John    Someone said a technique is a trick that works twice. I wanted to see if I could get anything interesting by turning the trick in the previous post...

John    Suppose you have an arc a, a portion of a circle of radius r, and you know two things: the length c of the chord of the arc, and the length b of the...

John    The previous post looks at the nonlinear pendulum equation and what difference it makes to the solutions if you linearize the equation. If the...

John    There’s a nice formula for the nth derivative of a product. It looks a lot like the binomial theorem. There is also a formula for the nth derivative...

John    The equation of motion for a pendulum is the differential equation where g is the acceleration due to gravity and is the length of the pendulum. When...

John    The previous post gave a simple and accurate approximation for the smaller angle of a right triangle. Given a right triangle with sides a, b, and c,...

John    Suppose you have a right triangle with sides a, b, and c, where a is the shortest side and c is the hypotenuse. Then the following approximation from...

Wayne Joubert    AI coding agents improved greatly last summer, and again last December-January. Here are my experiences since my last post on the subject. The models...

John    A few days ago I wrote a post on Newton’s diameter theorem. The theorem says to plot the curve formed by the solutions to f(x, y) 0 where f is a...

John Cook    The previous post looked at the FP4 4-bit floating point format. This post will look at another 4-bit floating point format, NF4, and higher...

John Cook    In ancient times, floating point numbers were stored in 32 bits. Then somewhere along the way 64 bits became standard. The C programming language...

John Cook    Let f(x, y) be an nth degree polynomial in x and y. In general, a straight line will cross the zero set of f in n locations [1]. Newton defined a...

John    If you know the distance d to a satellite, you can compute a circle of points that passes through your location. That’s because you’re at the...

John    The Wikipedia article on modern triangle geometry has an image labeled Artzt parabolas with no explanation. A quick search didn’t turn up anything...

John    Andrzej Odrzywolek recently posted an article on arXiv showing that you can obtain all the elementary functions from just the function and the...

John    The date of Easter The church fixed Easter to be the first Sunday after the first full moon after the Spring equinox. They were choosing a date in...

John    Today is Orthodox Easter. Western churches celebrated Easter last week. Why are the Eastern and Western dates of Easter different? Is Eastern Easter...

John    There’s a lot of mathematics just off the beaten path. You can spend a career in math and yet not know all there is to know about even the most basic...

John    Saw a post on X saying that the latitude of the Pyramid of Giza is the same as the speed of light. I looked into this, expecting it to be...

John    I recently ran across a post on X describing a process for creating a random fractal. First, pick a random point c inside a hexagon. Then at each...

John    I recently found out about Andrica’s conjecture: the square roots of consecutive primes are less than 1 apart. In symbols, Andrica’s conjecture says...

John    Last week I wrote about the orbit of Artemis II. The orbit of Artemis I was much more interesting. Because Artemis I was unmanned, it could spend a...

John    Landauer’s principle gives a lower bound on the amount of energy it takes to erase one bit of information: E log(2) kB T where kB is the Boltzmann...

John    The best way to run AI and remain HIPAA compliant is to run it locally on your own hardware, instead of transferring protected health information ...

John    This post will look at the problem of updating an average grade as a very simple special case of Bayesian statistics and of Kalman filtering. Suppose...

John    I used the term perilune in yesterday’s post about the flight path of Artemis II. When Artemis is closest to the moon it will be furthest from earth...

John    I was looking through an old geometry book [1] and saw a hyperbolic analog of Napier’s mnemonic for spherical trigonometry. In hindsight of course...

John    The Artemis II mission launched yesterday. Much like the Apollo 8 mission in 1968, the goal is to go around the moon in preparation for a future...

John    Pentagonal numbers are truncated triangular numbers. You can take the diagram that illustrates the nth pentagonal number and warp it into the base of...

John    I’m skeptical that quantum computing will become practical. However, if it does become practical before we’re prepared, the world’s financial system...