PICK A NUMBER: WHAT EDWIN ABBOTT DID NOT KNOW ABOUT FLATLAND
   
   
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   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)
   
   
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   Date Posted:  2/7/94  (ktb)