File spoon-archives/heidegger.archive/heidegger_2003/heidegger.0306, message 34


Date: Sat, 7 Jun 2003 22:12:45 -0400
From: Ed Wall <ewall-AT-umich.edu>
Subject: The dimension of a manifold


Michael E and Stuart

    For some reason I thought this might interest you:

Howard Eves _A Survey of Geometry_  (Boston: Allyn and Bacon, 1965),
volume 2 page 55 reads

Ideas like [circular coordinates] led to wide generalization in analytic
geometry, and, in 1865, caused Plucker [two dots over the u] to develop a
dimension theory.  If we define a manifold of geometrical elements to be a
set of geometrical elements that can be labeled with a real, continuous
coordinate system, then the dimension of the manifold is the number of
coordinates needed to determine a general element of the set... Although
[by this definition] space is three-dimensional in points, it may be
shown that it is four-dimensional in lines, and also in spheres.  It is
three-dimensional, however, in planes. In short, dimension depends 
not only on the "space," but also upon the fundamental elements that 
make up the space.  For the first time, geometers had defined 
perceivable spaces of more than three dimensions.


Ed Wall


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