Re: lengths are 'relative' in Euclidean geometry
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Tue, 26 Feb 2008 03:08:46 GMT
In article
<0d3301b7-2737-4fc8-8d11-888d2e4f1354@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Lax <Lax.Clarke@xxxxxxxxx> wrote:
In a book I'm reading it says that 'In Euclidean geometry lengths can
be measured only in terms of some arbitrary unit BECAUSE SIMILAR
NONCONGRUENT FIGURES EXIST.'
Can someone explain how the capitalized part is a determining factor
for whether length units are arbitrary or not for a geometry?
I'm not sure I can explain it, but I can tell you that in non-Euclidean
(Lobachevskian) geometry, similar figures are always congruent,
and there is an absolute unit of length. If you know a Euclidean
triangle has angles 60-60-60, then you don't know anything about
how big the triangle is - but if you know a Lobachevskian triangle
has angles 50-50-50 (say), then you know exactly how big the
triangle is. And you can use that to define an absolute unit of length.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
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