Iterations and Reflections

The title of this weblog has two meanings. The first mathematical and the second is intellectual. Mathematically speaking, an iteration is the act of replacing a value by some function applied to that value. For example, the iteration of is the way that you compute the Mandelbrot set. You start with a complex number and you iterate starting at 0 until becomes large or you get tired of iterating. In the latter case you say that is part of the Mandelbrot set, and you plot in the plane in black, as in the picture on the left.

A reflection is a reflection in the geometric sense, of reflection across some line of symmetry. Usually we are interested in a group of reflections and in pictures that are invariant under a group of reflections, such as the Circle Limit drawings of Escher, like the one shown on the right.

I also want to use the two words metaphorically. An iteration is one of many repeated attempts; a reflection is way of stepping back and looking at something from a new perspective. Just as the title has two meanings, the blog has two motivations. One is to explain some mathematics to both a general and a mathematical audience, and the other is to iterate and reflect on a large variety of subjects, especially ones in science, politics and religion. It is probably in the latter two subjects that I will not only inform and inspire, but provoke and offend.