It turns out that each positive real number \(x\in\mathbb{R}^+\) can be written in the form \(x=1/a_1+1/a_1a_2+1/a_1a_2a_3+\cdots\), where \({a_1,a_2,a_3,\ldots}\) is a unique non-decreasing sequence, finite or infinite, of postive integers. This is called the Engel expansion of \(x\), after…
A Voronoi diagram is a partition of the plane into a number of cells, where each cell consists of exactly those points that are closest to the site (or seed point) of that cell. An example with ten seed points is shown in…
Now that we are just past Pi Approximation Day (22/7 or July 22), I thought about how to find other fractional approximations of pi with small numerators and denominators. I decided that I was going to brute force this thing, just because I can…
There is really only one circle. Of course, you have to allow for translation and scaling for this to be true, but I think you’ll agree that a large circle is essentially the same as a small one. The geometrical term for this is similarity. Two objects are similar if…
Say that you have an ant that lives on an infinite plain with tiles that can be either black or white. The ant can move in the four main directions, and it keeps moving indefinitely, according to the following two simple rules…
Yes, you can do symbolic math in Python! The library to take a look at is SymPy. This article is not a SymPy tutorial, as I only want to walk you through some examples to show you the kinds of things that it can do. A good place to start…
As a variation on the Mandelbrot fractal that I’ve computed before using NumPy array operations, I’d like to show you a few Julia fractals with some color added. I’ll concentrate on the Julia sets that are generated using the same quadratic polynomial that is used to…
If you use NumPy for numerical computations, which I recommend if you program in Python, one of the key things to do is to process entire arrays in one go as much as possible (this is also true for MATLAB). Using these so-called vectorized operations makes sure that…
A Hadamard matrix is a matrix with all elements equal to \(+1\) or \(-1\), and for which the rows are mutually orthogonal. If you pick two rows from the matrix and write it as vectors \(\bf x\) and \(\bf y\), then these are orthogonal if…
What is jerk? Jerk is to acceleration as acceleration is to velocity. And, acceleration is to velocity as velocity is to position. So let’s start there. The instantaneous velocity at which you travel tells you…