John D. Cook

When mathematicians run out of symbols, they turn to other alphabets. Most math symbols are Latin or Greek letters, but occasionally you’ll run into Russian or Hebrew letters. Sometimes math uses a new font rather than a new alphabet, such as Fraktur. This is common in Lie groups when you want to associate related symbols to […]

I don’t understand homotopy groups of spheres, and it’s OK if you don’t either. Nobody fully understands them. This post is really more about information compression than homotopy. That is, I’ll be looking at ways to summarize what is known without being overly concerned about what the results mean. The task: map two integers to […]

Which has more mobility on an empty chessboard: a rook, a king or a knight? On an ordinary 8 by 8 chessboard, a rook can move to any one of 14 squares, no matter where it starts. A knight and a king both have 8 possible moves from the middle of the board, fewer moves […]

I’ve written before about a knight’s random walk on an ordinary chess board. In this post I’d like to look at the generalization to three dimensions (or more). So what do we mean by 3D chess? For this post, we’ll have a three dimensional lattice of possible positions, of size 8 by 8 by 8. […]

NASA has fueled the development of lots of spinoff technologies: smoke detectors, memory foam, infrared ear thermometers, etc. NASA didn’t pursue these things directly, but they were a useful side effect. Something analogous happens in mathematics. While pursuing one goal, mathematicians spin off tools that are useful for other purposes. Algebraic topology might be the […]

If all you know about a person is that he or she is around 5′ 7″, it’s a toss-up whether this person is male or female. If you know someone is over 6′ tall, they’re probably male. If you hear they are over 7″ tall, they’re almost certainly male. This is a consequence of heights […]

When I was in college, one of the professors seemed to lecture at a sort of quadratic pace, maybe even an exponential pace. He would proceed very slowly at the beginning of the semester, so slowly that you didn’t see how he could possibly cover the course material by the end. But his pace would […]

A friend asked me a question the other day that came out of a graphics application. He needed to trace out an ellipse in such a way that the length of curve traced out each second was constant. For a circle, the problem is simple: (cos(t), sin(t)) will trace out a circle covering a constant […]

The Catalan numbers Cn are closely related to the central binomial coefficients I wrote about recently: In this post I’ll write C(n) rather than Cn because that will be easier to read later on. Catalan numbers come up in all kinds of applications. For example, the number of ways to parenthesize an expression with n terms is the nth Catalan […]

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number. Specifically we have and so we have the following right triangle. The hypotenuse will always be irrational because the only Fibonacci numbers that are squares are 1 and 144, and 144 is the 12th Fibonacci number.

It’s hard to find bounds on binomial coefficients that are both simple and accurate, but in 1990, E. T. H. Wang upper and lower bounds on the central coefficient that are both, valid for n at least 4. Here are a few numerical results to give an idea of the accuracy of the bounds on a […]

The blog post that kicked off the recent series of posts looked at how far apart xy and yx are for quaternions. There I used the Euclidean distance, i.e. || xy – yx ||. This time I’ll look at the angle between xy and yx, and I’ll make some analogous graphs for octonions. For vectors x and y in three dimensions, the dot […]

An earlier post included code for multiplying quaternions, octonions, and sedenions. The code was a little clunky, so I refactor it here. def conj(x): xstar = -x xstar[0] *= -1 return xstar def CayleyDickson(x, y): n = len(x) if n == 1: return x*y m = n // 2 a, b = x[:m], x[m:] c, […]

Quaternion multiplication is associative but not commutative. An earlier post looked at the average size of the commutator xy – yx as a measure of how far quaternion multiplication is from being commutative. This post looks at an analogous question for octonions. Octonion mulitplication is neither commutative nor associative. So in this post we look at the […]

The previous post discussed octonions. This post will implement octonion multiplication in Python, and then sedenion multiplication. Cayley-Dickson construction There’s a way to bootstrap quaternion multiplication into octonion multiplication, so we’ll reuse the quaternion multiplication code from an earlier post. It’s called the Cayley-Dickson construction. For more on this construction , see John Baez’s treatise […]

A group is a set with a binary operation that closed associative, has an identity, and has inverses. This post will look at each of these properties and what happens if you modify or remove it. Magmas Closed means that applying the binary operation to any two elements of the set yields another elements of […]

