John D. Cook

A few days ago I wrote that circulant matrices all have the same eigenvectors. This post will show that it follows that circulant matrices commute with each other. Recall that a circulant matrix is a square matrix in which the rows are cyclic permutations of each other. If we number the rows from 0, then […]The post Circulant matrices commute first appeared on John D. Cook.

The previous post discussed an unusual algebraic structure on the real interval (-1, 1) inspired by (and applied to) special relativity. We defined an addition operator ⊕ by How might we extend this from the interval (-1, 1) to the unit disk in the complex plane? The definition won’t transfer over unmodified because it does […]The post Relativity, complex numbers, and gyrovectors first appeared on John D. Cook.

I usually think of an instructor as someone who unpacks things, such as unpacking the meaning of an obscure word or explaining a difficult concept. Last night I was trying to read some unbearably dry medical/legal material and thought about how an instructor might also pack things, wrapping dry material in some sort of story […]The post Packing versus unpacking first appeared on John D. Cook.

It’s well known that you cannot map a sphere onto the plane without distortion. You can’t map the entire sphere to the plane at all because a sphere and a plane are not topologically equivalent. But even if you want to map a relatively small portion of globe to paper, say France, with about 0.1% […]The post How faithful can a map be? first appeared on John D. Cook.

If a three-digit number looks like it might be prime, there’s about a 2 in 3 chance that it is. To be more precise about what it means for a number to “look like a prime,” let’s say that a number is obviously composite if it is divisible by 2, 3, 5, or 11. Then […]The post Recognizing three-digit primes first appeared on John D. Cook.

Suppose you randomly sample points from a unit cube. How far apart are pairs of points on average? My intuition would say that the expected distance either has a simple closed form or no closed form at all. To my surprise, the result is somewhere in between: a complicated closed form. Computing the expected value […]The post Expected distance between points in a cube first appeared on John D. Cook.

I was looking back at a post about the Soviet license plate game and was reminded of the amusing identity sin (n!)° = 0 for n ≥ 6. Would it be possible to find sin (n!)° in closed form for all positive integers n? For this post I’ll make an exception to my usual rule […]The post Sine of factorial degrees first appeared on John D. Cook.

Here’s a simple but surprising theorem from digital signal processing: linear, time-invariant (LTI) operators commute. The order in which you apply LTI operators does not matter. Linear in DSP means just you’d expect from seeing linear defined anywhere else: An operator L is linear if given any two signals x1 and x2, and any two […]The post LTI operators commute first appeared on John D. Cook.

This post presents a simple method of estimating monthly payments on a loan. According to [1] this is a traditional Persian method and still commonly used in Iran. A monthly payment amount is (principal + interest)/months but the total amount of interest over the course of a loan is complicated to compute. Initially you owe […]The post Approximate monthly loan payments first appeared on John D. Cook.

At the end of each month I write a newsletter highlighting the most popular posts of that month. When I looked back at my traffic stats to write this month’s newsletter I noticed that a post I wrote last year about how to memorize the ASCII table continues to be popular. This post is a […]The post How to memorize Unicode codepoints first appeared on John D. Cook.

Let φ be the golden ratio. The fractional parts of nφ bounce around in the unit interval in a sort of random way. Technically, the sequence is quasi-random. Quasi-random sequences are like random sequences but better in the sense that they explore a space more efficiently than random sequences. For this reason, Monte Carlo integration […]The post Golden integration first appeared on John D. Cook.

My years in graduate school instilled a Pavlovian response to PDEs: multiply by a test function and integrate by parts. This turns a differential equation into an integral equation [1]. I’ve been reading a book [2] on integral equations right now, and it includes several well-known techniques for turning certain kinds of integral equations into […]The post Moving between differential and integral equations first appeared on John D. Cook.

I was looking around in the Unicode block for miscellaneous symbols, U+2600, after I needed to look something up, and noticed there are four astrological symbols for angles: ⚹, ⚺, ⚻, and ⚼. These symbols are mysterious at first glance but all make sense in hindsight as I’ll explain below. Sextile The first symbol, ⚹, […]The post Symbols for angles first appeared on John D. Cook.

