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.

How far do you have to go down the decimal digits of π until you’ve seen all the digits 0 through 9? We can print out the first few digits of π and see that there’s no 0 until the 32nd decimal place. 3.14159265358979323846264338327950 It’s easy to verify that the remaining digits occur before the […]The post The coupon collector problem and π first appeared on John D. Cook.

Until yesterday, I was only aware of the traditional assignment of stars to constellations. In the comments to yesterday’s post I learned that H. A. Rey, best know for writing the Curious George books, came up with a new way of viewing the constellations in 1952, adding stars and connecting lines in order to make […]The post Adding stars to constellations first appeared on John D. Cook.

Suppose you wanted to write a program to plot constellations. This leads down some interesting rabbit trails. When you look up data on stars in constellations you run into two meanings of constellation. For example, Leo is a region of the night sky containing an untold number of stars. It is also a pattern of […]The post Plotting constellations first appeared on John D. Cook.

In 1881, astronomer Simon Newcomb noticed something curious. The first pages in books of logarithms were dirty on the edge, while the pages became progressively cleaner in later pages. He inferred from this that people more often looked up the logarithms of numbers with small leading digits than with large leading digits. Why might this […]The post Alien astronomers and Benford’s law first appeared on John D. Cook.

This morning Eric Berger posted a clip from The Hunt for Red October as a meme, and that made me think about the movie. I watched Red October this evening, for the first time since around the time it came out in 1990, and was surprised by a detail in one of the scenes. I […]The post Dutton’s Navigation and Piloting first appeared on John D. Cook.

An isomorphism is a structure-preserving function from one object to another. In the context of graphs, an isomorphism is a function that maps the vertices of one graph onto the vertices of another, preserving all the edges. So if G and H are graphs, and f is an isomorphism between G and H, nodes x […]The post The NBA and MLB trees are isomorphic first appeared on John D. Cook.

Last week I wrote about how to number MLB teams so that the number n told you where they are in the league hierarchy: n % 2 tells you the league, American or National n % 3 tells you the division: East, Central, or West n % 5 is unique within a league/division combination. Here […]The post Numbering minor league baseball teams first appeared on John D. Cook.

There are tricks for determining whether a number is divisible by various primes, but many of these tricks have to be applied one at a time. You can make a procedure for testing divisibility by any prime p that is easier than having to carry out long division, but these rules are of little use […]The post John Conway’s mental factoring method and friends first appeared on John D. Cook.

The previous post took a mathematical look at the National Football League. This post will do the same for Major League Baseball. Like the NFL, MLB teams are organized into a nice tree structure, though the MLB tree is a little more complicated. There are 32 NFL teams organized into a complete binary tree, with […]The post Major League Baseball and number theory first appeared on John D. Cook.

This post will look at the National Football League through the lens of graph theory, topology, and binary numbers. The NFL has a very nice tree structure, which isn’t too surprising in light of the need to make tournament brackets. The NFL is divided into two conferences, the American Football Conference and the National Football […]The post A mathematical look at the NFL first appeared on John D. Cook.

This post is a sequel to the post How Mr. Bidder calculated logarithms published a few days ago. As with that post, this post is based on an excerpt from The Great Mental Calculators by Steven B. Smith. Smith’s book says Arthur Benjamin squares large numbers using the formula n² = (n + a)(n − […]The post How Mr. Benjamin squares numbers first appeared on John D. Cook.

The sinc function is defined either as sin(x)/x or as sin(πx)/πx. We’ll use the former definition here because we’ll cite a paper that uses that definition. Here’s a plot of the sinc function and its first two derivatives. Thomas Grönwall proposed a problem to the American Mathematical Monthly in 1913 [1] bounding the derivatives of […]The post Bounding derivatives of the sinc function first appeared on John D. Cook.

A little while back I wrote about Napoleon’s theorem for triangles. A little later I wrote about Van Aubel’s theorem, a sort of analogous theorem quadrilaterals. This post presents another analog of Napoleon’s theorem for quadrilaterals. Napoleaon’s theorem says that if you start with any triangle, and attach equilateral triangles to each side, the centroids […]The post Another Napoleon-like theorem first appeared on John D. Cook.

The Playfair cipher was the first encryption technique to encrypt text two letters at a time. Instead of substituting one letter for another, it substitutes one pair of letters for another pair. This makes the method more secure than a simple substitution cipher, but hardly secure by modern standards. The Playfair cipher was used (and […]The post Playfair cipher first appeared on John D. Cook.

Simple substitution ciphers replace one letter with another. Maybe A goes to W, B goes to G, C goes to A, etc. These ciphers are famously easy to break, so easy that they’re common in puzzle books. Here’s one I made [1] for this post in case you’d like to try it. X RF SXIIXKW […]The post Simple substitution ciphers over a gargantuan alphabet first appeared on John D. Cook.

