Tie a knot in a rope and join the ends together. At each point in the rope, compute the curvature, i.e. how much the rope bends, and integrate this over the length of the rope. The Fary-Milnor theorem says the result must be greater than 4π. This post will illustrate this theorem by computing numerically […]
Category: Math
A sort of mathematical quine
Julian Havil writes what I think of as serious recreational mathematics. His books are recreational in the sense that they tell a story rather than cover a subject. They are lighter reading than a text book, but require more advanced mathematics than books by Martin Gardner. Havil’s latest book is Curves for the Mathematically Curious. […]
Predicted distribution of Mersenne primes
Mersenne primes are prime numbers of the form 2p – 1. It turns out that if 2p – 1 is a prime, so is p; the requirement that p is prime is a theorem, not part of the definition. So far 51 Mersenne primes have discovered [1]. Maybe that’s all there are, but it is […]
Collatz conjecture skepticism
The Collatz conjecture asks whether the following procedure always terminates at 1. Take any positive integer n. If it’s odd, multiply it by 3 and add 1. Otherwise, divide it by 2. For obvious reasons the Collatz conjecture is also known as the 3n + 1 conjecture. It has been computationally verified that the Collatz […]
Detecting typos with the four color theorem
In my previous post on VIN numbers, I commented that if a check sum has to be one of 11 characters, it cannot detect all possible changes to a string from an alphabet of 33 characters. The number of possible check sum characters must be at least as large as the number of possible characters […]
Vehicle Identification Number (VIN) check sum
A VIN (vehicle identification number) is a string of 17 characters that uniquely identifies a car or motorcycle. These numbers are used around the world and have three standardized formats: one for North America, one for the EU, and one for the rest of the world. Letters that resemble digits The characters used in a […]
Progress on the Collatz conjecture
The Collatz conjecture is for computer science what until recently Fermat’s last theorem was for mathematics: a famous unsolved problem that is very simple to state. The Collatz conjecture, also known as the 3n+1 problem, asks whether the following function terminates for all positive integer arguments n. def collatz(n): if n == 1: return 1 […]
Asimov’s question about π
In 1977, Isaac Asimov [1] asked how many terms of the slowly converging series π = 4 – 4/3 + 4/5 – 4/7 + 4/9 – … would you have to sum before doing better than the approximation π ≈ 355/113. A couple years later Richard Johnsonbaugh [2] answered Asimov’s question in the course of […]
Mickey Mouse, Batman, and conformal mapping
A conformal map between two regions in the plane preserves angles [1]. If two curves meet at a given angle in the domain, their images will meet at the same angle in the range. Two subsets of the plane are conformally equivalent if there is a conformal map between them. The Riemann mapping theorem says […]
Stone-Weierstrass on a disk
A couple weeks ago I wrote about a sort of paradox, that Weierstrass’ approximation theorem could seem to contradict Morera’s theorem. Weierstrass says that the uniform limit of polynomials can be an arbitrary continuous function, and so may have sharp creases. But Morera’s theorem says that the uniform limit of polynomials is analytic and thus […]
Distribution of zip code population
There are three schools of thought regarding power laws: the naive, the enthusiasts, and the skeptics. Of course there are more than three schools of thought, but there are three I want to talk about. The naive haven’t heard of power laws or don’t know much about them. They probably tend to expect things to […]
Landau kernel
The previous post was about the trick Lebesgue used to construct a sequence of polynomials converging to |x| on the interval [-1, 1]. This was the main step in his proof of the Weierstrass approximation theorem. Before that, I wrote a post on Bernstein’s proof that used his eponymous polynomials to prove Weierstrass’ theorem. This […]
Lebesgue’s proof of Weierstrass’ theorem
A couple weeks ago I wrote about the Weierstrass approximation theorem, the theorem that says every continuous function on a closed finite interval can be approximated as closely as you like by a polynomial. The post mentioned above uses a proof by Bernstein. And in that post I used the absolute value function as an […]
Proving that a choice was made in good faith
How can you prove that a choice was made in good faith? For example, if your company selects a cohort of people for random drug testing, how can you convince those who were chosen that they weren’t chosen deliberately? This is something I’ve helped companies with. It may be impossible to prove that a choice […]
Detecting a short period in an RNG
The last couple posts have been looking at the Cliff random number generator. Yesterday I posted about testing the generator with the DIEHARDER test suite, the successor to George Marsaglia’s DIEHARD battery of RNG tests. This morning I discovered something about the Cliff RNG which led to discovering something about DIEHARDER.The latter is more important: […]
Testing Cliff RNG with DIEHARDER
My previous post introduced the Cliff random number generator. The post showed how to find starting seeds where the generator will start out by producing approximately equal numbers. Despite this flaw, the generator works well by some criteria. I produced a file of 100 million 32-bit integers by multiplying the output values [1], which were […]
Fixed points of the Cliff random number generator
I ran across the Cliff random number generator yesterday. Given a starting value x0 in the open interval (0, 1), the generator proceeds by xn+1 = | 100 log(xn) mod 1 | for n > 0. The article linked to above says that this generator passes a test of randomness based on generating points on […]
Uniform approximation paradox
What I’m going to present here is not exactly a paradox, but I couldn’t think of a better way to describe it in the space of a title. I’ll discuss two theorems about uniform convergence that seem to contradict each other, then show by an example why there’s no contradiction. Weierstrass approximation theorem One of […]
Nearly parallel is nearly transitive
Let X, Y, and Z be three unit vectors. If X is nearly parallel to Y, and Y is nearly parallel to Z, then X is nearly parallel to Z. Here’s a proof. Think of X, Y, and Z as points on a unit sphere. Then saying that X and Y are nearly parallel means […]
Angles in the spiral of Theodorus
The previous post looked at how to plot the spiral of Theodorus shown below. We stopped the construction where we did because the next triangle to be added would overlap the first triangle, which would clutter the image. But we could certainly have kept going. If we do keep going, then the set of hypotenuse […]