100 books like Fractals

This isn’t a book about biology, but it makes the list anyway. Genes, chemistry, and mechanics stretch our conception of life. Still, there’s more needed to make sense of it all: the emerging understanding that dynamic interactions, feedback loops, and a bit of randomness make life alive. We need more books that lay this out; this was one of my motivations for writing my book.

Philip Ball’s very recent How Life Works seems masterful, and I’m sure it would make this list if I were writing a month or two later, after I’ve finished reading it. But since I can’t list either my book or his, I’ll list Gleick’s book, one of the best popular science books ever. It dives into the science of order and chaos, from the turbulence of Jupiter’s red spot to the unpredictability of weather, with a bit of biological pattern formation in the mix.… show more.

While I’ve spent much of my career debunking the Random Walk Theory and the Efficient Market Hypothesis, I believe you need to understand where we’re coming from in order to develop new ideas.

Cootner’s anthology contains all of the important early writings that became Capital Market Theory. You will find Bachilier’s original paper from 1900 postulating using statistics to analyze markets. Mandelbrot’s original paper suggesting that markets are not a random walk is here, too. 

It’s an important time capsule of the development of market theory. I’ve read it extensively because you can’t criticize widely held ideas unless you know them yourself.

Fractals are a combination of chance and rules, randomness and order. This book gives us multiple examples of how important randomness and uncertainty are to evolution, adaptability, and innovation.

This book is not about the markets, but it does show us how the element of uncertainty that cannot be eliminated from investing is crucial to the functioning of markets and other non-linear dynamical systems. Since risk and uncertainty are closely tied, understanding how chance functions in the development of markets will help in the development of market strategies.

The Austrian School of Economics has been completely non-mathematical. Yet their concepts have direct analogies to complexity theory. Reading this book gives you fundamental economic ideas that tie back to the mathematical foundations of complexity, chaos, and fractals.

When dealing with markets, which are made up of people, after all, I have always found that basing your ideas on behavior rather than pure mathematics is always the way to go. Otherwise, you will not know when the markets have changed, and your ideas may no longer work.

This book also brings us back to the idea of cycles and explains them in an intuitive way. O’Driscoll and Rizzo also write well, so this isn’t a dry academic overview. It is a book that gives you both theory and intuition.

Visual representations are not the only pathway to creative acts in art and science, but they are responsible for large territories of creativity – including, and surprisingly, the mathematical. Arthur Miller shows how ‘seeing the unseen’ becomes possible from atoms to the conservation of energy in science, and from modernism to cubism in art. The book itself is as visually striking as its contents and helped me to think through why the visual metaphor – ‘Oh, I see!’ – becomes the standard description of the moment of insight.

This is a sequel to Alex Bellos's bestseller Alex's Adventures in Numberland, but more focused on applications of mathematics to the real world, especially through physics. Many of these were known to me, particularly when they involved calculus, but I greatly enjoyed Alex's distinctive and novel way of putting across sophisticated ideas, in part by interspersing them with personal interviews with mathematicians of all kinds.  

This book is full of beautiful puzzles on a mathematical topic called pursuit evasion. Its author Paul Nahin has written tens of books in physics and mathematics.

Nahin's writing can be described as a captivating reading experience pulling readers into his world like a whirlpool. His appreciation of mathematics, physics, and the people who discover them is unmatchable. It seems like the physics of motion and the mathematics of calculus is inseparable, as can be witnessed in the book.

The calculus used in this book is heavy. Nevertheless, Nahin makes his readers fall in love with this big beast. Almost every puzzle in this book is aesthetically beautiful and gives readers a deep sense of satisfaction. My favorites include Lady in the Lake, and Lion and Man.

I love this book! Of course, it teaches the mathematics of the Lagrangian and Hamiltonian approaches to classical mechanics. But its real uniqueness lies in how the author explains the basic concepts of the subject with a constant emphasis on intuitive physics. It was only after reading Lanczos that I finally understood what variational calculations were really about. Ditto for virtual work, non-holonomic constraints, Liouville’s theorem and Hamilton-Jacobi theory. I barely noticed that I was learning about the history and philosophy of classical mechanics at the same time.

MIT scientist Tegmark directly connects contemporary physics and cosmology with our story of the Presocratic natural philosophers.

In his view the universe is not only described in the language of mathematics, it is a huge purely mathematical object. This was precisely the view of the Pythagoreans. In the course of expounding his theory of a mathematical universe, Tegmark brings the lay reader up to date on the latest developments in natural philosophy (aka theoretical physics and cosmology) and demonstrates their continuity with those of their ancient predecessors.   

Are you still here? Good. Because by now you are probably reading to tackle some university-level courses in quantum physics, right? Well, with your background in pop quantum physics all you need to get there is a little more abstraction. So, if you have the stomach for a bit of mathematics, Quantum Mechanics: The Theoretical Minimum by Leonard Susskind is your ticket to the big show! (Don’t say I didn’t warn you about the math, though.)