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One Two Three... Infinity
View on WikipediaOne 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[update]) 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.[1] The book includes many handmade illustrations by Gamow.
Key Information
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Synopsis
[edit]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
[edit]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
[edit]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
[edit]
Part III is the longest (150 pages) and begins with the "descending staircase" and the classical elements. "Plants take the largest part of the material used in the growth of their bodies ... from the air." 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 is stated in plain English (p. 123) as a "fundamental law of chemistry". The molecular structure of matter was uncovered with molecular beams by Otto Stern, and 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, and reference to J. J. Thomson, the electron is introduced, having mass 1/1840 of the mass of a hydrogen atom. The Rutherford model of the atom, an analogy to the 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 uses flower petals with stems at the inert gasses. The "utmost precision" of celestial mechanics is contrasted with the quantum of action, which leads to the uncertainty principle. Diffraction phenomena not explicable with geometric optics necessitated the wave mechanics of Louis de Broglie and Erwin Schrödinger.
In the chapter "The Riddle of Life" the states of matter in an automobile body, engine, and radiator are also present in living systems, but homogeneity of biological tissue is of a different sort. A human is estimated to have more than hundreds of thousands of billions of cells. To eat, grow, and multiply are posited as life characteristics. Dismissed are crystal accretion in a super-saturated solution, and the molecular reaction
![{\displaystyle {3\,\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} {}+{}2\,\mathrm {CO} {}+{}\mathrm {C} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{5}}\mathrm {OH} {}\mathrel {\longrightarrow } {}2\,\mathrm {C} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{5}}\mathrm {OH} {}+{}3\,\mathrm {O} {\vphantom {A}}_{\smash[{t}]{2}}\,{\cdot }\,}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/142be53b569f8d3d3f70c57e4b23d2f16bfdf002)
On the other hand, virus reproduction is the "missing link" between non-living and living organisms. The eight chromosomes of Drosophila melanogaster are acknowledged for their contribution to science. Growth by mitosis and reproduction by meiosis with gametes performing syngamy show the function of chromosomes. Growth and accretion are started with blastula and gastrula.
Part IV: Macrocosmos
[edit]Aristotle's On the Heavens founded cosmology. Earth's circumference was found by Eratosthenes, presuming Aswan is on the boundary of the Northern Tropic. Extra-terrestrial distances use 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 scale is introduced where the Sun is pumpkin-sized, Earth is pea-sized, and Moon poppy-sized. This scale proportions an astronomical unit to 200 feet. Friedrich Bessel measured the parallax of 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. Cephid variables are pulsating stars that have a period-luminosity relation, exploited by Harlow Shapley to estimate distances to globular clusters. The interstellar dust in the direction of the Galactic Center obscures the view except through Baade's Window.
Sources
[edit]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
- p. 49: What Is Mathematics? (1941) by Richard Courant and Herbert Robbins
- p. 146: Mr. Thompkins in Wonderland by himself
- p. 156: Atomic Physics (1935) by Max Born
- p. 156: Modern Physics (1940) by T. B. Brown
- p. 187: Explaining the Atom (1947) by Selig Hecht
- p. 216: "The Gold-Bug" by Edgar Allan Poe
- p. 272: On the Heavens by Aristotle
- p. 303: Exposition du Systeme du Monde by 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
[edit]Science writer Willy Ley praised Gamow's book, describing it as an "admittedly rare ... book which entertains by way of instruction".[2] Kirkus Reviews declared it "a stimulating and provocative book for the science-minded layman".[3] Theoretical physicist Sean M. Carroll credited One Two Three... Infinity with setting the trajectory of his professional life.[4] Cognitive scientist 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 identified One Two Three... Infinity as one of two books which had the greatest impact on him, the other being Mathematics and the Imagination by Edward Kasner and James R. Newman.[6]
In 1956, Gamow was awarded the Kalinga Prize by UNESCO for his work in popularizing science, including his book One, Two, Three... Infinity, as well as other works.[7]
References
[edit]- ^ One, Two, Three...Infinity (1947, revised 1961), Viking Press (copyright renewed by Barbara Gamow, 1974), reprinted by Dover Publications, ISBN 978-0-486-25664-1, illustrated by the author; eBook edition, Dover, 2012 ISBN 9781306350099; other editions and translations
- ^ Willy Ley, "Book Review", Astounding Science Fiction, June 1948, pp.158-61.
- ^ "One Two Three...Infinity by George Gamow". Kirkus Reviews. Retrieved 6 January 2015.