There is no polynomial with rational coefficients that evaluates to 0 at π. That is, π is a transcendental number, not an algebraic number. This post will prove this fact as a corollary of a more advanced theorem. There are proof that are more elementary and direct, but the proof given here is elegant. A […]The post Overpowered proof that π is transcendental first appeared on John D. Cook.

It is well-known that you can approximate a binomial distribution with a normal distribution. Of course there are a few provisos … It is also well-known that you can approximate a beta distribution with a normal distribution as well. This means you could directly approximate a binomial distribution with a beta distribution. This is a […]The post Beta approximation to binomial first appeared on John D. Cook.

Suppose you have a database of personally identifiable information (PII) and you want to allow someone else to query the data while protecting the privacy of the individuals represented by the data. There are two approaches: Deidentify, then query Query, then deidentify The first approach is to do whatever is necessary to deidentify the data—remove […]The post Query, then deidentify first appeared on John D. Cook.

“The only use I know for a confidence interval is to have confidence in it.” — L. J. Savage Can you have confidence in a confidence interval? In practice, yes. In theory, no. If you have a 95% confidence interval for a parameter θ, can you be 95% sure that θ is in that interval? […]The post Can you have confidence in a confidence interval? first appeared on John D. Cook.

Linear systems can show sensitive dependence on initial conditions, but they cannot be chaotic. Only nonlinear systems can be chaotic. George Datseris and Ulrich Parlitz explain this well in their book Nonlinear Dynamics: … Sensitive dependence is not sufficient for a definition of chaos. … the state space is first stretched and then folded within […]The post The essence of chaos first appeared on John D. Cook.

After writing the previous post on credit card numbers, I intended to link to a previous post that discussed credit card check sums. But I couldn’t find such a post. I’ve written about other kinds of checksums, such as the checksum scheme used in Vehicle Identification Numbers, but apparently I haven’t written about credit card […]The post Luhn checksum algorithm first appeared on John D. Cook.

The first 4 to 6 digits of a credit card number are the bank identification number or BIN. The information needed to decode a BIN is publicly available, with some effort, and so anyone could tell from a credit card number what institution issued it, what bank it draws on, whether its a personal or […]The post What can you learn from a credit card number? first appeared on John D. Cook.

Literal alphabets Natural language alphabets are all within an order of magnitude of the size of the Roman alphabet. The Hebrew alphabet has a few less letters and Russian has a few more. The smallest alphabet I’m aware of is Hawaiian with 13 letters. Syllabaries are larger than alphabets, but not an order of magnitude […]The post Tradeoff between alphabet size and word size first appeared on John D. Cook.

You cannot exactly construct an 11-sided regular polygon (called a hendecagon or an undecagon) using only a straight edge and compass. Gauss fully classified which regular n-gons can be constructed, and this isn’t one of them [1]. However, Albrecht Dürer [2] came up with a good approximate construction for a hendecagon. To construct an eleven-sided […]The post How Albrecht Dürer drew an 11-sided figure first appeared on John D. Cook.

The previous post showed that if you inscribe a hexagon and a decagon in the same circle, the ratio of the sides of the two polygons is the golden ratio. After writing the post I wondered whether you could construct the silver ratio or bronze ratio in an analogous way. Metallic ratios To back up […]The post Gold, silver, and bronze ratios first appeared on John D. Cook.

The new book A Panoply of Polygons cites a theorem Euclid (Proposition XIII.10) saying If a regular pentagon, a regular hexagon, and a regular decagon are inscribed in congruent circles, then their side lengths form a right triangle. This isn’t exactly what Euclid said, but it’s an easy deduction from what he did say. Here’s […]The post A pentagon, hexagon, and decagon walk into a bar … first appeared on John D. Cook.

This evening I ran across a trig identity I hadn’t seen before. I doubt it’s new to the world, but it’s new to me. Let A, B, and C be the angles of an arbitrary triangle. Then sin² A + sin² B + sin² C = 2 + 2 cos A cos B cos C. […]The post A new trig identity first appeared on John D. Cook.

