PICK A NUMBER: WHAT EDWIN ABBOTT DID NOT KNOW ABOUT FLATLAND This article also appears in the Oak Ridge National Laboratory Review (Vol. 25, No. 2), a quarterly research and development magazine. If you'd like more information about the research discussed in the article or about the Review, or if you have any helpful comments, drop us a line at: electronic mail, krausech@ornl.gov or pearcejw@ornl.gov; fax, 615/574-1001; phone, 615/574-7183 or 615/574-6774; or mail, ORNL Review, Oak Ridge National Laboratory, 4500-S 6144, Oak Ridge, TN 378312-6144. Thanks for reading the Review. Imagine living in a world of two, rather than three, dimensions. Edwin Abbott (1838-1926)--a prominent schoolmaster, scholar, and theologian who lived in England--anonymously published a book on this fantasy in 1880. It's called _Flatland, A Romance of Many Dimensions_. From our three-dimensional perspective, the inhabitants of Flatland appear in a variety of flat geometric shapes, including triangles, circles, and squares. Viewed from their perspective, Flatland was populated by lines of varying lengths and points for very slender people. The book focuses on the impact of a major event in Flatland's history in which a three-dimensional sphere slowly passes through the country, first driving the Flatlanders away from its expanding circular intersection with their world, then slowly retreating back to a point, and finally disappearing. The Flatlanders would view this event as time-related. Abbott seeks to broaden our conception of time by using this physical analogy. According to Abbott, Flatland was a difficult and dangerous place in which to live. Some of its inhabitants were so thin that, when angry, they could actually run through others, breaking them up into little pieces. Moreover, the Flatland government was highly intolerant of dissenting opinions, jailing the hero of the story for believing that the universe was not really flat. Thanks to insights by the Dutch mathematician, physicist, and astronomer Christian Huygens (1629-1695), however, we may better understand the intolerance and difficult personalities of Flatland's people. In a series of works, Huygens showed that sound waves in our three-dimensional world move "sharply" through space and time. That is, if we are standing at a distance from the source of a gunshot, for a time we hear nothing. Then we hear a rapid report followed again by silence. Not so in Flatland. Here there would also be silence until the sound of the shot reached us. But forever after that time, we would continue to hear sound arising from the shot--a kind of eternal reverberation. Because our world is full of sound, the result would be a constant background noise of varying volume--perhaps resembling the world of a parent who has bought a very powerful stereo system for his or her teenager--and lasting forever! Further support for the assertion that sound propagation is not "sharp" in two dimensions, as it is in three, arises from the analysis of a particular partial differential equation, the so-called "wave equation," by a number of applied mathematicians over the past 300 years. More generally, for spaces having an even number of dimensions (2, 4, 6,...) sound propagation would not be sharp, while in spaces having an odd number of dimensions (1, 3, 5,...), it would be. Knowing this, perhaps a future visitor to an "Evenland" of an even number of dimensions will find many more interesting things to tell us. Abbott's Flatland was published in paperback in its fifth edition by Barnes & Noble Books in 1983. Those interested in learning more about Huygens' Principle, or sound propagation in general, can turn to the book Methods of Mathematical Physics, Vol. II, by R. Courant and D. Hilbert (Interscience Publishers, 1962) or to me for more information. Alan D. Solomon (keywords: sound propagation, Flatland) ------------------------------------------------------------------------ Please send inquiries or comments about this gopher to the mail address: gopher@gopher.ornl.gov Date Posted: 2/7/94 (ktb)