- ^ Carroll, Sean M. (April 16, 2008). "Life-changing books: One, Two, Three... Infinity". New Scientist. Retrieved 6 January 2015.
- ^ "Up Front". The New York Times. May 27, 2007. Retrieved 6 January 2015.
- ^ "Neil deGrasse Tyson: By the Book". The New York Times. December 19, 2013. Retrieved 6 January 2015.
- ^ "Kalinga 1956". www.unesco.org. United Nations Educational, Scientific and Cultural Organization. Retrieved 2020-05-16.
One Two Three... Infinity
View on GrokipediaOverview
Author and Background
George Gamow was born on March 4, 1904, in Odessa, Russia (now Ukraine), into a family of educators; his father was a teacher of Russian literature and history. He displayed an early interest in science, particularly astronomy and physics, and pursued higher education at the University of Odessa before transferring to the University of Leningrad (now Saint Petersburg), where he studied under physicists Yakov Krutkov and Vladimir Fock. Gamow earned his doctorate in 1928 after completing his dissertation on the structure of atomic nuclei. Facing political pressures in the Soviet Union, he emigrated in 1933, first to England and then to the United States, where he joined the faculty at George Washington University in Washington, D.C. Gamow's scientific career was marked by groundbreaking contributions to nuclear physics and cosmology. In 1928, while still in Leningrad, he developed the quantum tunneling theory of alpha decay, explaining how alpha particles escape atomic nuclei, a model that revolutionized understanding of radioactive processes and earned him international recognition. He later advanced theories on beta decay and collaborated on early work in Big Bang nucleosynthesis, predicting the abundance of light elements like helium in the universe, which laid foundational ideas for modern cosmology. Throughout the 1930s and 1940s, Gamow's research extended to stellar nucleosynthesis and the expansion of the universe, influencing fields from nuclear reactions to astrophysics. Beyond his technical research, Gamow became a pioneering popular science communicator, using wit and simplicity to bridge complex ideas with general audiences. His writing career took off with the "Mr. Tompkins" series, beginning with Mr Tompkins in Wonderland in 1940, which imagined relativistic and quantum effects through everyday scenarios in a banker's dreams. Subsequent books like Mr Tompkins Explores the Atom (1944) and One Two Three... Infinity (1947) exemplified his signature style: blending humor, analogies, and self-illustrated diagrams to demystify science. Gamow personally drew the figures for his books, including the hand-sketched illustrations in One Two Three... Infinity that enhanced its accessibility. The context for Gamow's authorship of One Two Three... Infinity emerged in the post-World War II era, a time of heightened public fascination and anxiety about atomic energy following the Manhattan Project and Hiroshima bombings. As a physicist who had consulted for the U.S. government on nuclear matters, Gamow sought to popularize science amid the "atomic age," emphasizing wonder over fear to foster informed public discourse on topics from mathematics to cosmology.Publication History and Editions
One Two Three... Infinity was first published in 1947 by Viking Press as a popular science book aimed at a lay audience, spanning 340 pages and featuring 128 black-and-white illustrations drawn by the author himself.[6][7] A revised edition appeared in 1961, also from Viking Press, incorporating updates to reflect scientific progress since the original publication.[4] Key changes included additions on thermonuclear reactions and controlled energy release in Chapter VII, an updated estimate of the universe's age to over five billion years, revisions to the astronomical distance scale using data from the 200-inch Hale telescope, a redrawn Figure 101, and new material on synthetic organisms in Chapter IX.[4] The 1961 version maintained the 340-page length and the author's illustrations.[8] Subsequent printings under Viking included minor updates in a 1965 edition, preserving the core content of the 1961 revision.[8] In 1988, Dover Publications released an affordable paperback reprint of the 1961 revised edition, which remains unchanged and serves as the standard version today, with 340 pages and the original 128 illustrations.[2][9] The book has been translated into multiple languages, including Russian (with commentaries in post-Cold War editions), Chinese, and German, extending its international reach.[10][11][12] As of 2025, no major new editions have been issued since 1965, despite significant scientific advances in fields like cosmology and biology, leaving the 1988 Dover edition as the primary accessible format.[13] The book continues to be available in print-on-demand and digital formats, including a 2024 Kindle release on Amazon.[14][15]Synopsis
Part I: Playing with Numbers
