The previous post mentioned the dual of a tetrahedron is another tetrahedron. The dual of a cube is an octahedron and the dual of an octahedron is a cube. And the dual of a dodecahedron is an icosahedron, and the dual of an icosahedron is a dodecahedron. So if you …

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 …

Jacobi functions are complex-valued functions of a complex variable z and a parameter m. Often this parameter is real, and 0 ≤ m < 1. Mathematical software libraries, like Python’s SciPy, often have this restriction. However, m could be any complex number. The previous couple of posts spoke of the …

Thanks to the quadratic equation, it’s easy to tell whether a quadratic equation has a double root. The equation has a double root if and only if the discriminant is zero. The discriminant of a cubic is much less known, and the analogs for higher order polynomials are unheard of. …

Here are three curves that have interesting names and interesting shapes. The fish curve The fish curve has parameterization x(t) = cos(t) – sin²(t)/√2 y(t) = cos(t) sin(t) We can plot this curve in Mathematica with ParametricPlot[ {Cos[t] - Sin[t]^2/Sqrt[2], Cos[t] Sin[t]}, {t, 0, 2 Pi}] to get the following. …

Suppose you write down a number and take the sum of its digits. In what base will this sum be the smallest on average? Let’s do a couple examples comparing base 10 and base 2. The number 2022 in base 10 has digit sum 6, but its binary equivalent 11111100110 …

If you ask Mathematica for a list of Mexican states via CountryData["Mexico", "RegionNames"] you will get a list of strings: "Aguascalientes", "Baja California", ..., "Zacatecas"} However, when you try to turn this into a list of objects representing these states via states = Entity["AdministrativeDivision", {#, "Mexico"}] & /@ CountryData["Mexico", "RegionNames"] …

A couple days ago I wrote about Costas arrays. In a nutshell, a Costas array of size n is a solution to the n rooks problem, with the added constraint that if you added wires between the rooks, no two wires would have the same length and slope. See the …

A week ago I posted some pairs of similar flags on Twitter, and later I found that Mathematica’s CountryData database contains flag descriptions. So I thought I’d use the flag descriptions to see which flags Mathematica things are similar. For example, the FlagDescription attribute for Chad in Mathematica is Three …

Suppose you want to color a map with no two bordering regions having the same color. If this is a map on a plane, you can do this using only four colors, but maybe you’d like to use more. You can reduce the problem to coloring the nodes in a …

Suppose you’d like to tour Africa, visiting each country once, then returning to your starting point, minimizing the distance traveled. Here’s my first attempt at a solution using Mathematica, based on an example in the documentation for FindShortestTour. africa = CountryData["Africa"] pos = GeoPosition[CountryData[#, "CenterCoordinates"]] & /@ africa FindShortestTour[africa] GeoGraphics[{Thick, …

A Blaschke product is a function that is the product of Blaschke factors, functions of the form b(z; a) = |a| (a – z) / a (1 – a*z) where the complex number a lies inside the unit circle and a* is the complex conjugate of a. I wanted to …

Inversion in the unit circle is a way of turning the circle inside-out. Everything that was inside the circle goes outside the circle, and everything that was outside the circle comes in. Not only is the disk turned inside-out, the same thing happens along each ray going out from the …

Will Buckner sent me an email with the following question recently. (I’m sharing this with permission.) I am building a kiln using a catenary arch. The rear wall and front wall/door will be vertical and fill in the space under the arch, which has the dimensions of 41″W x 39.5″H. …

A few days ago I wrote about Kneser’s theorem. This theorem tells whether the differential equation u ″(x) + h(x) u(x) = 0 will oscillate indefinitely, i.e. whether it will have an infinite number of zeros. Today’s post will look at another theorem that gives more specific information about the …

This post looks at an exercise from Special Functions, exercise 6 in Appendix E. Suppose that xm+1 + ax – b = 0. Show that Use this formula to find a solution to x5 + 4x + 2 = 0 to four decimal places of accuracy. When m = 0 …

The previous post used the Lambert W function to solve an equation that came up in counting partitions of an integer. The first solution found didn’t make sense in context, but another solution, one on a different branch, did. The default branch k = 0 wasn’t what we were after, …

This post will present a remarkable theorem of Euler which makes a connection between integer partitions and pentagons. Partitions A partition of an integer n is a way of writing n as a sum of positive integers. For example, there are seven unordered partitions of 5: 5 4 + 1 …