The Mandelbrot set M defined by:
P-sub-c: C --> C, where C is the complex plane
Where:
P-sub-c: z --> z^2+c, for some c in C, typically starting at z=0
(P is a map from the complex plane to itself given by the complex quadratic polynomial z^2+c iterated from z0=0, z1=0^2+c=c, z2=c^2+c, z3=(c^2+c)^2+c and continued to find which values for c cause the iteration to eventually increase unbounded to infinity and which are finitely bounded over infinite iterations. This is all super simple to type into computer software, as in two lines of instruction)
M is the set of complex arguments for c that are bounded. The boundary forms one of the mind-blowing images ever conceived. It is globally self-similar with increasing complexity, and zooming deep into a fractal provides the best visual tool for appreciating and abstracting infinity itself and infinite complexity as well as the beautiful complex order and symmetry hiding inside said complexity. And it arises from a function that is understandable to a large extent to high school students, at least the basics of the complex plane and functions with complex arguments. The resulting complexity has to be seen to believe, and it is this complexity-from-simplicity that in many ways provides deep insight into nature, complex systems and structures, and the universe itself...
Fractals and their applications and implications, as analytical tools for chaotic dynamical systems and nonlinear dynamics, have revolutionized modern scientific thought.
If the entire book was written in the style of the final chapter, then I imagine something that could've been a five-star read. The language and voice is much more organic, thoughtful, reflective, and illuminating. The above description of the now-ubiquitous (rightly so) Mandelbrot set and his discovery and work in the theory of fractal geometry (comprising the latter half of the book) and its innumerable applications defines his deeply influential career. I wanted less of Benoit merely stating these things were major discoveries and more explaining more clearly and descriptively why they matter so much, and the same holds for the many depictions of contemporary geniuses encountered in his life focusing on their personalities and that they did great work but what that work was we are rarely given more than a superficial mention. This is straight-up autobiography, and even though Mandelbrot is one of the most influential (what we would now call) mathematical scientists of the last half-century plus, and he found himself in several fascinating environments surrounded by geniuses, he's simply not very good at writing about them. Memoirs are not a real thing, they are typically bad autobiographies with a supposed slight emphasis on reflection and possibly covering a shorter period of time than a straight historical, event/fact-based narrative. This is too often a boring, literal recounting of the chronological story of someone's life with some ancillary commentary or digression. Granted, his life was incredible and entirely unique, and by all accounts he was a wonderful human being (sadly he passed away recently though remarkably remained active in the scientific community his entire life), so it's frustrating that he frequently makes it dull to read about. Oh well, I've read parts of his "The Fractal Geometry of Nature" and will read the whole thing soon due to transcendent brilliance, and this book is such a quick read that I don't regret picking it up.