Given two quaternions x and y, the product xy might equal the product yx, but in general the two results are different. How different are xy and yx on average? That is, if you selected quaternions x and y at random, how big would you expect the difference xy – yx to be? Since this difference would increase proportionately if you increased the length of x […]

Today’s exponential sum is appropriate for a hot July day. Each day the exponential sum page shows a different exponential sum with the month, day, and year in the denominators of an exponential sum. The graphs are formed by plotting the partial sums and connecting the dots. For example, today’s sum, is based on the […]

Suppose you have an odd prime p and an integer a, with a not a multiple of p. Does a have a square root mod p? That is, does there exist an integer x such that x² is congruent to a mod p? Half the time the answer is yes, and half the time it’s no. (We will remove the restriction that p is prime near […]

The relationship between programming and computer science is hard to describe. Purists will say that computer science has nothing to do with programming, but that goes too far. Computer science is about more than programming, but it’s is all motivated by getting computers to do things. With few exceptions. students major in computer science in […]

A while back I wrote about how planets are evenly spaced on a log scale. I made a bunch of plots, based on our solar system and the extrasolar systems with the most planets, and said noted that they’re all roughly straight lines. Here’s the plot for our solar system, including dwarf planets, with distance […]

Objectives and constraints are symmetrical in a mathematical sense but are asymmetrical in a psychological sense. By taking dual formulations, you can reverse the mathematical role of objectives and constraints, but in application objectives are more obvious than constraints. In the question “What is the minimum value of x² over the interval [1, 5]?” the […]

In the previous post I said that the Jacobi functions are like trig functions. That’s true if you look along the real axis. If you look at the rest of the complex plane you’ll see how they can be very different. The sine function is periodic along the real axis, but it grows exponentially along […]

Jacobi’s elliptic functions are sorta like trig functions. His functions sn and cn have names that reminiscent of sine and cosine for good reason. These functions come up in applications such as the nonlinear pendulum (i.e. when θ is too large to assume θ is a good enough approximation to sin θ) and in conformal […]

The complex numbers make a field out of pairs of real numbers. The quaternions almost make a field out of four-tuples of numbers, though multiplication is not commutatative. Technically, quaternions form a division algebra. Frobenius’s theorem says only real vector spaces that can be made into division algebras are the real numbers, complex numbers, and […]

Here are three things about dominoes, two easy and one more advanced. Counting First, how many pieces are there in a set of dominoes? A domino corresponds to an unordered pair of numbers from 0 to n. The most popular form has n = 6, but there are variations with other values of n. You can show that […]

I asked two questions on twitter yesterday. The previous post summarized the results for a question about books that I asked from my personal Twitter account. This post will summarize the results of a question I asked from @AnalysisFact. If a genie offered to give you a thorough understanding of one theorem, what theorem would […]

I asked on Twitter today for books that people would like to have read, but don’t want to put in the time and effort to read. What’s a book you would like to have read but don’t want to read? — John D. Cook (@JohnDCook) June 15, 2018 Here are the responses I got, organized […]

Here are a couple of linear algebra identities that can be very useful, but aren’t that widely known, somewhere between common knowledge and arcane. Neither result assumes any matrix has low rank, but their most common application, at least in my experience, is in the context of something of low rank added to something of […]

Here are two apparently unrelated things you may have seen before. The first is an observation going back to Euler that the polynomial produces a long sequence of primes. Namely, the values are prime for n = 1, 2, 3, …, 40. The second is that the number is extraordinarily close to an integer. This number […]

There’s a proposal for California to split into three states. If that happens, what would happen to the US flag? The US flag has had 13 stripes from the beginning, representing the first 13 states. The number of stars has increased over time as the number of states has increased. Currently there are 50 stars, […]

The partition function p(n) counts the number of ways n unlabeled things can be partitioned into non-empty sets. (Contrast with Bell numbers that count partitions of labeled things.) There’s no simple expression for p(n), but Ramanujan discovered a fairly simple asymptotic approximation: How accurate is this approximation? Here’s a little Matheamtica code to see. p[n_] := PartitionsP[n] approx[n_] […]

I like Perl’s pattern matching features more than Perl as a programming language. I’d like to take advantage of the former without having to go any deeper than necessary into the latter. The book Minimal Perl is useful in this regard. It has chapters on Perl as a better grep, a better awk, a better […]

