# One Two Three... Infinity > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/One_Two_Three..._Infinity > Markdown URL: https://mediated.wiki/source/One_Two_Three..._Infinity.md > Source: https://en.wikipedia.org/wiki/One_Two_Three..._Infinity > Source revision: 1332124017 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) 1947 book by George Gamow One Two Three... Infinity First edition Author George Gamow Illustrator George Gamow Language English Subjects Science, mathematics Published 1947 (Viking Press) Publication place United States Media type Print Pages 340 ISBN 978-0486256641 LC Class Q162.G23 ***One Two Three... Infinity: Facts and Speculations of Science*** is a popular science book by theoretical physicist [George Gamow](/source/George_Gamow), first published in 1947, but still (as of 2020[\[update\]](https://en.wikipedia.org/w/index.php?title=One_Two_Three..._Infinity&action=edit)) 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](/source/Middle_school) students up through "intelligent layman" adults.[1] 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](/source/Heinz_Fraenkel-Conrat) and [Robley Williams](/source/Robley_Williams) separated [tobacco mosaic virus](/source/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](/source/Georg_Cantor) and infinity, and the [imaginary unit](/source/Imaginary_unit). After disparaging the [Roman numeral](/source/Roman_numeral) system for being limited to thousands (M), *[The Sand Reckoner](/source/The_Sand_Reckoner)* system of [myriads](/source/Myriad) 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](/source/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](/source/Generalized_continuum_hypothesis) holds, which Gamow fails to mention.) As for [prime numbers](/source/Prime_number), the sieve of Eratosthenes is shown. The [Fermat numbers](/source/Fermat_number) are given and related to primes. [Goldbach's conjecture](/source/Goldbach's_conjecture) is stated: "Every even number can be written as the sum to two primes." It was an epithet of [Gerolamo Cardano](/source/Gerolamo_Cardano) that stuck: square roots of negative numbers are *imaginary*. The [Argand diagram](/source/Argand_diagram) is displayed, and multiplication by *i* rotates the diagram counter-clockwise by a right angle. The study of [complex numbers](/source/Complex_number) then deviates into [treasure hunting](/source/Treasure_hunting). ### Part II: Space, Time & Einstein Part II opens with "unusual properties of space" and touches on "transformation of coordinates" and [polar coordinates](/source/Polar_coordinates) before taking up [topology](/source/Topology). [Euler's polyhedral formula](/source/Euler's_polyhedral_formula) for polyhedrons projected onto a sphere is illustrated and proven. Modification of the formula for the doughnut ([torus](/source/Torus)) and other holed surfaces is mentioned. The [four-color problem](/source/Four-color_problem) (solved 1976) is explained, and the fact that seven colors are necessary and sufficient on the doughnut. [Sphere eversion](/source/Sphere_eversion) is described in terms of two separate wormholes filling an apple. Reminding the reader of [gastrulation](/source/Gastrulation) in embryonic development, and interpreting a person as a doughnut, one of the illustrations depicts a person turned inside-out. The [chirality](/source/Chirality_(mathematics)) property of three-dimensional space is missing on the [Möbius strip](/source/M%C3%B6bius_strip) and [Klein bottle](/source/Klein_bottle). Turning to the temporal extension of space, there are [worldlines](/source/Worldline) and in the world-bars of beings "most of the fibers stay together as a group". [Rømer's determination of the speed of light](/source/R%C3%B8mer's_determination_of_the_speed_of_light) is recounted, leading to the [lightyear](/source/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](/source/Luminiferous_ether), the failure of the [Michelson–Morley experiment](/source/Michelson%E2%80%93Morley_experiment) in 1887 is described as a blow to classical physics and [absolute space and time](/source/Absolute_space_and_time). Speculating on future high-velocity travel, a trip after breakfast to [Sirius](/source/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](/source/Pr%C3%ADncipe) by a [1919 solar eclipse expedition](/source/Eddington_experiment). Given that the average curvature of the universe may be positive, negative or zero, the mass distribution may provide a resolution. ### Part III: Microcosmos Mendeleev flower-style periodic table Part III is the longest (150 pages) and begins with the "descending staircase" and