Parabolas and focal points II

Yesterday I wrote about parabolic mirrors. I pointed out that parallel rays hitting a parabola typically do not reflect back to a focal point. That happens only when the rays are parallel to the axis of the parabola. Then my friend Dan sent me an email encouraging me to look at the envelope of the…

Parabolas and focal points

A colleague from another department stopped by my office this morning with a question about parabolas and parabolic mirrors. In the simplest terms, his question was the following: When parallel rays reflect off a parabola, do they always converge to a focal point? As we can see in the diagram below, rays that are parallel…

The Japanese Theorem

[Update: I’ve written quite a bit more about this theorem since 2009. See this page for more details.] I’ve been playing with GeoGebra for the last few days. As an exercise I decided to create applets to demonstrate the extremely beautiful Japanese Theorem. The first appearance of the Japanese theorem was as a Sangaku problem….

Pentagrams and quartic polynomials

I’m still enjoying my new-found freedom that comes with the end of the semester. I’ve gotten some research done and I’ve been able to catch up on some reading. One article that I found particularly interesting was “Quartic Polynomials and the Golden Ration,” by Harland Totland, from the June 2009 issue of Mathematics Magazine. This…