I just finished listening to the latest episode of Twenty Thousand Hertz, the story behind “Deep Note,” the THX logo sound. There are a couple mathematical details of the sound that I’d like to explore here: random number generation, and especially Pythagorean tuning. Random number generation First is that part of the construction of the […]

Stirling numbers are something like binomial coefficients. They come in two varieties, imaginatively called the first kind and second kind. Unfortunately it is the second kind that are simpler to describe and that come up more often in applications, so we’ll start there. Stirling numbers of the second kind The Stirling number of the second […]

Start with the sequence of positive integers: 1, 2, 3, 4, … Now take partial sums, the nth term of the new series being the sum of the first n terms of the previous series. This gives us the triangular numbers, so called because they count the number of coins at each level of a […]

The nth Bell number is the number of ways to partition a set of n labeled items. It’s also equal to the following sum. You may have to look at that sum twice to see it correctly. It looks a lot like the sum for en except the roles of k and n are reversed in […]

If you average a large number independent versions of the same random variable, the central limit theorem says the average will be approximately normal. That is the absolute error in approximating the density of the average by the density of a normal random variable will be small. (Terms and conditions apply. See notes here.) But […]

I was playing around with something this afternoon and stumbled on something like Gibbs phenomena or Runge phenomena for the Central Limit Theorem. The first place most people encounter Gibbs phenomena is in Fourier series for a step function. The Fourier series develops “bat ears” near the discontinuity. Here’s an example I blogged about before […]

Let me say up front that relying on the normal distribution as an accurate model of extreme events is foolish under most circumstances. The main reason to calculate the probability of, say, a 40 sigma event is to show how absurd it is to talk about 40 sigma events. See my previous post on six-sigma […]

I saw on Twitter this afternoon a paraphrase of a quote from Nassim Taleb to the effect that if you see a six-sigma event, that’s evidence that it wasn’t really a six-sigma event. What does that mean? Six sigma means six standard deviations away from the mean of a probability distribution, sigma (σ) being the […]

Calendars are based on three frequencies: the rotation of the Earth on its axis, the rotation of the moon around the Earth, and the rotation of the Earth around the sun. Calendars are complicated because none of these periods is a simple multiple of any other. The ratios are certainly not integers, but they’re not […]

Erik Erlandson sent me a note saying he found my post on computing the soft maximum helpful. (If you’re unfamiliar with the soft maximum, here’s a brief description of what it is and how you might use it.) Erik writes I used your post on practical techniques for computing smooth max, which will probably be […]

NASA’s Galileo mission was primarily designed to explore Jupiter and its moons. In 1989, the Galileo probe started out traveling away from Jupiter in order to do a gravity assist swing around Venus. About a year later it also did a gravity assist maneuver around Earth. Carl Sagan suggested that when passing Earth, the Galileo […]

Most basic combinatorial problems can be solved in terms of multiplication, permutations, and combinations. The next step beyond the basics, in my experience, is counting selections with replacement. Often when I run into a problem that is not quite transparent, it boils down to this. Examples of selection with replacement Here are three problems that […]

It’s easy to create rational approximations for π. Every time you write down π to a few decimal places, that’s a rational approximation. For example, 3.14 = 314/100. But that’s not the best approximation. Think of the denominator of your fraction as something you have to buy. If you have enough budget to buy a three-digit […]

Here’s an interesting problem that came out of a logistic regression application. The input variable was between 0 and 1, and someone asked when and where the logistic transformation f(x) = 1/(1 + exp(a + bx)) has a fixed point, i.e. f(x) = x. So given logistic regression parameters a and b, when does the logistic curve given by y […]

A new video from 3Blue1Brown is about visualizing derivatives as stretching and shrinking factors. Along the way they consider the function f(x) = 1 + 1/x. Iterations of f converge on the golden ratio, no matter where you start (with one exception). The video creates a graph where they connect values of x on one […]

I just made one of those O’Reilly parody book covers. It’s a joke on Judea Pearl, expert in causal inference, and the Perl programming language, known for its unusual, terse syntax. Related:

When I was a teenager, my uncle gave me a calculus book and told me that mastering calculus was the most important thing I could do for starting out in math. So I learned the basics of calculus from that book. Later I read Michael Spivak’s two calculus books. I took courses that built on […]