the [classical elements](/source/Classical_element). "Plants take the largest part of the material used in the growth of their bodies ... from the air." [Rust](/source/Rust) is oxidation of iron. The question "How large are the atoms?" calls for an experiment to obtain an oil film just one molecule thick. "1 cu mm of oil can cover 1 sq m of water." The [law of definite proportions](/source/Law_of_definite_proportions) is stated in [plain English](/source/Plain_English) (p. 123) as a "fundamental law of chemistry". The molecular structure of matter was uncovered with [molecular beams](/source/Molecular_beam) by [Otto Stern](/source/Otto_Stern), and [Lawrence Bragg](/source/Lawrence_Bragg) invented "atomic photography" with X-rays. The section "Dissecting the atom" begins by considering oxygen to be doughnut-shaped, fitting the atoms of hydrogen forming water. Dismissing the notion, Gamow asserts that atoms are "complex mechanisms with a large number of moving parts". Through [ionization](/source/Ionization), and reference to [J. J. Thomson](/source/J._J._Thomson), the [electron](/source/Electron) is introduced, having mass 1/1840 of the mass of a hydrogen atom. The [Rutherford model](/source/Rutherford_model) of the atom, an analogy to the [Solar System](/source/Solar_System), is supported with reference to the percentage of mass at the center: 99.87% for the Sun and 99.97% for the nucleus. Gamow's version of the [periodic table of the elements](/source/Periodic_table_of_the_elements) uses flower petals with stems at the inert gasses. The "utmost precision" of [celestial mechanics](/source/Celestial_mechanics) is contrasted with the [quantum of action](/source/Quantum_of_action), which leads to the [uncertainty principle](/source/Uncertainty_principle). [Diffraction](/source/Diffraction) phenomena not explicable with geometric optics necessitated the wave mechanics of [Louis de Broglie](/source/Louis_de_Broglie) and [Erwin Schrödinger](/source/Erwin_Schr%C3%B6dinger). In the chapter "The Riddle of Life" the [states of matter](/source/States_of_matter) in an automobile body, engine, and radiator are also present in living systems, but homogeneity of [biological tissue](/source/Biological_tissue) is of a different sort. A human is estimated to have more than hundreds of thousands of billions of [cells](/source/Cell_(biology)). To eat, grow, and multiply are posited as life characteristics. Dismissed are crystal accretion in a super-saturated solution, and the molecular reaction - 3 H 2 O + 2 CO + C 2 H 5 OH ⟶ 2 C 2 H 5 OH + 3 O 2 ⋅ {\displaystyle {\ce {3 H2O + 2 CO + C2H5OH -> 2 C2H5OH + 3 O2 .}}} On the other hand, [virus](/source/Virus) reproduction is the "missing link" between non-living and living organisms. The eight [chromosomes](/source/Chromosome) of *[Drosophila melanogaster](/source/Drosophila_melanogaster)* are acknowledged for their contribution to science. Growth by [mitosis](/source/Mitosis) and reproduction by [meiosis](/source/Meiosis) with [gametes](/source/Gamete) performing [syngamy](/source/Syngamy) show the function of chromosomes. Growth and accretion are started with [blastula](/source/Blastula) and [gastrula](/source/Gastrula). ### Part IV: Macrocosmos Aristotle's *[On the Heavens](/source/On_the_Heavens)* founded [cosmology](/source/Cosmology). [Earth's circumference](/source/Earth's_circumference) was found by [Eratosthenes](/source/Eratosthenes), presuming [Aswan](/source/Aswan) is on the boundary of the Northern Tropic. Extra-terrestrial distances use [stellar parallax](/source/Stellar_parallax), which Gamow relates to human binocular vision working to push the end of a thread through the eye a needle. A solar-[pumpkin](/source/Pumpkin) scale is introduced where the Sun is pumpkin-sized, Earth is pea-sized, and Moon poppy-sized. This scale proportions an [astronomical unit](/source/Astronomical_unit) to 200 feet. [Friedrich Bessel](/source/Friedrich_Bessel) measured the parallax of [61 Cygni](/source/61_Cygni), concluding a distance of 10 light years, making him "the first man who with a yardstick stepped into interstellar space". In the solar-pumpkin scale, 61 Cygni is 30,000 miles away. Our own galaxy, the Milky Way, measures 100,000 light years in diameter with 5 to 10 light years thickness, totaling 4 × 1010 stars. [Cepheid variables](/source/Cepheid_variable) are pulsating stars that have a [period-luminosity relation](/source/Period-luminosity_relation), exploited by [Harlow Shapley](/source/Harlow_Shapley) to estimate distances to globular clusters. The interstellar dust in the direction of the [Galactic Center](/source/Galactic_Center) obscures