Mandelbrot Butterfly Identification – Classifying Misiurewicz Points of the Second Type

Abstract

My fortuitous discovery more than 30 years ago, of a figure I called the Mandelbrot Butterfly, came out of efforts to find a calculational shortcut.  This figure was seen as relevant to Cosmology and to Physics in a broad way, but much study was required to understand why.  This relevance is now seen to be related to the Misiurewicz points in M, most especially to those of the second type which are hard to find otherwise, but easy to identify and to label by their pre-period using the Butterfly figure as a guide.