The piranha problem says an event cannot be highly correlated with a large number of independent predictors. If you have a lot of strong predictors, they must predict each other, analogous to having too many piranhas in a small body of water: they start to eat each other. The piranha problem is subtle. It can […]The post Piranhas and prime factors first appeared on John D. Cook.

Sean Connolly asked in a comment yesterday about the density of safe primes. Safe primes are so named because Diffie-Hellman encryption systems based on such primes are safe from a particular kind of attack. More on that here. If q and p = 2q + 1 are both prime, q is called a Sophie Germain prime and p is a […]The post Density of safe primes first appeared on John D. Cook.

My son in law was playing around with Midjourney v5 and I asked him to try to redo some of the images I’ve made with DALL-E 2. Back in August i wrote a post about using DALL-E 2 to generate mnemonic images for memorizing the US presidents using the Major mnemonic system. To memorize that […]The post Redoing images in Midjourney first appeared on John D. Cook.

Suppose you’re using a program that takes a regular expression as an argument. You didn’t get the match you expected, then you realize you’d like your search to be case-insensitive. If you were using grep you’d go back and add a -i flag. If you were writing a Perl script, you could add a /i […]The post Bringing regex modifiers into the regex first appeared on John D. Cook.

The previous post mentioned the dual of a tetrahedron is another tetrahedron. The dual of a cube is an octahedron and the dual of an octahedron is a cube. And the dual of a dodecahedron is an icosahedron, and the dual of an icosahedron is a dodecahedron. So if you take the dual of a […]The post Double duals of polyhedra first appeared on John D. Cook.

I was looking back at the book The Concrete Tetrahedron and wondered what would happen if I used the title as a prompt to DALL-E. Could I get it to create an image of a concrete tetrahedron? It seemed to understand concrete but not tetrahedron, which is a little surprising since tetrahedron is such a […]The post Hints of regular solid duality first appeared on John D. Cook.

Secure hash functions often do not take a key per se, but they can be used with a key. Adding a key to a hash is useful, for example, to prevent a rainbow table attack. There are a couple obvious ways to incorporate a key K when hashing a message M. One is to prepend […]The post How to turn an unkeyed hash into a keyed hash first appeared on John D. Cook.

Is the woman in this photograph more likely to be named Esther or Caitlin? Yesterday Mark Jason Dominus published wrote about statistics on first names in the US from 1960 to 2021. For each year and state, the data tell how many boys and girls were given each name. Reading the data “forward” you could […]The post First names and Bayes’ theorem first appeared on John D. Cook.

Like the previous post, this post riffs on a photo [1] I stumbled on while looking for something else. Would it be easier to identify the man in this photo or the man whose photo appeared in the previous post, copied below. I think it would be easier for a human to recognize the person […]The post Identifiable to man or machine? first appeared on John D. Cook.

I ran across the image below [1] when I was searching for something else, and it made me think of a few issues in data privacy. The green lights cut across the man’s body like tomographic imaging. No one of these green lines would be identifiable, but maybe the combination of the lines is. We […]The post Privacy and tomography first appeared on John D. Cook.

My wife and I were talking about the engine coolant temperature symbol on our car dashboard yesterday. I said I expect there’s a Unicode code point for this symbol. Now I don’t think there is. But there is an ISO Standard name for the symbol. It’s part of ISO 7000: Graphical symbols for use on […]The post Engine coolant temperature symbol first appeared on John D. Cook.

Cryptographic hash functions are also known as one-way functions because given an input x, one can easily compute its hashed value f(x), but it is impractical to recover x from knowing f(x). However, if we know that x comes from a small universe of possible values, our one-way function can effectively become a two-way function, […]The post Privacy implications of hashing data first appeared on John D. Cook.

When I left academia [1] my first job was working as a programmer. I was very impressed by a new programmer we hired who hit the ground running. His first week he looked at some problem we were working on and said “Oh, you need a topological sort.” I’d never heard of a topological sort […]The post Topological sort first appeared on John D. Cook.

