---
title: "One Two Three... Infinity"
chunk: 1/2
source: "https://en.wikipedia.org/wiki/One_Two_Three..._Infinity"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:20:41.547970+00:00"
instance: "kb-cron"
---

One Two Three... Infinity: Facts and Speculations of Science is a popular science book by theoretical physicist George Gamow, first published in 1947, but still (as of 2020) available in print and electronic formats. The book explores a wide range of fundamental concepts in mathematics and science, written at a level understandable by middle school students up through "intelligent layman" adults. The book includes many handmade illustrations by Gamow.

== Synopsis ==
The 340-page book has four parts (marked I, II, III, and IV) and eleven chapters. In the preface, the shortness of the last part is attributed to the prior coverage in Gamow's previous books The Birth and Death of the Sun and Biography of the Earth. There are 128 illustrations that Gamow drew, "topologically transformed" from works by "numerous artists and illustrators", thanked by Gamow in the preface. A four-page index is included.  
In 1961 a new edition was published. In its preface, Gamow says that by luck the 1947 edition was "written just after a number of important scientific advances", so that "relatively few changes and additions were necessary". For example, Heinz Fraenkel-Conrat and Robley Williams separated tobacco mosaic virus into lifeless molecules and then recombined them into active virus. A 1965 edition speculated on assembly of a "man-made virus particle" (p. 267).

=== Part I: Playing with Numbers ===
Part I is mainly concerned with expressing large numbers, Georg Cantor and infinity, and the imaginary unit. After disparaging the Roman numeral system for being limited to thousands (M), The Sand Reckoner system of myriads and octades is described. In terms of one-to-one correspondences, in the world of infinity "a part may be equal to the whole". Aleph number zero is described, with aleph one related to points in a plane, and aleph two to curves. (These latter associations are not true unless the generalized continuum hypothesis holds, which Gamow fails to mention.) As for prime numbers, the sieve of Eratosthenes is shown. The Fermat numbers are given and related to primes. Goldbach's conjecture is stated: "Every even number can be written as the sum to two primes." It was an epithet of Gerolamo Cardano that stuck: square roots of negative numbers are imaginary. The Argand diagram is displayed, and multiplication by i rotates the diagram counter-clockwise by a right angle. The study of complex numbers then deviates into treasure hunting.

=== Part II: Space, Time & Einstein ===
Part II opens with "unusual properties of space" and touches on "transformation of coordinates" and polar coordinates before taking up topology. Euler's polyhedral formula for polyhedrons projected onto a sphere is illustrated and proven. Modification of the formula for the doughnut (torus) and other holed surfaces is mentioned. The four-color problem (solved 1976) is explained, and the fact that seven colors are necessary and sufficient on the doughnut. Sphere eversion is described in terms of two separate wormholes filling an apple. Reminding the reader of gastrulation in embryonic development, and interpreting a person as a doughnut, one of the illustrations depicts a person turned inside-out. The chirality property of three-dimensional space is missing on the Möbius strip and Klein bottle. 
Turning to the temporal extension of space, there are worldlines and in the world-bars of beings "most of the fibers stay together as a group". Rømer's determination of the speed of light is recounted, leading to the lightyear and the light-foot (1.1×10−9 seconds) as space-time equivalents. Then space-time intervals are measured with the Pythagorean theorem modified with a negative term for the square of the temporal separation. A bus going down Fifth Avenue in New York City represents a moving point of reference, and requires a "rotation of the four-dimensional axis-cross", with the separation "invariant with respect to rotation". Considering the luminiferous ether, the failure of the Michelson–Morley experiment in 1887 is described as a blow to classical physics and absolute space and time. Speculating  on future high-velocity travel, a trip after breakfast to Sirius to land on a planet for lunch and the return to Earth for dinner is described. Curvature of starlight beams was confirmed with photographs taken at Príncipe by a 1919 solar eclipse expedition. Given that the average curvature of the universe may be positive, negative or zero, the mass distribution may provide a resolution.

=== Part III: Microcosmos ===