*Flatland*, Edwin Abbott’s story of a two-dimensional universe, as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to understand and appreciate the many ’dimensionsâ? of this classic satire. Mathematical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show 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.

This book was selected for *Choice *magazine's annual list of Outstanding Academic Titles in 2011. Full article.

### Table of Contents

Acknowledgments

Introduction

Flatland with Notes and Commentary

Part I This World

Part II Other Worlds

Epilogue by the Editor

Continued Notes

Appendix A: A Critical Reaction to Flatland

Appendix B: The Life and Work of Edwin Abbott Abbott

Recommended Reading

References

Index of Defined Words

Index

### Excerpt

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.

### 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*.