When a company promises not to sell your personal data, this promise alone doesn’t mean much. “We will not sell your personal data, but … We might get hacked. We might give it to a law enforcement or intelligence agency. We might share or trade your data without technically selling it. We might alter our […]The post “We won’t sell your personal data, but …” first appeared on John D. Cook.

A couple weeks ago I wrote about how H. A. Rey introduced a new way of looking at the constellations, making them look more like their names. That post used Leo as an example. This post looks at Boötes (The Herdsman) [1]. Here is the constellation using the connections indicated in the IAU star chart. […]The post Connecting the dots differently first appeared on John D. Cook.

The Gumbel distribution, named after Emil Julius Gumbel (1891–1966), is important in statistics, particularly in studying the maximum of random variables. It comes up in machine learning in the so-called Gumbel-max trick. It also comes up in other applications such as in number theory. For this post, I wanted to point out how a couple […]The post Famous constants and the Gumbel distribution first appeared on John D. Cook.

Markov’s inequality is very general and hence very weak. Assume that X is a non-negative random variable, a > 0, and X has a finite expected value, Then Markov’s inequality says that In [1] the author gives two refinements of Markov’s inequality which he calls Hansel and Gretel. Hansel says and Gretel says Related posts […]The post Strengthen Markov’s inequality with conditional probability first appeared on John D. Cook.

The other day I was talking with someone I met while I was doing my postdoc at Vanderbilt and Eric Schechter’s book Handbook of Analysis and its Foundations came up. Eric was writing that book while we were there. He kindly listed me in the acknowledgements for having reviewed a few pages of the book. […]The post There’s a woset in my reposit first appeared on John D. Cook.

The apparent size of a distant object can be measured by projecting the object onto a unit sphere around the observer and calculating the area of the projected image. A unit sphere has area 4π. If you’re in a ship far from land, the solid angle of the sky is 2π steradians because it takes […]The post Solid angle of a star first appeared on John D. Cook.

This previous post looked at the hyperbolic secant distribution. This distribution has density and characteristic function sech(t). It’s curious that the density and characteristic function are so similar. The characteristic function is essentially the Fourier transform of the density function, so this says that the hyperbolic secant function, properly scaled, is a fixed point of […]The post Fixed points of the Fourier transform first appeared on John D. Cook.

I hadn’t run into the hyperbolic secant distribution until I saw a paper by Peng Ding [1] recently. If C is a standard Cauchy random variable, then (2/π) log |C| has a hyperbolic secant distribution. Three applications of this distribution are given in [1]. Ding’s paper contains a plot comparing the density functions for the hyperbolic […]The post Hyperbolic secant distribution first appeared on John D. Cook.

Countries and regions have two letter abbreviations (ISO 3166-1) as do languages (ISO 639-1). So do chemical elements, US states, and books filed using the Library of Congress system. I was curious how many of the 676 possible two-letter combinations are used by the abbreviation systems above. About two thirds, not as many as I […]The post Two-letter abbreviations. first appeared on John D. Cook.

Naive interpolation of rotation matrices does not produce a rotation matrix. That is, if R1 and R2 are rotation (orthogonal) matrices and 0 < t < 1, then is not in general a rotation matrix. You can represent rotations with unit quaternions rather than orthogonal matrices (see details here), so a reasonable approach might be […]The post Interpolating rotations with SLERP first appeared on John D. Cook.

The shuffle product of two words, w1 and w2, written w1 Ш w2, is the set of all words formed by the letters in w1 and w2, preserving the order of each word’s letters. The name comes from the analogy with doing a riffle shuffle of two decks of cards. For example, bcd Ш ae, […]The post Shuffle product first appeared on John D. Cook.

The previous post commented that although the digits in the decimal representation of π are not random, it is sometimes useful to think of them as random. Similarly, it is often useful to think of prime numbers as being randomly distributed. If prime numbers were samples from a random variable, it would be natural to […]The post Prime numbers and Taylor’s law first appeared on John D. Cook.