John D. Cook

In a high school algebra class, you learn how to solve polynomial equations in one variable, and systems of linear equations. You might reasonably ask “So when do we combine these and learn to solve systems of polynomial equations?” The answer would be “Maybe years from now, but most likely never.” There are systematic ways to […]

Different colors of noise are named by analogy with colors of light. Pink noise is between white noise and red noise. White noise has equal power at all frequencies, just as white light is a combination of all the frequencies of the visible spectrum. The components of red noise are weighted toward low frequencies, just […]

Yesterday a friend told me about a software project whose owners said “We’re going to do this the right way.” I told him I have two opposite reactions when I hear that: Ooh, that sounds like fun! Run away! I’ve been on several projects where the sponsors have identified some aspect of the status quo […]

This is the third, and last, of a series of posts on Benford’s law, this time looking at a famous open problem in computer science, the 3n + 1 problem, also known as the Collatz conjecture. Start with a positive integer n. Compute 3n + 1 and divide by 2 repeatedly until you get an odd […]

Introduction Samples from a Cauchy distribution nearly follow Benford’s law. I’ll demonstrate this below. The more data you see, the more confident you should be of this. But with a typical statistical approach, crudely applied NHST (null hypothesis significance testing), the more data you see, the less convinced you are. This post assumes you’ve read the […]

Introduction to Benford’s law In 1881, Simon Newcomb noticed that the edges of the first pages in a book of logarithms were dirty while the edges of the later pages were clean. From this he concluded that people were far more likely to look up the logarithms of numbers with leading digit 1 than of […]

The golden angle is related to the golden ratio, but it is not as well known. And the relationship is not quite what you might think at first. The golden ratio φ is (1 + √5)/2. A golden rectangle is one in which the ratio of the longer side to the shorter side is φ. […]

There are many ways to divide people into four personality types, from the classical—sanguine, choleric, melancholic, and phlegmatic—to contemporary systems such as the DISC profile. The Myers-Briggs system divides people into sixteen personality types. I just recently ran across the “enneagram,” an ancient system for dividing people into nine categories. There’s one thing advocates of […]

Last night I was driving toward the Denver airport and the airport reminded me of the cover of Abramowitz and Stegun’s Handbook of Mathematical Functions. Here’s the airport: And here’s the book cover: I’ve written about the image on book cover before. Someone asked me what function it graphed and I decided it was probably […]

As much as we admire simplicity and strive for simplicity, something in us isn’t happy when we achieve it. Sometimes we’re disappointed with a simple solution because, although we don’t realize it yet, we didn’t properly frame the problem it solves. I’ve been in numerous conversations where someone says effectively, “I understand that 2+3 = […]

A Menger sponge is created by starting with a cube a recursively removing chunks of it. Draw a 3×3 grid on one face of the cube, then remove the middle square, all the way through the cube. Then do the same for each of the eight remaining squares. Repeat this process over and over, and do it […]

The harmonic numbers are defined by Harmonic numbers are sort of a discrete analog of logarithms since As n goes to infinity, the difference between Hn and log n is Euler’s constant γ = 0.57721… [1] How would you compute Hn? For small n, simply use the definition. But if n is very large, there’s a way […]

I’ve written quite a few pages that are separate from the timeline of the blog. These are a little hidden, not because I want to hide them, but because you can’t make everything equally easy to find. These notes cover a variety of topics: Math diagrams Numerical computing Probability and approximations Differential equations Python Regular expressions […]

I’ve updated the icons for my Twitter accounts.

The study of the planet Mercury provides two examples of the bandwagon effect. In her new book Worlds Fantastic, Worlds Familiar, planetary astronomer Bonnie Buratti writes The study of Mercury … illustrates one of the most confounding bugaboos of the scientific method: the bandwagon effect. Scientists are only human, and they impose their own prejudices […]

This post serves two purposes. It will empirically explore a question in number theory and demonstrate quantile-quantile (q-q) plots. It will shed light on a question raised in the previous post. And if you’re not familiar with q-q plots, it will serve as an introduction to such plots. The previous post said that for almost all x > […]

A few days ago I wrote about how powers of the golden ratio are nearly integers but powers of π are not. This post is similar but takes a little different perspective. Instead of looking at how close powers are to the nearest integers, we’ll look at how close they are to their floor, the largest […]