Gamow opens Part I by inviting readers to "play with numbers," transforming abstract mathematical concepts into engaging narratives that reveal their beauty and strangeness. He begins with the allure of ever-larger finite numbers, using historical vignettes to illustrate humanity's fascination with scale. For instance, the ancient Greek mathematician Archimedes, in his work The Sand Reckoner, estimated that the grains of sand required to fill the visible universe number fewer than , a feat achieved by devising a numeral system for vast quantities. Gamow also recounts the Indian legend of Sissa Ben Dahir, who, as a reward for inventing chess, requested one grain of wheat on the first board square, two on the second, doubling each time up to 64 squares, yielding a total exceeding grains—enough to cover the Earth many times over. These examples highlight exponential growth, setting the stage for modern large numbers like Avogadro's constant, approximately , which counts the molecules in one gram-mole of any substance and underscores the atomic world's immensity. To push boundaries further, Gamow introduces the googol, defined as or 1 followed by 100 zeros, a term coined in 1938 by mathematician Edward Kasner with input from his nine-year-old nephew Milton Sirotta during a discussion on enormous quantities. The googolplex, $1010^{10^{100}}$), defies physical realization, as writing it out would require more space than the observable universe contains.[16] Transitioning from colossal finites to the infinite, Gamow explores Georg Cantor's revolutionary set theory, which posits that not all infinities are equal. He explains countable infinities, such as the set of natural numbers , with cardinality (aleph-null), where elements can be paired one-to-one with integers despite endless extent. In contrast, the real numbers form an uncountable infinity, with cardinality , known as the continuum, larger than . To demonstrate, Gamow presents Cantor's diagonal argument: suppose all real numbers between 0 and 1 are listed as infinite decimals (e.g., 0., 0., etc.); form a new number 0. where differs from (say, by adding 1 modulo 10, avoiding 9s for convergence). This diagonal number mismatches every listed one, proving no complete enumeration exists. This proof, from Cantor's 1891 paper, establishes transfinite cardinals like for the continuum, opening a hierarchy of infinities beyond intuition.[16] To convey infinity's paradoxes, Gamow describes David Hilbert's Grand Hotel thought experiment, introduced in his 1924 lectures on the infinite. Imagine a hotel with infinitely many rooms, all occupied (guest in room ). A new arrival prompts shifting each guest from room to , freeing room 1. For infinitely many new guests (say, from a bus numbered by naturals), assign them to even-numbered rooms after shifting originals to odds, or use prime/composite distinctions for multiple infinite buses. This illustrates that infinite sets can be placed in one-to-one correspondence with proper subsets, a property absent in finite sets, challenging everyday arithmetic.[16] Gamow then introduces imaginary numbers to solve equations without real solutions, defining , the imaginary unit. Complex numbers take the form , where and are real, and are represented geometrically on the complex plane with real parts along the horizontal axis and imaginary along the vertical—like plotting vectors from the origin. He illustrates with a whimsical treasure hunt: instructions like "go 3 units east, then units north" (equivalent to 3 + ) or multiplications that rotate and scale, revealing complex arithmetic's intuitive vector interpretation. This foundation hints at broader utility, as complex numbers elegantly describe physical phenomena like electrical circuits and wave propagation in physics, where real and imaginary components capture amplitude and phase without lengthy derivations.[16] The section culminates with prime numbers, the "atoms" of integers, whose distribution Gamow portrays as irregular yet infinite. He details Euclid's proof from Elements (Book IX, Proposition 20, ca. 300 BCE): assume finitely many primes ; form . exceeds all and is not divisible by any, so either is prime or its prime factors are new, contradicting finitude. Primes thin out (density roughly by the prime number theorem), but puzzles persist. Gamow highlights Fermat's Last Theorem, conjectured in 1637, stating no positive integers satisfy —a claim unsolved in 1947 despite proofs for specific up to 269. He also discusses the Goldbach conjecture, proposed by Christian Goldbach in a 1742 letter to Leonhard Euler, asserting every even integer greater than 2 is the sum of two primes (e.g., ), verified computationally for vast numbers but unproven. These open problems exemplify number theory's enduring allure.[16][17]Part II: Space, Time, and Einstein
