In 1884, Edwin Abbott Abbott wrote a mathematical adventure set in a two-dimensional plane world, populated by a hierarchical society of regular geometrical figures-who think and speak and have all too human emotions. Since then Flatland has fascinated generations of readers, becoming a perennial science-fiction favorite. By imagining the contact of beings from different dimensions, the author fully exploited the power of the analogy between the limitations of humans and those of his two-dimensional characters. A first-rate fictional guide to the concept of multiple dimensions of space, the book will also appeal to those who are interested in computer graphics. This field, which literally makes higher dimensions seeable, has aroused a new interest in visualization. We can now manipulate objects in four dimensions and observe their three-dimensional slices tumbling on the computer screen. But how do we interpret these images? In his introduction, Thomas Banchoff points out that there is no better way to begin exploring the problem of understanding higher-dimensional slicing phenomena than reading this classic novel of the Victorian era. "One of the most imaginative, delightful and, yes, touching works of mathematics, this slender 1884 book purports to be the memoir of A. Square, a citizen of an entirely two-dimensional world." ― The Washington Post Book World " Flatland has remained of interest for over a century precisely because of its ability to engage its readers on so many different planes in so many different dimensions." ― Victorian Studies "This reprint of Abbott's Flatland adventures contains an Introduction by Thomas Banchoff which is worth reading on its own. So if you don't have yet this book at home, go ahead and buy this edition." ― Zentralblatt MATH Edwin Abbott Abbott (1838-1926), the author of more than fifty books on classics, theology, history, and Shakespeare, was headmaster of the City of London School and one of the leading educators of his time. Thomas Banchoff is professor emeritus of mathematics at Brown University and author of Beyond the Third Dimension . Flatland A Romance of Many Dimensions By Edwin Abbott Abbott PRINCETON UNIVERSITY PRESS Copyright © 1991 Princeton University Press All rights reserved. ISBN: 978-0-691-16555-4 Contents PREFACE TO THE SECOND AND REVISED EDITION, ix, INTRODUCTION, xiii, Part I This World, SECTION, 1 Of the Nature o f Flatland, 3, 2 Of the Climate and Houses in Flatland, 4, 5 Concerning the Inhabitants of Flatland, 6, 4 Concerning the Women, 8, 5 Of our Methods of Recognizing one another, 22, 6 Of Recognition by Sight, 16, 7 Concerning Irregular Figures, 20, 8 Of the Ancient Practice of Painting, 22, 9 Of the Universal Colour fill, 24, 10 Of the Suppression of the (Chromatic Sedition, 27, 11 Concerning our Priests, 30, 12 Of the Doctrine of our Priests, 32, Part II Other Worlds, 13 How I had a Vision of Lineland, 39, 14 How fn my Vision I endeavoured to explain the nature of Flatland, but could not, 42, 15 Concerning a Stranger from Spaceland, 46, 16 How the Granger vainly endeavoured to reveal to me in words the mysteries of Spaceland, 49, 17 How the Sphere, having in vain tried words, resorted to deeds, 58, 18 How I came to Spaceland and what I saw there, 57, 19 How, though the Sphere showed me other mysteries of Spaceland, I still desired more; and what came of it, 61, 20 How the Sphere encouraged me in a Vision, 69, 21 How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success, 68, 22 How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result, 70, CHAPTER 1 Part I This World "Be patient, for the world is broad and wide." § 1: Of the Nature of Flatland I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard and with luminous edges—and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things. In such a country, you will perceive at once that it is impossible that there should be anything of what you call a "solid" kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonst