This is a book of majesty and genius. What it does is enable the reader to see the whole range of the cosmos from the atom to the universe in a coherent way. With his theory, Dr. Nottale has accomplished something truly great. Even if his vision is ultimately deemed a failure, one has to regard the attempt as a magnificent success.

I'm on my second (slower) reading of the book so what is this book about? At root it's a program to unify biology and physics (and you thought Einstein was ambitious!) It's difficult to imagine a more challenging goal (one Nottale himself thinks it will take centuries to complete.) The central question is understanding morphogenesis, i.e. how order comes about, how structure is formed resulting in things, assemblies (e.g. galactic clusters), organisms; how form comes into being and evolves. Studying the scale of these phenomena at each level of their existence marks the beginning of a far deeper science than what we have had before. One might label it Generalized Quantum Mechanics.

Scale is the key. We've all noticed the similarities in nature at various levels: e.g. an overhead picture of a hurricane vs. a spiral galaxy. It's clear that at some point Nottale encountered Mandlebrot's seminal work, Fractals: Form, Chance, and Dimension (1977) (Nottale would have been 25 at the time) and a revelation occurred. Fractals, of course, are quite familiar now. They reside in the realm of continuous, non-differentiable functions, so bizarre that mathematicians once referred to them as "monsters." But familiarity can breed fondness and Nottale concluded that if physics could be extended into that realm, one far larger than the domain of differentiable functions -- specifically second-order differential equations which have been the mainstay of physics since Newton -- fractals would be in their natural home. And far more could be done with the problems of structure, shape, and form. The fit seemed to be natural, though the mathematics more difficult. Scale and its relativization (keep in mind there are no absolute scales) could be studied in their natural home.

What Nottale discovered, after the mathematical preliminaries were in place and explaining them takes up a good third of the book, was that generalized Schrodinger equations are the key tool in understanding the emergence of structure. They are the engines of order, call it "negentropy." Each of the levels/scales from the atom up (at jumps of 10 to the 4th - 5th power) yields schrodinger-type analyses and associated equations. In other words, physics beyond micro-physics -- the Quantum domain -- is recast into something kind of like QM, but with striking differences. The good news is that the strange behaviors of quantum mechanics (the woo-woo stuff that has so cluttered our thinking regarding a consistent interpretation of QM) would not for the most part be carried over into these new and extended domains, though there would still be quantization. The better news is that new phenomena would be predicted, new aspects of nature would be revealed (e.g. Helium II).

This is perhaps most strikingly illustrated in Nottale's work on the solar system(s) where he shows that this macro-system is, unlike the atom, quantized by velocity. The new velocity quantum-unit is 144.7 kilometers/second which appears to be a new universal constant. Orbits are either in some integer multiple or integer fraction of this value. He is then able to derive the spread of possible planetary orbits -- given the size of the central sun. And as noted this is for all solar systems. Building upon observational work that has only been possible in the past 20 years, he has been able to confirm these predictions.

It's not that all these ideas are new nor that Nottale is some sort of solitary genius working alone. Some of the ideas, like applying the Schrodinger equation to the solar system, go back to the 1920's. [They weren't successful.]

Not meaning to detract from his collaborators, but I think it can be said fairly that Nottale is the goto guy in these explorations. Please understand that other people are working in the program as well (for example, a few years back the Schrodinger equation was applied to the problem of rogue waves). But what is crucial is that these these type of problems are being addressed as part of a scientific research program, one that begins with the underlying fractal geometry (about which there remain many outstanding questions) and builds through each successive level/scale, extending and applying the scale relativity physics.

The book ultimately extends these scale relationships from galactic clusters to the structure of flowers, the beginnings of a true science of biology. Unfortunately, the life sciences section in the book comes across as rushed.

Note: Since Scale Relativity and Fractal Space-Time (2011) is essentially an overview of his work, my hope is that he and his colleagues cover much more regarding the morphogenesis of life in Schrodinger`s Flowers (Des fleurs pour Schrödinger: La relativité d'échelle et ses applications - 2009). That book, however, has not been translated into English. [Why?]

Scale Relativity and Fractal Space-Time is not an easy read, yet if you aced Advanced Calculus, you can keep up with it. Even if you find yourself struggling, you should still be able to see and appreciate much that is new.

Whatever your feelings regarding the math, it is my hope that people will be drawn to the sheer wonder of this book. That they will find not only the beauty of Nottale's grand vision, but the thrill of seeing things in a radically different way. To see the solar system or a flower as an extended quantum system is nothing short of astounding. I urge everyone who is interested in the beauty and mystery of the world to invest time in this book. Granted, it will take more than one reading to absorb it's many details, but there is so much to awe and inspire that I am confident the dedicated reader will be richly rewarded.

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