## Make Your Own Pythagorean Cup

My parents recently went to Greece. They brought me back a souvenirâ€”a practical joke cup called a “Pythagorean cup.” The legend behind the cup is that Pythagoras or one of the Pythagoreans invented this cup to prevent gluttony. The vessel looks like a cup with an odd pillar in the center. When you fill it…

## Why do mirrors reverse right and left but not up and down?

[I apologize to those of you who have been reading my blog for more than a year. I’m reposting something I wrote last year at this time. I was then, and am now, teaching Calculus III, and we just finished discussing the cross product. I ended the conversation by telling my classes how the cross…

## 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…

## Why do mirrors reverse right and left but not up and down?

Stand in front of a mirror and hold up your right hand. The person standing in the mirror holds up her left hand. Why is that? Why does a mirror reverse left and right? After all, it does not reverse up and down. Before we answer that question, we have to ask a more basic…