an interest in modal logic. Recently these two worlds collided: I became aware of work that uses modal logic to reason about data privacy. The story begins with Helen Nissenbaum's [...]The post Privacy policies and modal logic first appeared on John D. Cook.

The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function (s) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to . The Quasi Riemann Hypothesis (QRH) says that there exists a constant < 1 [...]The post Consequences of progress toward the Riemann Hypothesis first appeared on John D. Cook.

The Fast Fourier Transform (FFT) algorithm can compute the discrete Fourier transform of a sequence of lengthn in time O(n logn). OpenAI recently posted a paper saying there is an algorithm that could compute the discrete Fourier transform in O(n (logn)1 - ) time for = 10-13. This result is amazing. It seemed that [...]The post Faster Fourier Transform first appeared on John D. Cook.

For a real numberx, the irrationality index (x) is a way of measuring how wellx can be approximated by rational numbers. Ifx is rational, (x) = 1. Ifx is irrational, (x) >= 2. OpenAI recently published a proof that () = 2. Almost all real numbers have irrationality exponent 2, so the new result says [...]The post Irrationality exponent of first appeared on John D. Cook.

I was skimming through David Well's book The Penguin Dictionary of Curious and Interesting Geometry and was surprised by this entry: Lissajous figures orBowditch curves First discussed by Nathaniel Bowditch in 1815, and later by Jules Antoine Lissajous in 1857 ... I wrote a couple posts about Lissajous curves years ago, and recently I wrote [...]The post Lissajous and Bowditch first appeared on John D. Cook.

Godel's incompleteness theorem illustrated the need to distinguish between what is true and what is provable. There are true statements that cannot be proven. Let p denote the assertion thatp is provable in Peano arithmetic. The logic with this interpretation for the operator is the Godel-Lob logic, also called provability logic. This is a [...]The post A topological model for provability logic first appeared on John D. Cook.

An earlier post presented an elegant plane geometry theorem discovered by the 19th century school teacher Auguste Miquel. This post presents his pentagon theorem. Start with a pentagon. It may be irregular, but it needs to be convex. Extend each of the sides of the pentagon to form a star, then draw give circles, one [...]The post Miquel's pentagon theorem first appeared on John D. Cook.

The previous post discussed a superficial connection between modal logic and topology, that both use the terms regular andnormal to indicate added sets of axioms. McKinsey and Tarski developed a deeper connection between modal logic and topology that we'll discuss here. Starting with a topological spaceX and a proposition p, define [[p]] as the set [...]The post Topological models of modal logic first appeared on John D. Cook.

You can't say much about modal logic in general. You have to be more specific to get anywhere. You have to choose some axioms. Ideally the axioms you need for your application correspond to a named set of axioms that has been studied before. The situation is similar in point-set topology. You can't say very [...]The post Modal logic and topology first appeared on John D. Cook.

Euclidean geometry dates back at least to Euclid (circa 300 BC), and so you might think it's been pretty well picked over by now. And yet people still occasionally discover new plane geometry theorems. Some of these new theorems are complicated, asking question that the ancients would not have asked. But once in a while [...]The post Miquel's pivot theorem first appeared on John D. Cook.

Despite the predictions that no one would ever put build data centers in space, Google is starting on Thursday. Google's prototype satellite will be one of 130 payloads on SpaceX's Transporter 18 mission on October 1. The server will follow a dawn-dusk orbit, a special case of a sun-synchronous orbit (SSO), following the terminator line [...]The post Servers in dawn-dusk orbit first appeared on John D. Cook.

In the novel Carry On, Mr. Bowditch, a sailor asked Nathaniel Bowditch to teach him how to do navigational calculations, but the man only knows how to add and subtract by counting on his fingers. He had not heard of multiplication. Bowditch is surprised, but realizes if he made tables of logs of trig functions, [...]The post Navigation with only addition, subtraction, and tables first appeared on John D. Cook.

A couple days ago a friend told me about the book Carry On, Mr. Bowditch, a fictional account of the life of Nathaniel Bowditch (1773-1838). I've been listening to the book on Audible, and apparently it's only lightly fictionalized. Bowditch was a self-educated mathematician and astronomer, best known for his book The American Practical Navigator, [...]The post Nathaniel Bowditch first appeared on John D. Cook.

Suppose you want to solve a triangle. You know two sides and the angle between them. Then you can solve for the third side using the law of cosines. Now suppose you want to solve abig triangle, a triangle on the surface of the earth so large that the curvature of the earth matters. You [...]The post Haversine law first appeared on John D. Cook.

Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve. Suppose you're collecting data on the left side of the curve. If there's even a small amount of error in your data, you won't be able to predict the [...]The post Why fitting a logistic is nearly impossible from early data first appeared on John D. Cook.

There's a common saying in discussion of fractals that the length of a coastline depends on how small a device you use to measure it. I thought this was a hypothetical, say as applied to the steps in the construction of the Koch snowflake. But the saying has its roots in actually surveying. Lewis Fry [...]The post Empirical fractal first appeared on John D. Cook.

I recently bought a copy of Los Alamos Rolodex, a book displaying business cards from Los Alamos Nation Labs from 1967 to 1978. You can find some examples of the cards here. One of the cards in the book is for Eugene Frank, President of B & F Instruments. His card lists his phone number [...]The post Phone words first appeared on John D. Cook.

I've written three posts on cosine similarity lately. The first looked at interpreting cosine similarity. The second looked at an approximation related to the first. The third looked at how ranking according to cosine similarity works better than cosine similarity itself. Normalized word vectors are points on a high dimensional sphere, and geometry in high [...]The post Converting between cosine similarity and concentration ratio first appeared on John D. Cook.

Yesterday I wrote about the canonical example of how vector embeddings of words add: king" - man" + woman" queen" This should be interpreted as saying that the word vector forking, minus the word vector forman, plus the word vector forwoman, is in some sense close to the word vector forqueen. This post will [...]The post Coffee + milk latte first appeared on John D. Cook.

The product of four consecutive Fibonacci numbers equals the product of two consecutive integers. For example, 3 * 5 * 8 * 13 = 39 * 40. I ran across this theorem in a note [1] that says The product of any four consecutive Fibonacci numbers is twice a triangular number." Since triangular numbers have [...]The post Fibonacci product first appeared on John D. Cook.

The previous post looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly chosen vectors are likely nearly perpendicular, and so relatively large angles, such as 50, indicate very closely related words. Another way to look at this, as explained in the previous post, is that in [...]The post Simple approximation for spherical cap area first appeared on John D. Cook.

Machine learning represents words as vectors and measures the similarity of words by the angles between the vectors. For vectorsxandy, where is the angle between the vectors, and so This is the cosine similarity between the words represented by x and y. Small angles have large cosines, and so words with larger cosine similarities [...]The post What counts as a large cosine similarity? first appeared on John D. Cook.

Suppose I give you ann-digit number and ask you what it represents. This seems impossible, and in theory itis impossible. But in practice it's often possible. Apps on a phone may automatically interpret a 10-digit number as a phone number or a 16-digit number as a package tracking number. And very often these interpretations are [...]The post Guessing the meaning of a number first appeared on John D. Cook.