George Parker Bidder (1806–1878) was a calculating prodigy. One of his feats was mentally calculating logarithms to eight decimal places. This post will explain his approach. I’ll use “log” when the base of the logarithm doesn’t matter, and add a subscript when it’s necessary to specify the base. Bidder was only concerned with logarithms base […]The post How Mr. Bidder calculated logarithms first appeared on John D. Cook.

The sine function has period 2π, an irrational number. and so if we take the sines of the integers, we’re going to get a somewhat random sequence. (I’m assuming, as always that we’re working in radians. The sines of integer numbers of degrees are much less interesting.) Here’s a plot of the sines of 0, […]The post Sine of integers first appeared on John D. Cook.

A lot of smart people have a rule not to memorize anything that they can derive on the spot. That’s a good rule, up to a point. But past that point it becomes a liability. Most students err on the side of memorizing too much. For example, it’s common for students to memorize three versions […]The post Derive or memorize? first appeared on John D. Cook.

To test whether a number is divisible by 11, add every other digit together and subtract the rest of the digits. The result is divisible by 11 if and only if the original number is divisible by 11. For example, start with n = 31425. Add 3, 4, and 5, and subtract 1 and 2. […]The post Divisibility by base + 1 first appeared on John D. Cook.

The Maidenhead locator system divides the earth into fields, squares, and subsquares. The first two characters in a Maidenhead locator specify the square. These are letters A through R representing 20 degrees of longitude or 10 degrees of latitude. Latitude A runs from the South Pole to 80° south of the equator. Latitude R runs […]The post How large is a Maidenhead field? first appeared on John D. Cook.

What do we mean by rectangle? This post will derive the area of a spherical region bounded by lines of latitude and longitude. Such a region corresponds to an actual rectangle on a Mercator projection map, with sides aligned with the coordinate axes, and is approximately a rectangle on a sphere if the rectangle is […]The post Area of a “rectangle” on a globe first appeared on John D. Cook.

I was looking at frequencies of pitches and saw something I hadn’t noticed before: F# and G have very nearly integer frequencies. To back up a bit, we’re assuming the A above middle C has frequency 440 Hz. This is the most common convention now, but conventions have varied over time and place. We’re assuming […]The post F# and G first appeared on John D. Cook.

Suppose you’re going to fit a spline s to a function f by interpolating f at a number of points. What can you know a priori about how well s will approximate f? This question was thoroughly resolved five decades ago [1], but the result is a bit complicated, so we’ll incrementally work our way […]The post How well does a spline fit a function? first appeared on John D. Cook.

The previous post was about 12 probability distributions named after Irving Burr. This post is about 12 probability distributions named after Karl Pearson. The Pearson distributions are better known, and include some very well known distributions. Burr’s distributions are defined by their CDFs; Pearson’s distributions are defined by their PDFs. Pearson’s differential equation The densities […]The post The Pearson distributions first appeared on John D. Cook.

As I mentioned in the previous post, there are 12 distributions named for Irving Burr, known as Burr Type I, Burr Type II, Burr Type III, …, Burr Type XII. [1] The last of these is by far the most common, and the rest are hard to find online. I did manage to find them, […]The post The other Burr distributions first appeared on John D. Cook.

Irving Burr came up with a set of twelve probability distributions known as Burr I, Burr II, …, Burr XII. The last of these is by far the best known, and so the Burr XII distribution is often referred to simply as the Burr distribution [1]. See the next post for the rest of the […]The post Burr distribution first appeared on John D. Cook.

This post will distinguish between periodic, almost periodic, and quasiperiodic functions, and give examples of the latter. Definitions A function f is periodic with period T if f(x + T) = f(x) for all x. For example, trig functions are periodic. A function f is almost periodic with period T if f(x + T) ≈ […]The post Quasiperiodic functions first appeared on John D. Cook.

If a three-digit number is divisible by 37, it remains divisible by 37 if you rotate its digits. For example, 148 is divisible by 37, and so are 814 and 481. This rotation property could make it easier to recognize multiples of 37 or easier to carry out trial division. Before proving the theorem, I’ll […]The post Rotating multiples of 37 first appeared on John D. Cook.

Suppose you’re given a number and you’d like to tell whether its a square, or at least you’d like to be able to determine quickly if it’s not a square. This post began as a thread I wrote on Twitter. For starters, the last digit of a square in base 10 must be 0, 1, […]The post Recognizing squares first appeared on John D. Cook.

This post will reproduce a three plots from a paper of Hénon on dynamical systems from 1969 [1]. Let α be a constant, and pick some starting point in the plane, (x0, y0), then update x and y according to xn+1 = xn cos α − (yn − xn²) sin α yn+1 = xn sin […]The post Hénon’s dynamical system first appeared on John D. Cook.

