# South Dakota Travel Problem

This light-weight problem arose from a newspaper article that had me looking at a Google map of the area near Sioux Falls, South Dakota. What I saw was an excellent example of the Taxicab Geometry, allegedly first considered by Hermann Minkowski, mathematical friend of Albert Einstein. The map configuration was perpetrated by the great Public Land Survey System (PLSS) that originated with Thomas Jefferson and spread from Ohio (more or less) west to the California coast. This scheme overlaid the country basically with a 1 mile x 1 mile square grid of roads, and South Dakota is a prime example.

I first confronted the PLSS doing genealogy research, where the grid became a main method for locating the farms of ancestors. I had no idea it was so extensive. A fabulous book about the system and the history of land surveying in America is Andro Linklater’s Measuring America: How the United States Was Shaped By the Greatest Land Sale in History (2002). I learned all about perches and 17th century English mathematician Edmund Gunter’s survey chain, which became an essential tool for so vast a survey undertaking. Regarding the implications of the PLSS in South Dakota, see the South Dakota Travel Problem.

# Point Set Topology

Probably the most satisfying article I have put together is a recent one on point set topology. An old friend of mine, who studied math and physics in college but ended up getting a doctorate in English, asked me, what was topology? Knowing that there were two main branches of topology (general or point set topology and algebraic topology), I chose to describe point set topology first, especially since it was what I was most familiar with and had worked with most in my graduate work.

The essay turned out to have a surprising structure more like a musical theme and variations. The theme was the geometric series. I found it to be a wonderful medium to show the evolution of ideas (acting as variations) from the early Greeks (Zeno’s Paradoxes) through the development of calculus, decimal expansions of real numbers, to power series, metric spaces, and finally general topologies.

There was an additional benefit to this series of transformations of an initial idea: one of the major aspects of true mathematics became evident, namely, the extension of an idea into new territories that reveal unexpected connections to other forms of mathematics. Treating complicated functions as points in a topological space was a wonderful idea developed over the end of the 19th and beginning of the 20th centuries and became the basis of the field of functional analysis. See Point Set Topology (revised).

(Update 6/3/2021) Slightly revised version.

I happened to review this article and noticed I made a mistake in my integration example.  I have no idea what I was thinking at the time, so I corrected it.  As I reviewed the rest of the article, I noticed a bunch of “typos” that would make the text confusing, so I corrected those as well.  And finally I rephrased wording in a couple of places to try to make things clearer.

# Power of 2 Problem

Virtually the very first “math” problem I got interested in involved a 7th grade homework problem in 2005 that a colleague at work said her son had been given. I ended up commenting and helping on a number of further problems, which gave me some insight into the state of current public school teaching in mathematics. It was both encouraging and discouraging at the same time. I will join the math education commentary at a later date.

The problem was not that bad: What is the largest power of 2 that divides 800! without a remainder? (where “!” means “factorial”, for example, 5! = 5 x 4 x 3 x 2 x 1). I solved it in my usual pedestrian way. I showed it to a friend of mine (an algebraist!) and he of course had a nifty approach. He showed it to a colleague of his at NSF (a physicist) and he had the niftiest solution of all! (Most humbling.) See the Power of 2 Problem.