Topology, vector fields and indexes
This first article appeared in the New York Times. It talks about vector fields and topology, and uses finger prints as an example of a foliation derived from the flow of a vector field on a smooth surface.
You might read the comments too.
Note: the author of the quoted article made a welcome correction:
small point that I finessed in the article, and maybe shouldn’t have: it’s about orientation fields (sometimes called line fields or director fields), not vector fields. Think of the elements as undirected vectors (ie., the ridges don’t have arrows on them). The singularities for orientation fields are different from those for vector fields. You can’t have a triradius in a continuous vector field, for example.
Comment by Steven Strogatz
Our local paper had a nice piece by Brian Gaines on political polls. Of interest to statistics students is the following:
1. Pay little attention to “point estimates.”
Suppose a poll finds that Candidate X leads Y, 52 percent to 48 percent. Those estimates come with a margin of error, usually reported as plus or minus three or four percentage points. It is tempting to ignore this complication, and read 52 to 48 as a small lead, but the appropriate conclusion is “too close to call.”
2. Even taking the margins of error into account does not guarantee accurate estimates.
For example, 52 percent +/- 4 percent represents an interval of 48 to 56 percent. Are we positive that the true percentage planning to vote for X is in that range? No. When we measure the attitudes of millions by contacting only hundreds, there is no escaping uncertainty. Usually, we compute intervals that will be wrong five times out of 100, simply by chance.
Note: a consistent lead of 4 points is significant, but doesn’t mean much for an isolated poll.