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For reviews that make pretty round numbers in count from when I started, I like to re-read something I enjoyed before I started writing reviews and then review it. This is the 500th book review that I've written, which seems like a good time to re-read and review the book that's probably had the most influence on me of any single book I've read.

One Two Three... Infinity, subtitled Facts and Speculations of Science, was originally written by George Gamow in 1947 and then revised somewhat in 1961. It's an amazing tour of some of the more interesting bits of physics, with a bit of mathematics and biology thrown in, covering in some depth more separate topics in under 350 pages of any non-fiction book I can recall. I'm not sure how old I was when I first read this book, but I doubt I was more than eight and I know for certain that I was younger than twelve. I read and re-read it many times up through college and even gave copies of it to friends. It's been republished in various editions and appears to still be in print from Dover (the second copy that I own), but for the sake of this review, I re-read my original Bantam paperback. It's ragged, the cover is a bit torn, and it's full of underlining where I marked particularly interesting sections. Seeing what I decided to underline twenty years ago is part of the nostalgia.

Even though it's been more than a decade since I last read this book, I still know it almost by heart. It opens with a discussion of mathematics, specifically very large numbers, with two examples of the power of exponential growth that I still remember vividly. The first is about a vizier who (in legend) invented the game of chess. When told by a king that he could have any reward he wished, he asked for one grain of wheat for the first square of a chessboard, two for the second, four for the third, and so forth. The king happily agreed, only to discover that the total amount of wheat would be more than the entire world's wheat production for two thousand years. The second is a legend of a temple containing a game of Towers of Hanoi with 64 disks which, when complete, will mean the end of the world. At a move per second, this would take significantly longer than the estimated age of the universe.

One Two Three... Infinity is full of stories like that, and charming examples of whatever point Gamow is discussing, all illustrated with wonderful and often whimsical line drawings by the author. (I'm particularly fond of the drawing of creation of a two-dimensional projection of a horse, featuring a horse lying in a giant mechanical press labelled "wrong" and a shadow projection labelled the "right.") Sometimes the illustrations are just there for a smile, but often they provide a useful and sometimes surprising clue for how to visualize some bit of physics. There are more precise and more detailed pictures in a modern physics textbook, but Gamow's drawings have a charm that's hard to match.

After large numbers, Gamow discusses infinities (one of my favorite portions of the book), including how to compare sizes of infinities. He then moves on to imaginary numbers and uses them to solve a wonderful pirate treasure map problem. From there, he goes into topology, a discussion of the map coloring problem, chirality (although he doesn't use that word), Möbius strips, and Klein bottles. Then it's on to time as a dimension, relativity (presented as a coordinate transform using imaginary numbers), and a fascinating discussion of the history of measurements of the speed of light. This is sprinkled with several classic physics limericks I can still recite by heart.

And that's only half the book. Still remaining is a wonderful explanation of the history of our understanding of atomic structure, including classic experiments to determine the size of a molecule. Gamow walks through discoveries about atomic structure up to the level of the proton, neutron, and electron model, with some discussion of the at-the-time theoretical neutrino. (This book predates the theory of quarks and some of the more exotic particles, and Gamow is incorrectly optimistic that nucleons, electrons, and neutrinos are the only fundamental particles.) Mixed in is a bit of quantum physics, although not more than a basic introduction to Heisenberg uncertainty and wave/particle duality in light. Gamow spends more time on radioactivity and nuclear reactions, including the solar fusion cycle. And I haven't yet mentioned the chapter on cells, which moves into a discussion of genetics, or the discussion of entropy that includes an excellent introduction to probability and even cryptographic analysis of a simple substitution cipher. Gamow skips, for this book, most of the detailed discussion of astrophysics, since he wrote other books on that topic, but he concludes with a brief discussion of planetary formation theories, the main sequence of stars, the Big Bang, and the expanding universe. All crammed into a short and very readable book.

One Two Three... Infinity was my foundational course in both physics and mathematics. I relied on things I'd learned from Gamow to understand physics well into third-semester college physics, and his introduction to infinities and Cantor's method for comparing them was the foundation of my understanding of that topic even in graduate school. Gamow doesn't shy away from equations or high-school algebra, although he does avoid calculus entirely. But because of the wonderful diagrams and lucid text, even people who don't want to understand the equations can pick up a great deal from this book.

Unfortunately, George Gamow died in 1968 and One Two Three... Infinity was last revised in 1961. It holds up remarkably well for a cutting-edge science book that's over fifty years old, but the age is showing enough that I can't quite recommend it unaccompanied as an introduction. Some of the changes are minor (computer proof of the four-color map coloring theorem, the recent proof of Fermat's Last Theorem), but particularly in atomic physics and quantum mechanics, it needs a great deal of supplementary material to be brought up to date. Gamow still uses essentially a solar system model of the atom and has an only cursory explanation of the strong force and no mention of the weak force, and that plus the absence of quarks and the fermion/boson Standard Model makes the section on the atom useful mostly as a lesson in the history of science (although it's an excellent history lesson). The section on cells is similarly missing subsequent discoveries about DNA and gene sequencing, and the brief section on astrophysics lacks black holes, the Chandrasekhar limit, neutron stars, pulsars, quasars, dark matter, and the various controversies and questions around the expanding universe. Relying only on this book now would give someone a dated and somewhat skewed picture, with in places outdated terminology.

That's a shame, since Gamow remains one of the best science writers I've ever read, and One Two Three... Infinity is a brilliant bit of broad-ranging science education. It's a hard book for me to review since it's so deeply embedded in how I think about science and I've read it so many times. Even after the nth re-read, it can still suck me in. The sections on mathematics hold up the best, and I will recommend Gamow's explanations of imaginary numbers and infinities to anyone.

If you're already familiar with physics but would like to see basic concepts explained from a different angle, usually following the history and evolution of scientific understanding, I can still recommend One Two Three... Infinity. If you're not already familiar, it sadly can no longer be relied upon and needs to be supplemented by more recent material, but I think it could still be helpful with concepts that can require multiple explanations from different angles before they make sense. And regardless of whether it's still useful for its original purpose today, I will always be grateful to Gamow for sparking my interest in physics and mathematics and giving me a solid foundation for all my subsequent scientific reading and education.

Reviewed: 2009-11-06

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