Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians

This collection of 39 short stories gives the reader a unique opportunity to take a look at the scientific philosophy of Vladimir Arnold, one of the most original contemporary researchers. Topics of the stories included range from astronomy, to mirages, to motion of glaciers, to geometry of mirrors and beyond. In each case Arnold's explanation is both deep and simple, which makes the book interesting and accessible to an extremely broad readership. Original illustrations hand drawn by the author help the reader to further understand and appreciate Arnold's view on the relationship between mathematics and science. Arnold's talent for exposition shines in this collection of short chapters on a miscellany of topics. I could not stop reading until I reached the end of the book. This book will entertain and enrich any curious person, whether a layman or a specialist”. - Mark Levi, Penn State University, USA author of The Mathematical Mechanic. “This book, which fits all mathematical ages, provides a glimpse into the “laboratory” of one of the most influential mathematicians of our time. Its genre is absolutely unique. A kaleidoscope of intriguing examples illustrating applications of mathematics to real life, intertwines with entertaining and often wildly funny mathematical anecdotes, as well as with profound insights into modern research areas. A brilliant informal exposition, complemented by artful drawings by the author, makes the book a fascinating read”. - Leonid Polterovich, Tel-Aviv University, Israel.

Table of Contents:
The eccentricity of the Keplerian orbit of Mars Rescuing the empennage Return along a sinusoid The Dirichlet integral and the Laplace operator Snell's law of refraction Water depth and Cartesian science A drop of water refracting light Maximal deviation angle of a beam The rainbow Mirages Tide, Gibbs phenomenon, and tomography Rotation of a liquid What force drives a bicycle forward? Hooke and Keplerian ellipses and their conformal transformations The stability of the inverted pendulum and Kapitsa's sewing machine Space flight of a photo camera cap The angular velocity of a clock hand and Feynman's “self-propagating pseudoeducation” Planetary rings Symmetry (and Curie's principle) Courant's erroneous theorems Ill-posed problems of mechanics Rational fractions of flows Journey to the center of the earth Mean frequency of explosions (according to Ya. B. Zel'dovich) and de Sitter's world The Bernoulli fountains of the Nikologorsky bridge Shape formation in a three-liter glass jar Lidov's moon landing problem The advance and retreat of glaciers The ergodic theory of geometric progressions The Malthusian partitioning of the world Percolation and the hydrodynamics of the universe Buffon's problem and integral geometry Average projected area The mathematical notion of potential Inversion in cylindrical mirrors in the subway Adiabatic invariants Universality of Hack's exponent for river lengths Resonances in the Shukhov tower, in the Sobolev equation, and in the tanks of spin-stabilized rockets The theory of rigid body rotation and hydrodynamics

Review :
The remarks provide a glimpse into the background of the history and the culture of science and humanity. This book will entertain and make the reader think. I recommend this fascinating book to any curious person." - László Csizmadia, ACTA Sci Math "Examples teach no less than rules, and errors, more than correct but abstruse proofs. Looking at the pictures in this book, the reader will understand more than learning by rote dozens of axioms (even together with their consequences about what sea the Volga River falls into and what horses eat). Most essays in the book are quite short, and their level of difficulty varies significantly -- some require only knowledge of a high school mathematics and some may be viewed as a serious challenge even for an experienced mathematician. As most texts written by Arnold, the book under review is a quite demanding but very stimulating and inspiring reading featuring original author's illustrations." - Zentralblatt Math "This is a wonderful book for browsing, for anyone drawn to physical applications of mathematics or to Arnold himself and the breadth of his interests." - MAA Reviews