"Möbius and his Band: Mathematics and Astronomy in Nineteenth-Century Germany" John Fauvel,Robin Wilson,Raymond Flood, editors

"Simple shadow projection from a point light source L casts a picture drawn on one translucent screen onto a second screen.  The image of a point is a point, and the image of a line is a line.  The image of two intersecting lines is usually two intersecting lines, but need not always be so."

Jeremy Gray, "Möbius's Geometrical Mechanics"

I can remember the first time I made a Möbius strip.  My grandmother liked to give me mathy kinds of books, puzzle books mostly, and I would devour them.  One book that tickled me pink as an adolescent was "The Metric Book ... of Amusing Things to Do."  The characters of the book were the Metric Mice and they taught about the metric system through activities, like recipes and games.  It came with its own meter stick made of card stock.  I still have the book (and the meter stick, surprisingly).  One of the activities is to make a Möbius strip, and of course, to cut it in various ways to get connected and disconnected loops.

Other than some topics in textbooks along the way, my next connection with Möbius was in graduate school.  I wrote an expository dissertation titled:   Intersection Numbers:  An application of algebraic geometry to computer-aided geometric design.  One topic I covered as I prepared for this work was Barycentric coordinates.  Although they only played a small part of that work, I have had undergraduates do various things with projective space and Barycentric coordinates since then as part of their senior research projects in mathematics.  It is through that connection that I ran across this book, because the popularization of Barycentric coordinates and an understanding of projective space is due to Möbius.  I have found the chapter by Jeremy Gray particularly useful, and the conversational tone has been palatable to some, if not all, of my students.

The book is really a history of math and astronomy in Germany in the 1800s.  The six chapters do not focus on Möbius, per se, but rather his band, that is, the scientists and mathematicians and the developments they made before and during his time.  With little introduction, the book begins with a chapter by John Fauvel that places Möbius in history.  Gert Schubring writes about the development of the mathematics community in Germany in the Nineteenth Century, which I was surprised to learn had lagged behind Britain and France at the time, especially since I think of Germany of a mathematical power.  Chapter 3 is a history of astronomical practice at the time told by Allan Chapman, and by far my favorite chapter.

The first three chapters are non-mathematical in nature, but the next two discuss mathematics specifically, and the casual reader might find this daunting.  Bear with it for the history and significance.  Jeremy Gray discusses Möbius's most important work, the Barycentric calculus and Norman Biggs discusses Möbius's role in the development of topology as a mathematical discipline.  In the last chapter, Ian Stewart writes about the mathematical and scientific legacy of Möbius.

Even though this book had been on my shelf at school for a long time, I had only pulled it down for reference and to lend to students.  I'm glad I got a chance to read it completely and was pleased to read about these episodes in the history of mathematics.