One of the case studies in Michael Beirut’s book How to is the graphic design for the planned community Celebration, Florida. The logo for the town’s golf course is an illustration of the bike shed principle. C. Northcote Parkinson observed that it is easier for a committee to approve a nuclear power plant than a bicycle […]

Last week I wrote a blog post showing that powers of the golden ratio are nearly integers. Specifically, the distance from φn to the nearest integer decreases exponentially as n increases. Several people pointed out that the golden constant is a Pisot number, the general class of numbers whose powers are exponentially close to integers. […]

When I first saw ring theory, my impression was that there were dozens of kinds of rings with dozens of special relations between them—more than I could keep up with. In reality, there just a few basic kinds of rings, and the relations between them are simple. Here’s a diagram that shows the basic kinds of […]

In his paper Mindless statistics, Gerd Gigerenzer uses a Freudian analogy to describe the mental conflict researchers experience over statistical hypothesis testing. He says that the “statistical ritual” of NHST (null hypothesis significance testing) “is a form of conflict resolution, like compulsive hand washing.” In Gigerenzer’s analogy, the id represents Bayesian analysis. Deep down, a […]

This morning I was reading Terry Tao’s overview of the work of Yves Meyer and ran across this line: The powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers: for instance, φ11 = 199.005… is unusually close to 199. I’d never heard that before, so I wrote a little code […]

In a recent interview, Tyler Cowen discusses complacency, (neruo-)diversity, etc. Let me give you a time machine and send you back to Vincent van Gogh, and you have some antidepressants to make him better. What actually would you do, should you do, could you do? We really don’t know. Maybe he would have had a […]

Radim Řehůřek of Rare Technologies just posted an interview he and I did a few days ago. We mostly talk about consulting: getting started, finding work, good and bad leads, etc.

The canonical examples of natural and unnatural transformations come from linear algebra, namely the relation between a vector space and its first and second duals. We will look briefly at the finite dimensional case, then concentrate on the infinite dimensional case. Two finite-dimensional vector spaces over the same field are isomorphic if and only if […]

The ladder of abstractions in category theory starts with categories, then functors, then natural transformations. Unfortunately, natural transformations don’t seem very natural when you first see the definition. This is ironic since the original motivation for developing category theory was to formalize the intuitive notion of a transformation being “natural.” Historically, functors were defined in order […]

Eight short, accessible blog posts based on the work of the intriguing mathematician Srinivasa Ramanujan: Ramanujan series for computing π Ramanujan π approximation Ramanujan’s nested radical Ramanujan approximation for circumference of an ellipse Ramanujan’s most beautiful identity Ramanujan’s factorial approximation Sums of fourth powers Algorithm used for world record pi calculations

Larry Wall, creator of the Perl programming language, created a custom degree plan in college, an interdisciplinary course of study in natural and artificial languages, i.e. linguistics and programming languages. Many of the features of Perl were designed as an attempt to apply natural language principles to the design of an artificial language. I’ve been […]

A blog post by Evelyn Lamb yesterday introduced Thomas Baruchel’s web site by of images from complex analysis. I wondered what a collage of these images would look like, so I used the ImageQuilts software by Edward Tufte and Adam Schwartz to create the image below. Related: Applied complex analysis

In my previous post, I discussed how number theory and topology relate to other areas of math. Part of that was to show a couple diagrams from Jean Dieudonné’s book Panorama of Pure Mathematics, as seen by N. Bourbaki. That book has only small star-shaped diagrams considering one area of math at a time. I’ve created a […]

Areas of math all draw on and contribute to each other. But there’s a sort of trade imbalance between areas. Some, like analytic number theory, are net importers. Others, like topology, are net exporters. Analytic number theory uses the tools of analysis, especially complex analysis, to prove theorems about integers. The first time you see […]

Yesterday I ran across Askar Safin’s blog post The Collapse of the UNIX Philosophy. Two quotes from the post stood out. One was from Rob Pike about the Unix ideal of little tools that each do one job: Those days are dead and gone and the eulogy was delivered by Perl. The other was a […]

The trapezoid rule is the most obvious numerical integration technique. It comes directly from the definition of a definite integral, just a Riemann sum. It’s a very crude technique in general; you can get much more accuracy with the same number of function evaluations by using a more sophisticated method. But for smooth periodic functions, […]

I’d never given any thought to the order of polar coordinates until yesterday. They’re written (r, θ). That’s just how it is. But a friend pointed out two reasons why this bothers him. First, r is typically a function of θ, just as y is typically a function of x. But in rectangular coordinates, the […]