Suppose you are trying to approximate some number x and you’ve got it sandwiched between two rational numbers: a/b < x < c/d. Now you’d like a better approximation. What would you do? The obvious approach would be to take the average of a/b and c/d. That’s fine, except it could be a fair amount […]The post Mediant approximation trick first appeared on John D. Cook.

Suppose you need to optimize, i.e. maximize or minimize, a function f(x). If this is a practical problem and not a textbook exercise, you probably need to optimize f(x) subject to some constraint on x, say g(x) = 0. Hmm. Optimize one function subject to a constraint given by another function. Oh yeah, Lagrange multipliers! […]The post Lagrange multiplier setup: Now what? first appeared on John D. Cook.

Suppose f(x) and g(x) are functions that are each proportional to their second derivative. These include exponential, circular, and hyperbolic functions. Then the integral of f(x) g(x) can be computed in closed form with a moderate amount of work. The first time you see how such integrals are computed, it’s an interesting trick. I wrote […]The post Avoid having to integrate by parts twice first appeared on John D. Cook.

I’m not sure whether automatic text completion on a mobile device is a net good. It sometimes saves a few taps, but it seems like it’s at least as likely to cause extra work. Although I’m ambivalent about autocomplete on my phone, I like it in my text editor. The difference is that in my […]The post Good autocomplete first appeared on John D. Cook.

Saving keystrokes is overrated, but maintaining concentration is underrated. This post is going to look at automating small tasks in order to maintain concentration, not to save time. If a script lets you easily carry out some ancillary task without taking your concentration off your main task, that’s a big win. Maybe the script only […]The post Small-scale automation first appeared on John D. Cook.

I typically announce new blog posts from my most relevant twitter account: data science from @DataSciFact, algebra and miscellaneous math from @AlgebraFact, TeX and typography from @TeXtip, etc. If you’d like to be sure that you’re notified of each post, regardless of what algorithms Twitter applies to your feed, you can subscribe to this blog […]The post Remove algorithmic filters from what you read first appeared on John D. Cook.

When I think of bit twiddling, I think of C. So I was surprised to read Paul Khuong saying he thinks of Common Lisp (“CL”). As always when working with bits, I first doodled in SLIME/SBCL: CL’s bit manipulation functions are more expressive than C’s, and a REPL helps exploration. I would not have thought […]The post Number of bits in a particular integer first appeared on John D. Cook.

The lemniscate of Bernoulli came up in a post a few days ago. This shape is a special case of a Cassini oval: ((x + a)² + y²) ((x – a)² + y²) = a4. Here’s another way to arrive at the lemniscate. Draw a hyperbola (blue in the figure below), then draw circles centered […]The post Lemniscate of Bernoulli first appeared on John D. Cook.

Van Aubel’s theorem is analogous to Napoleon’s theorem, though not a direct generalization of it. Napoleon’s theorem says to start with any triangle and draw equilateral triangles on each side. Connect the centers of the three new triangles, and you get an equilateral triangle. Now suppose you start with a quadrilateral and draw squares on […]The post Van Aubel’s theorem first appeared on John D. Cook.

Can a Pythagorean triangle have one size of length 2023? Yes, one possibility is a triangle with sides (2023, 6936, 7225). Where did that come from? And can we be more systematic, listing all Pythagorean triangles with a side of length 2023? Euclid’s formula generates Pythagorean triples by sticking integers m and n into the […]The post Pythagorean triangles with side 2023 first appeared on John D. Cook.

The density function of a normal distribution with mean 0 and standard deviation √(2kt) satisfies the heat equation. That is, the function satisfies the partial differential equation You could verify this by hand, or if you’d like, here’s Mathematica code to do it. u[x_, t_] := PDF[NormalDistribution[0, Sqrt[2 k t]], x] Simplify[ D[u[x, t], {t, […]The post Heat equation and the normal distribution first appeared on John D. Cook.

Here’s something surprising: You can apply a symmetric function to a symmetric shape and get something out that is not symmetric. Let f(z) be the average of z and its reciprocal: f(z) = (z + 1/z)/2. This function is symmetric in that it sends z and 1/z to the same value. It’s also symmetric in […]The post Airfoils first appeared on John D. Cook.

Johannes Kepler thought that planetary orbits were ellipses. Giovanni Cassini thought they were ovals. Kepler was right, but Cassini wasn’t far off. In everyday speech, people use the words ellipse and oval interchangeably. But in mathematics these terms are distinct. There is one definition of an ellipse, and several definitions of an oval. To be […]The post Oval orbits? first appeared on John D. Cook.