In Part II of One Two Three... Infinity, George Gamow delves into the geometric and topological peculiarities of space, illustrating how mathematical abstractions can reshape our understanding of the physical universe. He begins with one-sided surfaces, such as the Möbius strip, independently discovered by August Ferdinand Möbius and Johann Benedict Listing in 1858, which features only one side and one edge when formed by twisting and joining the ends of a rectangular strip.[18] Gamow explains that traversing the Möbius strip reverses orientation, as demonstrated by a drawing of a left-facing profile that appears right-facing upon return, and notes that cutting it lengthwise yields a single doubled-length loop rather than two separate pieces. He extends this to the Klein bottle, conceptualized by Felix Klein in 1882 as a non-orientable, boundaryless surface that cannot exist without self-intersection in three-dimensional space but models a four-dimensional analogue. These topologies, including the projective plane as another one-sided surface, challenge Euclidean intuitions and suggest possible structures for a finite yet unbounded cosmos, akin to a worm-eaten apple where channels form a self-contained volume without edges.[19] Gamow transitions to practical applications in cartography, highlighting distortions in mapping curved surfaces like the Earth onto flat planes and the historical quest for accurate projections. He discusses map coloring problems, where adjacent regions must differ in hue, and presents the four-color theorem, conjecturing that four colors suffice for any planar map—a claim unproven in 1947 but later verified in 1976 by Kenneth Appel and Wolfgang Haken through computer-assisted analysis of reducible configurations.[20] While five colors were already known to be adequate via Euler's formula relating vertices (V), edges (E), and faces (F) as V - E + F = 2, Gamow emphasizes the theorem's topological roots in graph theory, using it to bridge everyday puzzles with deeper spatial properties.[19] Building on these ideas, Gamow introduces non-Euclidean geometries, where space deviates from flatness due to intrinsic curvature, as pioneered by Carl Friedrich Gauss, Nikolai Lobachevsky, and János Bolyai in the early 19th century. In hyperbolic geometry, with negative Gaussian curvature like a saddle, the sum of a triangle's angles is less than 180° and multiple parallels exist through a point to a line; conversely, elliptic geometry on a sphere exhibits positive curvature, zero parallels, and angle sums exceeding 180° (e.g., 210° for a large spherical triangle). Gamow defines "straight lines" as geodesics—the shortest paths, such as great circles on a sphere—and uses the worm-eaten apple analogy to envision a three-dimensional hyperbolic space that curves back on itself, finite in extent yet edgeless, prefiguring models of a closed universe. These geometries underscore that space's properties are not absolute but depend on measurement scales and curvature.[19] Gamow then bridges mathematics to physics with Albert Einstein's special relativity, formulated in his 1905 paper, which unifies space and time into a four-dimensional Minkowski spacetime. The theory arose from the null result of the 1887 Michelson-Morley experiment, which sought but failed to detect Earth's velocity relative to the luminiferous ether using an interferometer, implying no preferred frame.[21] Einstein postulated the constancy of light speed c (approximately 3 × 10^8 m/s) in all inertial frames, leading to the Lorentz transformation and effects like time dilation, where proper time τ for a moving clock relates to coordinate time t by or equivalently, dilated time t' = t / \sqrt{1 - v^2/c^2}, as derived from the invariance of the spacetime interval ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2.[22] Length contraction follows similarly, with moving length l' = l \sqrt{1 - v^2/c^2}. Gamow illustrates with examples, such as muons reaching Earth's surface due to dilated lifetimes or the twin paradox, where one twin's high-speed journey results in less elapsed time upon return; the resolution lies in the asymmetry of acceleration, breaking inertial symmetry, as Einstein clarified in 1918.[23] As velocity nears c, relativistic mass increases without bound, prohibiting superluminal travel.[19] Shifting to general relativity, Einstein's 1915 theory, Gamow describes gravity not as a force but as spacetime curvature induced by mass-energy, governed by the Einstein field equations. The equivalence principle equates gravitational and inertial mass, implying that free fall follows geodesics in curved spacetime. Light paths bend near massive bodies, a prediction confirmed during the 1919 solar eclipse by expeditions led by Arthur Eddington and Frank Dyson, measuring starlight deflection of 1.61″ ± 0.30″ against Einstein's 1.75″ forecast for rays grazing the Sun.[24] Gamow speculates on extreme outcomes, like light trapped in dense stars forming "frozen stars" (precursors to black holes), and notes E = mc^2 from special relativity, linking mass to energy release. He touches on wormholes as theoretical bridges in spacetime, akin to topological tunnels, though unresolved as causal loops.[19] Gamow concludes the part with relativistic implications for time travel, highlighting paradoxes like the grandfather paradox, where altering the past (e.g., killing one's ancestor) precludes one's existence, or superluminal signals inverting cause and effect. Such scenarios, potentially enabled by closed timelike curves, remain speculative and physically barred by the light-speed limit, as mass diverges asymptotically. Gamow resolves the twin paradox via the non-inertial frame of the traveler, where simultaneity shifts during turnaround, ensuring consistency without backward causation. These discussions underscore relativity's unification of space, time, and gravity, transforming abstract math into cosmic reality.[19]Part III: Microcosmos
