Here’s something I only recently realized: the mobius band arises naturally when you consider the space of all lines in the plane.
To begin with, the open mobius band (i.e. the usual mobius band without its boundary) is the same as the space of all lines in the plane. The easiest way to see that is Continue reading “The Mobius Band Revisited”
I was thinking through a probability situation the other day, and I noticed I was making an unspoken assumption, namely that if two events and are independent, then their complements and are independent.
At first I thought that this probably wasn’t true, because I’ve never seen that particular result explicitly stated in any probability or statistics book I’ve ever read (and I’ve read a few over the years), and because of the fact that human intuition for probability is notoriously unreliable (e.g. even Paul Erdos got the Monty Hall problem wrong when he first heard it).
Well, it turns out to be correct. In fact, a stronger result is true, namely Continue reading “Complementary Independence”
This is the latest incarnation of me (Walter Vannini) writing about mathematics and related topics. Previous versions were the essays on my website Continue reading “Introduction”