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.

An ellipse can be defined as the set of points such that the sum of the distances to two fixed points, the foci, has a constant value. A Cassini oval is the set of points such that the product of the distances to two foci has a constant value. You can write down an equation […]The post Cassini ovals first appeared on John D. Cook.

Let f be an analytic function on the unit disk with f(0) = 0 and derivative f ′(0) = 1. If f is one-to-one (injective) then this puts a strict limit on the size of the series coefficients. Let an be the nth coefficient in the power series for f centered at 0. If f is one-to-one […]The post Bounds on power series coefficients first appeared on John D. Cook.

In the process of creating a Pratt certificate to prove that a number n is prime, you have to find a number a that seems kinda arbitrary. As we discussed here, a number n is prime if there exists a number a such that an-1 = 1 mod n and a(n-1)/p ≠ 1 mod n […]The post Probability problem with Pratt prime proofs first appeared on John D. Cook.

The previous post illustrated a technique for finding factors of number of the form bn – 1. This post will look at an analogous, though slightly less general, technique for numbers of the form bn + 1. There is a theorem that says that if m divides n then bm + 1 divides bn + […]The post Factoring b^n + 1 first appeared on John D. Cook.

Suppose you want to factor a number of the form bn – 1. There is a theorem that says that if m divides n then bm – 1 divides bn – 1. Let’s use this theorem to try to factor J = 22023 – 1, a 609-digit number. Factoring such a large number would be more difficult if it didn’t have […]The post Factoring b^n – 1 first appeared on John D. Cook.

Barycentric coordinates describe the position of a point relative to the three vertices of a triangle. Trilinear coordinates describe the position of a point relative to the three sides of a triangle. It’s surprisingly simple to convert from one to the other. Why should this be surprising? Because the distance from a point to a […]The post Converting between barycentric and trilinear coordinates first appeared on John D. Cook.

I’ve written lately about two general ways to prove that a number is prime: Pratt certificates for moderately-large primes and elliptic curve certificates for very large primes. If you can say more about the prime you wish to certify, there may be special forms of certificates that are more efficient. In particular, there are efficient […]The post Special primality proofs first appeared on John D. Cook.

The Riemann zeta function ζ(s) is given by an infinite sum and an infinite product for complex numbers s with real part greater than 1 [*]. The infinite sum is equal to the infinite product, but which would give you more accuracy: N terms of the sum or N terms of the product? We’ll take […]The post Zeta sum vs zeta product first appeared on John D. Cook.

In a paper on arXiv Simon Plouffe gives the formula which he derives from an equation in Abramowitz and Stegun (A&S). It took a little while for me to understand what Plouffe intended. I don’t mean my remarks here to be criticism of the author but rather helpful hints for anyone else who might have […]The post Approximating pi with Bernoulli numbers first appeared on John D. Cook.

This weekend I saw a sign in the window of a Burger King™ that made me think of an interesting problem. If you know the number of possibilities like this, how would you reverse engineer what the options that created the possibilities? In the example above, there are 211,184 = 213×33 possible answers, and so […]The post Reverse engineering options first appeared on John D. Cook.

Douglas Hofstadter, best known as the author of Godel, Escher, Bach, wrote the foreword to Clark Kimberling’s book Triangle Centers and Central Triangles. Hofstadter begins by saying that in his study of math he “sadly managed to sidestep virtually all of geometry” and developed an interest in geometry, specifically triangle centers, much later. The ancient […]The post Foreshadowing Page Rank first appeared on John D. Cook.

Most applied differential equations are second order. This probably has something to do with the fact that Newton’s laws are second order differential equations. Higher order equations are less common in application, and when they do pop up they usually have even order, such as the 4th order beam equation. What about 3rd order equations? […]The post Third order ordinary differential equations first appeared on John D. Cook.

Suppose you hire me to solve an optimization problem for you. You want me to find the value of x that minimizes f(x). I go off and work on finding the best value of x. I report back what I found, and you might say “Thanks, That’s a good value of x. But how do […]The post Proof of optimization first appeared on John D. Cook.

I’ve written recently about a simple kind of primality certificates, Pratt certificates. These certificates are easy to understand, and easy to verify, but they’re expensive to produce. In order to produce a Pratt certificate that n is a prime you have to factor n-1, and that can take a long time if n is large […]The post Elliptic curve primality certificates first appeared on John D. Cook.

Suppose a number n written in binary has two 1s and all the rest of its bits are zeros. If n is prime, then the 1s must be the first and last bits of n. The first bit is 1 because the first bit of every positive integer is 1. The last bit is 1 […]The post Primes with two non-zero bits first appeared on John D. Cook.

Last week I wrote about primailty certificates. These certificates offer a way to verify that a number is prime using less computation than was used to discover than the number was prime. This post gives a couple more examples of primality certificates using sonnet primes. As described here, These are primes of the form ababcdcdefefgg, […]The post Certified sonnet primes first appeared on John D. Cook.