In Part III of One Two Three... Infinity, George Gamow explores the atomic and subatomic realms, elucidating the foundational discoveries that reshaped our understanding of matter's building blocks. He begins with the atom's internal structure, drawing on Ernest Rutherford's pivotal 1911 gold foil experiment conducted at the University of Manchester. In this setup, a beam of alpha particles from a radioactive source was fired at an ultrathin sheet of gold foil, with a fluorescent screen detecting the particles' deflections. While most particles passed straight through, approximately one in 8,000 was scattered at angles greater than 90 degrees, defying the prevailing plum pudding model of the atom and revealing a minuscule, positively charged nucleus surrounded by mostly empty space.[25] This observation implied that the atom's mass and positive charge were concentrated in a tiny core, with electrons orbiting at a distance, akin to planets around the sun in a scaled-down solar system.[26] Rutherford's analysis of the scattering data yielded a quantitative formula describing the phenomenon, where the differential cross-section for alpha particle deflection is proportional to , with as the scattering angle; this hyperbolic dependence matched experimental observations and confirmed the nuclear model's electrostatic repulsion between the positively charged alpha particles and the gold nucleus.[27] Gamow emphasizes how this experiment, building on Geiger and Marsden's earlier measurements, provided empirical evidence for the atom's discrete structure, paving the way for subsequent quantum refinements.[28] Transitioning to the organization of elements, Gamow recounts Dmitri Mendeleev's 1869 periodic table, which arranged the known 63 elements by increasing atomic weight, revealing recurring patterns in chemical properties that he termed "periodicity."[29] Mendeleev boldly predicted undiscovered elements like gallium and germanium based on gaps in his table, but later work by Henry Moseley in 1913 established atomic number—the number of protons in the nucleus—as the fundamental ordering principle, correlating directly with an element's position and chemical behavior.[30] This shift highlighted the role of electron configurations, with elements in the same group sharing similar outer-shell electron arrangements that dictate reactivity; for instance, alkali metals in Group 1 have one valence electron, fostering their tendency to lose it readily.[31] Gamow introduces quantum mechanics by contrasting classical atomic models with Niels Bohr's 1913 quantized orbits for the hydrogen atom, where electrons occupy discrete energy levels rather than spiraling continuously into the nucleus. In Bohr's model, the energy of the electron in the nth orbit is given by where is the principal quantum number starting from 1, yielding the ground state energy of -13.6 eV for hydrogen and explaining spectral line emissions as transitions between these levels. This quantization resolved the instability of Rutherford's planetary atom and matched the Balmer series of hydrogen lines observed in spectroscopy. Gamow further discusses wave-particle duality, a cornerstone of quantum theory, where entities like electrons and photons exhibit both particle-like (localized impacts in detectors) and wave-like (interference patterns in double-slit experiments) properties, as demonstrated by Louis de Broglie's 1924 hypothesis and Clinton Davisson's 1927 electron diffraction confirmation. A key quantum limitation Gamow highlights is Werner Heisenberg's uncertainty principle, formulated in 1927, which states that the product of uncertainties in position and momentum satisfies , where and is Planck's constant.[32] This inequality arises from the wave nature of particles, implying that precise measurement of one property inherently disturbs the other; for example, localizing an electron's position with light scatters it, altering its momentum unpredictably. The principle underscores the probabilistic essence of quantum mechanics, prohibiting classical determinism at atomic scales and influencing phenomena like electron cloud orbitals rather than fixed paths.[33] Gamow, a pioneer in cosmology, devotes significant attention to the origins of chemical elements, presenting his own 1940s theory of primordial nucleosynthesis within the hot Big Bang model. In the universe's first minutes, when temperatures exceeded 10^9 K, neutrons and protons fused rapidly to form light nuclei: primarily hydrogen (about 75% by mass) through unbound protons, and helium-4 (about 25%) via deuterium bottlenecks and subsequent captures, with trace amounts of helium-3, deuterium, and lithium-7.[34] This process, detailed in Gamow's collaborations with Ralph Alpher and Robert Herman, predicted the observed cosmic abundances of these primordial elements, distinguishing them from heavier ones forged in stellar interiors, and provided early evidence for the expanding universe's thermal history.