In a comment to my previous post on quatrefoils, Jan Van lint suggested a different equation for quatrefoils: r = a + |cos(2θ)| Here are some examples of how these curves look for varying values of a. As a increases, the curves get rounder. We can quantify this by looking …

Most people have heard of word problems, but not as many have heard of the word problem. If you’re imagining that the word problem is some superlatively awful word problem, I can assure you it’s not. It’s both simpler and weirder than that. The word problem is essentially about whether …

Oct. 20, 2020, 12:19 a.m.

A Pythagorean triple is a set of positive integers that can be the lengths of sides of a right triangle, i.e. numbers a, b, and c such that a² + b² = c². A primitive Pythagorean triple (PPT) is a Pythagorean triple whose elements are relatively prime. For example, (50, …

A “rose” in mathematics is typically a curve with polar equation r = cos(kθ) where k is a positive integer. If k is odd, the resulting graph has k “petals” and if k is even, the plot has 2k petals. Sometimes the term rose is generalized to the case of …

I was reading The 99% Invisible City this evening, and there was a section on quatrefoils. Here’s an example of a quatrefoil from Wikipedia. There’s no single shape known as a quatrefoil. It’s a family of shapes that look something like the figure above. I wondered how you might write …

I’m working on a project these days where I’ve used four different kinds of matrix product, which made me wonder if there’s another kind of product out there that I could find some use for. In the process of looking around for other matrix products, I ran across the Kronecker …

Oct. 11, 2020, 11:25 p.m.

This post is a follow-on to the previous post on perfectly nonlinear functions. In that post we defined a way to measure the degree of nonlinearity of a function between two Abelian groups. We looked at functions that take sequences of four bits to a single bit. In formal terms, …

The other day I heard someone suggest that a good cocktail party definition of cryptography is puzzle making. Cryptographers create puzzles that are easy to solve given the key, but ideally impossible without the key. Linearity is very useful in solving puzzles, and so a puzzle maker would like to …

Yesterday I blogged about an exercise in the book The Cauchy-Schwarz Master Class. This post is about another exercise from that book, exercise 5.8, which is to prove Kantorovich’s inequality. Assume and for non-negative numbers pi. Then where is the arithmetic mean of m and M and is the geometric …

There’s a theorem that’s often used and assumed to be true but rarely stated explicitly. I’m going to call it “the baseball inequality” for reasons I’ll get to shortly. Suppose you have two lists of k positive numbers each: and Then This says, for example, that the batting average of …

A catenary with scale a is the graph of the function f(x; a) = a cosh(x/a) – a. The x and the a are separated by a semicolon rather than a comma to imply that we think of x as the variable and a as a parameter. This graph passes …

Sept. 28, 2020, 4:39 p.m.

There are a couple ways in which a measurement might not be straight. Yesterday I wrote a blog post about not measuring straight toward your target. You’d like to measure from (0, 0) to (x, 0), but something is in the way, and so you measure from (0, 0) to …

Sept. 26, 2020, 6:35 p.m.

This post will start with a motivating example, looking at measuring a room in inches and in feet. Then we will segue into a discussion of contravariance and covariance in the simplest setting. Then we will discuss contravariant and covariant tensors more generally. Using a tape measure In my previous …

Sept. 26, 2020, 2:50 p.m.

Suppose a contractor is measuring the length of a wall. He starts in one corner of the room, and lets out a tape measure heading for the other end of the wall. But something is in the way, so instead of measuring straight to the corner, he measures to a …

Sept. 26, 2020, 1:35 a.m.

The previous post mentioned a Math Overflow question about unexpected mathematical images, and reproduced one that looks like field grass. This post reproduces another set of images from that post. Start anywhere in the complex plane with integer coordinates and walk west one unit at a time until you run …

Sept. 25, 2020, 2:25 a.m.

Math Overflow has an interesting question about unexpected mathematical images. Here’s a response from Payam Seraji that was easy to code up. Here’s the code that produced the image. from numpy import * import matplotlib.pyplot as plt t = linspace(0, 39*pi/2, 1000) x = t*cos(t)**3 y = 9*t*sqrt(abs(cos(t))) + t*sin(0.2*t)*cos(4*t) …

Sept. 24, 2020, 11:35 p.m.

The previous post was a visual introduction to bilinear transformations, a.k.a. Möbius transformations or fractional linear transformations. This post is a short follow-up focused more on calculation. A bilinear transformation f has the form where ad – bc ≠ 0. Inverse The inverse of f is given by The transformation …

Sept. 24, 2020, 1:14 a.m.

This post expands on something I said in passing yesterday. I said in the body of the post that … the image of a circle in the complex plane under a Möbius transformation is another circle. and added in a footnote that For this to always be true, you have …

Sept. 23, 2020, 1:21 p.m.