The electricity went out for a few hours recently, and because the power was out, the internet was out. I was trying to do a little work on my laptop, but I couldn’t do what I intended to do because I needed a network connection to access some documentation. I keep offline documentation for just […]The post Self-documenting software first appeared on John D. Cook.

A few days ago a comment that a graph looked like a Maxwell-Boltzman density lead to an approximation of 1/Γ(x), possibly a useful approximation. Approximating Γ(x) is a well-known problem, and for large x the solution is to use Stirling’s approximation or a few more terms from the asymptotic series that Stirling’s approximation is a […]The post Approximating 1/Γ(x) first appeared on John D. Cook.

The previous post discussed the circumcenter and orthocenter of a triangle. Euler proved that the centroid, circumcenter, and orthocenter all fall on a common line, now called the Euler line. The centroid is the center of mass of a triangle. If you draw lines from each vertex to the midpoint of the opposite side, the […]The post Euler line first appeared on John D. Cook.

The previous post mentioned that the law of sines gives you the diameter of a circle through the vertices of a triangle. How would you find the center of this circle, the blue dot in the image above? If the angles of the triangle are α. β, and γ, then the trilinear coordinates of the […]The post Relating circumcenter and orthocenter first appeared on John D. Cook.

A few days ago I wrote about the law of cotangents. This law says that if we label the sides of a triangle a, b, c and label the angles opposite each side α. β, γ, then where s is the semi-parameter, i.e. and r is the radius of the incircle, the largest circle that […]The post Computing inscribed radius and circumscribed radius first appeared on John D. Cook.

When I shared an image from the previous post on Twitter, someone who goes by the handle Nonetheless made the astute observation that image looked like the Maxwell-Boltzmann distribution. That made me wonder what 1/Γ(x) would be like turned into a probability distribution, and whether it would be approximately like the Maxwell-Boltzmann distribution. (Here I’m […]The post Maxwell-Boltzmann and Gamma first appeared on John D. Cook.

A little while ago I wrote a post looking at how the infinite product for sine converges. The plot of the error terms is both mathematically and aesthetically interesting. This post will look at similar plots for the reciprocal of the gamma function. The reciprocal of the gamma function is an entire function, i.e. is […]The post Visualizing convergence of an infinite product first appeared on John D. Cook.

Rational trigonometry is a very different way of looking at geometry. At its core are two key ideas. First, instead of distance, do all your calculations in terms of quadrance, which is distance squared. Second, instead of using angles to measure the separation between lines, use spread., which turns out to be the square of […]The post Rational Trigonometry first appeared on John D. Cook.

Geoff Lindsey contacted me recently to ask whether he could use the sheet music from one of my blog posts in a video he was making on Morse code snippets hidden in music. The sheet music appears about a minute into the video. After watching the video, his previious video played, a video about words […]The post Hidden messages in music first appeared on John D. Cook.

The previous post commented that the law of tangents is much less familiar than the laws of sines and cosines. The law of cotangents is even more obscure. If you ask Google’s Ngram viewer to plot occurrences of “law of cotangents” over time, it will return “Ngrams not found: law of cotangents.” What is this […]The post Law of cotangents first appeared on John D. Cook.

I would have thought that the laws of sines, cosines, and tangents were all about equally familiar, but apparently that is not the case. Here’s a graph from Google’s Ngram viewer comparing the frequencies of law of sines, law of cosines, and law of tangents. As of 2019, the number of references to the laws […]The post Law of tangents first appeared on John D. Cook.

The previous post implicitly asserted that J = 8675309 is a prime number. Suppose you wanted proof that this number is prime. You could get some evidence that J is probably prime by demonstrating that 2J-1 = 1 mod J. You could do this in Python by running the following [1]. >>> J = 8675309 […]The post Pratt Primality Certificates first appeared on John D. Cook.

The quadratic reciprocity theorem addresses the question of whether a number is a square modulo a prime. For an odd prime p, the Legendre symbol is defined to be 0 if a is a multiple of p, 1 if a is a (non-zero) square mod p, and -1 otherwise. It looks like a fraction, but […]The post Quadratic reciprocity algorithm first appeared on John D. Cook.

How many groups are there with 2023 elements? There’s obviously at least one: Z2023, the integers mod 2023. Now 2023 = 7 × 289 = 7 × 17 × 17 and so we could also look at Z7 + Z17 + Z17 where + denotes direct sum. An element of this group has the form […]The post Groups of order 2023 first appeared on John D. Cook.