[35] Extending to the biology of the microcosmos, Gamow examines life's molecular foundations as they were understood in the mid-20th century, prior to the 1953 DNA double helix discovery. He describes chromosomes as thread-like structures in cell nuclei that carry hereditary information through genes, then believed to consist of proteins due to their complexity and abundance in chromatin. These proteinaceous genes were thought to replicate and mutate, directing protein synthesis for traits via enzyme chains, with early speculations on diamond-shaped "tetrahedral" coding schemes for amino acids.[36] Gamow portrays viruses as enigmatic entities on the life-matter boundary: crystalline outside hosts yet capable of reproducing by hijacking cellular machinery, lacking independent metabolism and thus classified as non-living reproducers rather than true organisms. In the 1961 edition, he adds discussion of synthetic viruses, referencing advances like Heinz Fraenkel-Conrat's 1950s reconstitution of tobacco mosaic virus from purified RNA and protein components, demonstrating infectivity and blurring distinctions between life and chemistry.[37]Part IV: Macrocosmos
In Part IV of One Two Three... Infinity, George Gamow explores the vast scales of the universe, beginning with ancient attempts to measure cosmic dimensions. Around 240 BCE, the Greek scholar Eratosthenes calculated the Earth's circumference to be approximately 40,000 kilometers by comparing the angle of the sun's rays at noon in Alexandria and Syene, using the known distance between the two cities and basic geometry.[38] This measurement demonstrated the power of observation and mathematics in probing planetary size without direct traversal. Later, Aristarchus of Samos proposed a heliocentric model in the third century BCE, placing the Earth in orbit around the Sun, and recognized that stellar parallax—the apparent shift in star positions due to Earth's motion—could measure distances to the stars, though the effect was too small to detect with ancient instruments, implying immense stellar separations.[39] Gamow then discusses methods for determining stellar distances, highlighting parallax as refined in the 19th and 20th centuries and the role of Cepheid variables as "standard candles." In 1908, Henrietta Swan Leavitt discovered the period-luminosity relation for Cepheids, showing that these pulsating stars' intrinsic brightness correlates directly with their pulsation period, allowing astronomers to gauge distances by comparing apparent and absolute magnitudes.[40] This relation enabled calibration of cosmic yardsticks beyond the solar neighborhood, transforming parallax-limited measurements into tools for galactic scales. Turning to galaxy structure, Gamow describes the Milky Way as an "island universe," a vast disk of stars, gas, and dust, with early telescopic observations revealing spiral nebulae as separate systems. In the 1920s, Edwin Hubble used the 100-inch Hooker telescope at Mount Wilson to observe Cepheids in the Andromeda nebula (M31), confirming in 1924 that it lay far beyond the Milky Way—about 900,000 light-years away—thus establishing spiral nebulae as distant galaxies comparable in size to our own.[41] These findings resolved debates over whether the universe consisted of a single galaxy or myriad independent ones, expanding humanity's view from a provincial stellar system to a cosmos teeming with billions of galaxies. Gamow delves into evidence for an expanding universe, drawing on redshift observations of distant galaxies. In 1929, Hubble published data showing that galaxies recede from us with velocities proportional to their distances, formalized as Hubble's law:where is the recession velocity, is the distance, and is the Hubble constant (initially estimated at around 500 km/s/Mpc).[42] This redshift, interpreted as Doppler stretching of light due to motion away from the observer, suggested a dynamic universe originating from a hot, dense state rather than a static one. Based on contemporary values in 1947, the universe's age was estimated at roughly 2 billion years, implying ongoing expansion from a primordial epoch. In the 1961 edition, this estimate was revised to over 5 billion years based on new astronomical data.[43] This expansion fueled cosmological debates, with the Big Bang model resolving Olbers' paradox—the puzzle of why the night sky is dark despite infinite stars—by limiting the universe's age, so light from distant sources has not yet reached us, and by the universe's finite expansion diluting stellar light over time.[43] He speculates on the "heat death," where expansion leads to ever-cooling temperatures and maximum entropy, ending in a dilute, lifeless void. Gamow ties these ideas to his own work, including pioneering Big Bang nucleosynthesis, where he and collaborators Ralph Alpher and Hans Bethe explained light element abundances (like helium) as products of nuclear reactions in the first minutes after the universe's birth, linking microscopic quantum processes to macroscopic cosmic evolution.[44]