Someone mailed me this afternoon asking if I knew what function was graphed on the cover of Abramowitz and Stegun’s famous Handbook of Mathematical Functions. Here’s a close-up of the graph from a photo of my copy of A&S. This was my reply. It looks like a complex function of a complex variable. I assume […]

Fourier series express functions as a sum of sines and cosines of different frequencies. Bessel series are analogous, expressing functions as a sum of Bessel functions of different orders. Fourier series arise naturally when working in rectangular coordinates. Bessel series arise naturally when working in polar coordinates. The Fourier series for a constant is trivial. […]

The web is all abuzz about how SHA1 is “broken”, “a failure,” “obsolete”, etc. It is supposed to be computationally impractical to create two documents that have the same secure hash code, and yet Google has demonstrated that they have done just that for the SHA1 algorithm. I’d like to make two simple observations to […]

In one section of his book The Great Good Thing, novelist Andrew Klavan describes how he bluffed his way through high school and college, not reading anything he was assigned. He doesn’t say what he majored in, but apparently he got an English degree without reading a book. He only tells of one occasion where […]

I posted a couple things on Twitter today about micro-resumés. First, here’s how I’d summarize my work in a tweet. What I’ve done: Math prof, programmer, statistician What I do now: Consulting #microresume — John D. Cook (@JohnDCook) February 21, 2017 (The formatting is a little off above. It’s leaving out a couple line breaks […]

You can associate a matrix with a graph and find the eigenvalues of that matrix. This gives you a spectrum associated with the graph. This spectrum tells you something about the graph analogous to the way the spectrum of a star’s light tells you something about the chemical composition of the start. So maybe it […]

In 1838, Rev. Henry Moseley discovered that a large number of mollusk shells and other shells can be described using three parameters: k, T, and D. First imagine a thin wire running through the coil of the shell. In cylindrical coordinates, this wire follows the parameterization r = ekθ z = Tt If T = 0 this is a […]

In his new book The Perfect Shape, Øyvind Hammer shows how to create a graph something like the poster for Alfred Hitchcock’s movie Vertigo. Hammer’s code uses a statistical language called Past that I’d never heard of. Here’s my interpretation of his code using Python. import matplotlib.pyplot as plt from numpy import arange, sin, cos, […]

As with the previous post, this post is a spinoff of a blog post by Brian Hayes. He considers the problem of determining whether a number n is a Fibonacci number and links to a paper by Gessel that gives a very simple solution: A positive integer n is a Fibonacci number if and only if either 5n2 – […]

The other day I ran across a blog post by Brian Hayes that linked to an article by David Cantrell on how to compute the inverse of the gamma function. Cantrell gives an approximation in terms of the Lambert W function. In this post we’ll write a little Python code to kick the tires on […]

Morse code was designed so that the most frequently used letters have the shortest codes. In general, code length increases as frequency decreases. How efficient is Morse code? We’ll compare letter frequencies based on Google’s research with the length of each code, and make the standard assumption that a dash is three times as long as a dot. […]

The following quote gives a good description of the value of abstract mathematics. The quote speaks specifically of “universal algebra,” but consistent with the spirit of the quote you could generalize it to other areas of mathematics, especially areas such as category theory. Universal algebra is the study of features common to familiar algebraic systems […]

There are a couple ways to measure how well a graph remains connected when nodes are removed. One ways is to consider nodes dropping out randomly. Another way, the one we look at here, assumes an adversary is trying to remove the most well-connected nodes. This approach was defined by Schneider et al [1]. It […]

In this post I interview GiveDirectly co-founder Paul Niehaus about charitable direct cash transfers and their empirical approach to charity. JC: Can you start off by telling us a little bit about Give Directly, and what you do? PN: GiveDirectly is the first nonprofit that lets individual donors like you and me send money directly to […]

Excerpt from the new book Big Data of Complex Networks: Big Data and data protection law provide for a number of mutual conflicts: from the perspective of Big Data analytics, a strict application of data protection law as we know it today would set an immediate end to most Big Data applications. From the perspective of […]

Suppose you have a linear dynamic system. That is, the function that predicts the next state from the current state to the next is linear. Suppose also that the states in your system are not known precisely but have some uncertainty modeled by a (multivariate) normal distribution. Then the uncertainty in the state at the […]