The Bessel functions Jn for even n look something like the sinc function. How well can you approximate the former by sums of the latter? To make things concrete, we’ll approximate J2. Here’s a plot of J2. And here’s a plot of sinc(x) = sin(πx)/πx. The sinc approximation for a function f(x) is given by […]The post Sinc approximation to Bessel function first appeared on John D. Cook.

Chess-related A knight’s tour magic square A king’s tour magic square Language-related Alphamagic squares in English Alphamagic squares in French Alphamagic squares in Spanish Planet-related Mars Jupyter More mathematical Magic square of squares Magic square of primes Magic squares as matrices Magical permutations Greco-Latin squares and magic squaresThe post A dozen magic square posts first appeared on John D. Cook.

It doesn’t take much imagination to understand why DEN is the IATA abbreviation for the Denver airport, but the abbreviation MCO for the Orlando airport is more of a head scratcher. Here is a list of the busiest airports in the US along with a brief indication of the reason behind their abbreviations. Some require […]The post Airport abbreviation origins first appeared on John D. Cook.

I recently ran into the following comic strip online: [Update: Thanks to Bryan Cantanzaro for letting me know via the comments that the image above was created by Hannah Hillam. The version I found had had her copyright information edited out. I will replace the image above with a legitimate version shortly.] [Update 2: I’m […]The post Visually symmetric words first appeared on John D. Cook.

Almost all binomial coefficients are divisible by their row number. This is a theorem from [1]. What does it mean? If you iterate through Pascal’s triangle, left-to-right and top-to-bottom, noting which entries C(m, k) are divisible by m, the proportion approaches 1 in the limit. The author proves that the ratio converges to 1, but […]The post Pascal’s triangle mod row number first appeared on John D. Cook.

In last week’s post on polynomial approximations for sine, I showed that the polynomial based on Chebyshev series was much better than a couple alternatives. I calculated a few terms of the Chebyshev series for sin(πx) but didn’t include the calculations in that blog post. I calculated the series coefficients numerically, but this post will […]The post Chebyshev series for sine first appeared on John D. Cook.

How long does it take the earth to complete one rotation on its axis? The answer depends on your frame of reference. A solar day is the time it takes for the sun to appear at the same position in the sky. A sidereal day is the time it takes for a distant star to […]The post Solar Day vs Sidereal Day first appeared on John D. Cook.

Here are some posts I’ve written that fall under the general heading of coding theory. Although coding theory can overlap with secret codes, it’s more concerned with things like Morse code, Reed-Solomon codes, and Unicode. Radio related Frequency Shift Keying Morse code numbers and abbreviations How efficient is Morse code? Algebraic coding theory Prefix codes […]The post Coding theory posts first appeared on John D. Cook.

These were the most popular posts on my site this year. #10: How is portable radio possible? The length of an antenna is typically 1/2 or 1/4 of the length of the radio wave it’s designed to receive. How does an AM radio not need an antenna as long as a football field? See also […]The post Top posts of 2022 first appeared on John D. Cook.

Taylor polynomials are terrific local approximations but poor global approximations. Taylor polynomials are optimal in some sense near their center, but are seldom the best choice over a large interval. This post will look at approximating sin(πx) over [-1, 1] with fifth degree polynomials. First, this plot compares the approximation error for a fifth order […]The post Polynomial approximations to sine first appeared on John D. Cook.

Euler’s product formula for sine is To visualize the convergence of the infinite product, let’s look at the error in approximating sin(πx) with the Nth partial product of the infinite product, i.e. Here’s a plot of the partial products. We knew before making the plot that the error had to go to zero as N […]The post Euler product for sine first appeared on John D. Cook.

World record marathon times have been falling in increments of roughly 30 seconds, each new record shaving roughly 30 seconds off the previous record. If someone were to set a new record, taking 20 seconds off the previous record, this would be exciting, but not suspicious. If someone were to take 5 minutes off the […]The post Surprisingly not that surprising first appeared on John D. Cook.

Suppose a spaceship is headed from the earth to the moon. At some point we say that the ship has left the earth’s sphere of influence is now in the moon’s sphere of influence (SOI). What does that mean exactly? Wrong explanation #1 One way you’ll hear it described is that the moon’s sphere of […]The post Sphere of infuence first appeared on John D. Cook.

Do fifth degree polynomial equations come up in applications? Yes, and this post will give an example. In general the three-body problem, describing the motion of three objects interacting under gravity, does not have a closed-form solution. However, Euler and Lagrange discovered a few special cases that do have closed-form solutions. We will look at […]The post Lagrange’s quintic and Descartes’ rule first appeared on John D. Cook.

Yesterday Terence Tao published a blog post on bounds for the Poisson probability distribution. Specifically, he wrote about Bennett’s inequalities and a refinement that he developed or at least made explicit. Tao writes This observation is not difficult and is implicitly in the literature … I was not able to find a clean version of […]The post Poisson distribution tail bounds first appeared on John D. Cook.