FlatlandEdwin A. Abbott (with notes and commentary by William F. Lindgren and Thomas F. Banchoff)
303 pp., paperback, 2010. Series: MAA Spectrum
Cambridge University Press, Mathematical Association of America
ISBN: 978-0-521-75994-6
December 16, 2009
Flatland
Edwin A. Abbott (with notes and commentary by William F. Lindgren and Thomas F. Banchoff)
303 pp., paperback, 2010. Series: MAA Spectrum
Cambridge University Press, Mathematical Association of America
ISBN: 978-0-521-75994-6
Flatland is Edwin Abbott's story of a two-dimensional universe, as told by one of its inhabitants, A Square, who is introduced to the mysteries of three-dimensional space.
This fully annotated edition enables modern-day readers to understand and appreciate the many dimensions of this classic satire, which has enjoyed popularity since its publication more than 125 years ago.
Mathematical notes and illustrations enhance the book's use as an elementary introduction to higher-dimensional geometry. Historical notes draw connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings, as well as the works of Plato and Aristotle, serve to interpret the text.
Commentary on language and literary style includes numerous definitions of obscure words. An appendix gives a comprehensive account of the life and work of Flatland's remarkable author.
Excerpt (p. 203): Extra-Solids
An "extra-solid" is an object in four-dimensional space; what the Square calls and Extra-Cube is now called a hypercube. C. Howard Hinton first called the four-dimensional analogue of a cube "four-square"; later, he settled on the name "tessaract" (now spelled "tesseract") (Hinton 1880; 1888). Higher-dimensional polyhedra are now called "polytopes," and the standard notation for a four-dimensional, regular polytope is "k-cell," where k indicates the number of three-dimensional boundary cells. Thus, a hypercube is an 8-cell.
In two-dimensional space, there are n-sided regular polygons for every integer n greater than two. In three-dimensional space, there are just five regular polyhedral. . . . In a paper written in 1852 but not published until 1901, the Swiss geometer Ludwig Schläfli determined all of the higher-dimensional regular polytopes. He showed that in four-dimensional space there are just six regular polytopes: the 5-cell, 8-cell, 16-cell, 120-cell, and 600-cell (analogues of the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively) as well as the 24-cell, which has no three-dimensional analogue. Further, he showed that for every n greater than 4, the only regular polytopes are the n-dimensional analogues of the tetrahedron, the cube, and the octahedron.
Contents:
Introduction. Flatland with Notes and Commentary. Part I. This World: 1. Of the Nature of Flatland; 2. Of the Climate and Houses in Flatland; 3. Concerning the Inhabitants of Flatland; 4. Concerning the Women; 5. Of our Methods of Recognizing one another; 6. Of Recognition by Sight; 7. Concerning Irregular Figures; 8. Of the Ancient Practice of Painting; 9. Of the Universal Colour Bill; 10. Of the Suppression of the Chromatic Sedition; 11. Concerning our Priests; 12. Of the Doctrine of our Priests. Part II. Other Worlds: 13. How I had a Vision of Lineland; 14. How in my Vision I endeavoured to explain the nature of Flatland, but could not; 15. Concerning a Stranger from Spaceland; 16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland; 17. How the Sphere, having in vain tried words, resorted to deeds; 18. How I came to Spaceland and what I saw there; 19. How, though the Sphere showed me other mysteries of Spaceland, I still desired more; and what came of it; 20. How the sphere encouraged me in a Vision; 21. How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success; 22. How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result. Epilogue by the Editor. Continued Notes. Appendix A: Critical Reaction to Flatland. Appendix B: The Life and Work of Edwin Abbott Abbott. Recommended Reading. References. Index of Defined Words. Index.
About the Commentators:
William F. Lindgren (Slippery Rock University, Slippery Rock, Pa.) is the coauthor of Quasi-Uniform Spaces.
Thomas F. Banchoff (Brown University) is the author of Beyond the Third Dimension and coauthor of Linear Algebra through Geometry. He is a former president of the MAA. In 1978, Banchoff and Louis H. Kauffman won the Lester R. Ford Award for their article "Immersions and Mod-2 Quadratic Forms," published in the American Mathematical Monthly.
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