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Math Plots and Plans: Edwin A. Abbott

Edwin Abbott Abbott

After 116 years, Edwin Abbott Abbott's Flatland is still the best introduction to the method of analogy used by virtually all mathematicians and physicists when describing the fourth dimension. In recent years there have been more than a dozen new editions in English, and translations into at least nine foreign languages.

Wait a minute! Wasn't "Flatland" sexist? The answer, everyone will be happy to know, is "No". The place, Flatland, was a sexist society, but the book "Flatland" is intended, and was received, as social satire and more. Its author was highly regarded as an educational reformer hailed by the premier women educators as a leader in the movement to bring educational opportunities to young women in Victorian England. For more information, see the section on social satire in the introduction by Thomas Banchoff to the Princeton Science Series edition of "Flatland". An article that provides background into the social satire of Flatland also is available on line.

A recent exhibit, Flatland: A Millennial Book, which was hosted by the Brown University library as a physical display, is now available on the web as a virtual experience.

Many authors cite Edwin Abbott Abbott as one of their first inspirations in the study of abstract mathematics, particularly in higher dimensions. One of these is Prof. Dirk Struik, who, at this writing, is 105 years old. A picture of him giving a lecture on his hundredth birthday is on-line [Image], as is the report of this event that appeared in the Notices of the AMS. Here is his reference to Flatland from his classic book Lectures in Classical Differential Geometry.

Wonderful news is the impending reissue of one of the best books about life in a modern two-dimensional universe, namely The Planiverse by A. K. Dewdney, featuring the hero of the story, Yendred. You can order your own copy of The Planiverse electronically. Dionys Burger's Sphereland follows the adventures of A Square's grandson as he tries to understand the global geometry of his two-dimensional world. It is available in a two book companion set along with Flatland.

Much of the popularity of Flatland in the US is thanks to Hayward Cirker, founder of Dover Publications, who passed away in March of 2000. In 1952, he personally chose Flatland as one of his first titles in mathematics. You can view a poster advertising that very first Dover edition [Image]. Since that time, it has remained one of their best sellers. Seven years ago, Flatland appeared as a Dover Thrift Edition, selling for $1, and guaranteeing that it will remain a favorite of teachers and students for some time to come. The Dover editions have sold more than half a million copies.

A fine treatment of many of the themes from Flatland and an introduction to the world of higher-dimensional "polytopes" is available in a final project by two freshmen and a junior, all from Paideia School in Atlanta, GA. An excellent project from the pre-internet days is the Dimensional Geographic of Suttirat Anne Larlarb, then a student at Brown University, for Math 8, "The Mathematical Way of Thinking". It is, of course, purely fictional, so please don't fill out the application!

For an excellent site showing the slices not only of the Hypercube but also other regular figures in four-dimensional space, see the site of Mark Newbold. It also contains a bibliography of additional links. Rudy Rucker has written a program for the IBM PC that displays rotationg hypercubes and more that you can download.

Here is another good site on regular polytopes in higher dimensions, and another featuring rotating stereo images. The classic reference in this field is Regular Polytopes by Prof. H.S.M. Coxeter still in print as a Dover paperback. For Mathematica users, you can download a mathematica notebook in which the sixteen regular polytopes in four dimensions are constructed. Russell Towle provides QuickTime animations of series of solid sections of some of the regular star-polytopes, probably the first such ever made.

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