Coiled logarithmic graph

How many ways can you select six points in the plane so that every subset of three points forms the vertices of an isosceles triangle? One solution is to choose the five vertices of a regular pentagon and the center. It’s easy to verify that this is a solution. The much harder part is to […] The post Solution to a problem of Erdős first appeared...

This post was motivated by an exercise in [1] that says Prove that for the hyperbolic functions … formulas hold similar to those in Section 2.3 with all the minuses replaced by pluses. My first thought was that this sounds like Osborn’s rule, a heuristic for translating between (circular) trig identities and hyperbolic trig identities. […] The...

The vis-viva equation greatly simplifies some calculations in orbital mechanics. It is reminiscent of how conservation of energy can sometimes trivialize what appears to be a complicated problem. In fact, the vis-viva equation is derived from conservation of energy, but the derivation is not trivial. Which is good: the effort required in the of...

The most efficient maneuver for transferring from one circular orbit to another circular orbit of roughly the same size is the Hohmann transfer orbit. It requires two burns: one to leave the initial circular orbit into an elliptical orbit, and another to leave the elliptic orbit for the new circular orbit. If the new orbit […] The post...

Shortly after I started using a MacBook I remapped the keys so that they function the same way on Mac OS, Windows, and Linux. The key in the lower left corner, for example, behaves the same way across operating systems, as does the key to the left of the space bar. Note that I’m going […] The post Rotating MacBook keys first appeared on John D....