Re: dimensions

Hara Ra (
Fri, 16 Jan 1998 14:40:28 -0800

> Anton wrote
> : The Cartesian product of a line (d=1) and a line (d=1) is a plane (d=2).
> : The Cartesian product of a line (d=1) and a point (d=0) is a line (d=1).

In a continuious manifold, the dimensionality is the number of real
numbers required to uniquely define a point therin. A line needs 1
number, a plane 2 numbers, etc. For flat eculidian space this is easily
used to calculate a lot of other things, for example, distance = sqrt
(x*x + y*y + ....). It gets harder for noneuclidian spaces - for example
the surface of a torus.

Though it is possible to map all the points on a plane to a line, and
all points in a 3 space to a plane and therefore to a line, such
mappings violate continuity, which requires that a small displacement of
the point result in a small change in the real number tuplet which
defines the location of the point.

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