the view except through [Baade's Window](/source/Baade's_Window). ### Sources Instead of a bibliography as an appendix, Gamow cites a dozen titles in the course of his exposition: - p. 9: *Mathematical Recreations and Essays* (1919) by [W. W. Rouse Ball](/source/W._W._Rouse_Ball) - p. 49: *[What Is Mathematics?](/source/What_Is_Mathematics%3F)* (1941) by [Richard Courant](/source/Richard_Courant) and [Herbert Robbins](/source/Herbert_Robbins) - p. 146: *Mr. Thompkins in Wonderland* by himself - p. 156: *Atomic Physics* (1935) by [Max Born](/source/Max_Born) - p. 156: *Modern Physics* (1940) by T. B. Brown - p. 187: *Explaining the Atom* (1947) by [Selig Hecht](/source/Selig_Hecht) - p. 216: "[The Gold-Bug](/source/The_Gold-Bug)" by [Edgar Allan Poe](/source/Edgar_Allan_Poe) - p. 272: *[On the Heavens](/source/On_the_Heavens)* by [Aristotle](/source/Aristotle) - p. 303: *Exposition du Systeme du Monde* by [Laplace](/source/Pierre-Simon_Laplace) - p. 304: *Birth and Death of the Sun* (1940) by himself - p. 304: *Biography of the Earth* by himself - p. 315: *A Planet Called Earth* by himself ## Reception Science writer [Willy Ley](/source/Willy_Ley) praised Gamow's book, describing it as an "admittedly rare ... book which entertains by way of instruction".[2] *[Kirkus Reviews](/source/Kirkus_Reviews)* declared it "a stimulating and provocative book for the science-minded layman".[3] Theoretical physicist [Sean M. Carroll](/source/Sean_M._Carroll) credited *One Two Three... Infinity* with setting the trajectory of his professional life.[4] Cognitive scientist [Steven Pinker](/source/Steven_Pinker) read the book as a child, and has cited it as contributing to his interest in popular science writing.[5] Astrophysicist and science popularizer [Neil deGrasse Tyson](/source/Neil_deGrasse_Tyson) identified *One Two Three... Infinity* as one of two books which had the greatest impact on him, the other being *[Mathematics and the Imagination](/source/Mathematics_and_the_Imagination)* by [Edward Kasner](/source/Edward_Kasner) and [James R. Newman](/source/James_R._Newman).[6] In 1956, Gamow was awarded the [Kalinga Prize](/source/Kalinga_Prize) by [UNESCO](/source/UNESCO) for his work in popularizing science, including his book *One, Two, Three... Infinity*, as well as other works.[7] ## References 1. **[^](#cite_ref-Gamow_1-0)** *One, Two, Three...Infinity* (1947, revised 1961), Viking Press (copyright renewed by Barbara Gamow, 1974), reprinted by Dover Publications, [ISBN](/source/ISBN_(identifier)) [978-0-486-25664-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-25664-1), illustrated by the author; eBook edition, Dover, 2012 [ISBN](/source/ISBN_(identifier)) [9781306350099](https://en.wikipedia.org/wiki/Special:BookSources/9781306350099); other editions and translations 1. **[^](#cite_ref-2)** Willy Ley, "Book Review", *[Astounding Science Fiction](/source/Astounding_Science_Fiction)*, June 1948, pp.158-61. 1. **[^](#cite_ref-kirkus_3-0)** ["One Two Three...Infinity by George Gamow"](https://www.kirkusreviews.com/book-reviews/george-gamow-2/one-two-threeinfinity/). Kirkus Reviews. Retrieved 6 January 2015. 1. **[^](#cite_ref-carroll_4-0)** [Carroll, Sean M.](/source/Sean_M._Carroll) (April 16, 2008). ["Life-changing books: One, Two, Three... Infinity"](https://www.newscientist.com/article/dn13705-lifechanging-books-one-two-three-infinity.html#.VKwpFzHF98E). [New Scientist](/source/New_Scientist). Retrieved 6 January 2015. 1. **[^](#cite_ref-5)** ["Up Front"](https://www.nytimes.com/2007/05/27/books/review/Upfront-t-1.html?pagewanted=print&_r=0). *The New York Times*. May 27, 2007. Retrieved 6 January 2015. 1. **[^](#cite_ref-nyt-tyson_6-0)** ["Neil deGrasse Tyson: By the Book"](https://www.nytimes.com/2013/12/22/books/review/neil-degrasse-tyson-by-the-book.html?pagewanted=all). *The New York Times*. December 19, 2013. Retrieved 6 January 2015. 1. **[^](#cite_ref-7)** ["Kalinga 1956"](http://www.unesco.org/new/en/natural-sciences/science-technology/sti-policy/global-focus/science-popularization/prizes/kalinga-prize/kalinga-winners/kalinga-1956/). *www.unesco.org*. United Nations Educational, Scientific and Cultural Organization. Retrieved 2020-05-16. ## External links - [One Two Three...Infinity](https://archive.org/details/B-001-001-760) via [Internet Archive](/source/Internet_Archive) Authority control databases Open Library --- Adapted from the Wikipedia article [One Two Three... Infinity](https://en.wikipedia.org/wiki/One_Two_Three..._Infinity) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/One_Two_Three..._Infinity?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.