Suppose you have a keypad that will unlock a door as soon as it sees a specified sequence of four digits. There’s no “enter” key to mark the end of a four-digit sequence, so the four digits could come at any time, though they have to be sequential. So, for example, if the pass code […]

# Category: Math

## Binary surprise

As mentioned in the previous post, the Gauss-Wantzel theorem says you can construct a regular n-gon with a straight edge and compass if and only if n has the form 2k F where k is a non-negative integer and F is a product of distinct Fermat primes. Let’s look at the binary representation of these […]

## Exact values of sine and cosine

If you know the sine of any angle, you can find its cosine from the Pythagorean theorem. And if you know the sine of an angle you can find the sine of any multiple of that angle using the identity for the sine of a sum. You can find the sine of 30 degrees from […]

## Golay codes

Suppose you want to sent pictures from Jupiter back to Earth. A lot could happen as a bit travels across the solar system, and so you need some way of correcting errors, or at least detecting errors. The simplest thing to do would be to transmit photos twice. If a bit is received the same […]

## Chinese character frequency and entropy

Yesterday I wrote a post looking at the frequency of Koine Greek letters and the corresponding entropy. David Littleboy asked what an analogous calculation would look like for a language like Japanese. This post answers that question. First of all, information theory defines the Shannon entropy of an “alphabet” to be bits where pi is […]

## Greek letter frequency and entropy

Would the letters in an ancient Greek text carry more or less information than in modern English? To address this question, I downloaded a copy of the Greek New Testament from Project Gutenberg and ran the word frequency script from my previous post. This lead to the follow table of letters and percent frequency. α […]

## Hyperexponential and hypoexponential distributions

There are a couple different ways to combine random variables into a new random variable: means and mixtures. To take the mean of X and Y you average their values. To take the mixture of X and Y you average their densities. The former makes the tails thinner. The latter makes the tails thicker. When […]

## Primes that don’t look like primes

Primes usually look odd. They’re literally odd [1], but they typically don’t look like they have a pattern, because a pattern would often imply a way to factor the number. However, 12345678910987654321 is prime! I posted this on Twitter, and someone with the handle lagomoof replied that the analogous numbers are true for bases 2, […]

## Harmonographs

In the previous post, I said that Lissajous curves are the result of plotting a curve whose x and y coordinates come from (undamped) harmonic oscillators. If we add a little bit of dampening, multiplying our cosine terms by negative exponentials, the resulting curve is called a harmonograph. Here’s a bit of Mathematica code to […]

## Lissajous curves and knots

Suppose that over time the x and y coordinates of a point are both given by a harmonic oscillator, i.e. x(t) = cos(nx t + φx) y(t) = cos(ny t + φy) Then the resulting path is called a Lissajous curve. If you add a z coordinate also given by harmonic oscillator z(t) = cos(nz […]

## Curvature of an ellipsoid

For an ellipsoid with equation the Gaussian curvature at each point is given by Now suppose a ≥ b ≥ c > 0. Otherwise relabel the coordinate axes so that this is the case. Then the largest curvature occurs at (±a, 0, 0), and the smallest curvature occurs at (0, 0, ±c). You could prove […]

## Fixed points

Take a calculator and enter any number. Then press the cosine key over and over. Eventually the numbers will stop changing. You will either see 0.99984774 or 0.73908513, depending on whether your calculator was in degree mode or radian mode. This is an example of a fixed point, a point that doesn’t change when you […]

## Number of real roots in an interval

Suppose you have a polynomial p(x) and in interval [a, b] and you want to know how many distinct real roots the polynomial has in the interval. You can answer this question using Sturm’s algorithm. Let p0(x) = p(x) and letp1(x) be its derivative p‘(x). Then define a series of polynomials for i ≥ 1 […]

## Total curvature of a knot

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 […]

## 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 […]