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Topic: Can the Lobachevsky plane be embedded into R^3 ?
Replies: 26   Last Post: Mar 28, 2007 4:03 PM

 Messages: [ Previous | Next ]
 Hero Posts: 1,052 Registered: 12/13/04
Re: Can the Lobachevsky plane be embedded into R^3 ?
Posted: Mar 12, 2007 7:31 AM

Thomas Mautsch wrote:
> Hero wrote:
>

> > On 5 Mrz., 22:07, k...@msiu.ru wrote:
> >> Can the Lobachevsky plane be embedded into R^3 ?
......
......
>
> > The answer is no.
> > Lobachevsky is contradicting Euclid's fifth postulate.

>
> This is completely irrelevant nonsense!

,,Lobachevsky would instead develop a geometry in which the fifth
postulate was not true
...
Lobachevsky replaced Euclid's parallel postulate with the postulate
that there is more than one parallel line through any given point; a
famous consequence is that the sum of angles in a triangle must be
less than 180 degrees. ,,
http://en.wikipedia.org/wiki/Nicolai_Ivanovich_Lobachevsky

> Euclid's fifth postulate for *PLANE* geometry does
> not prevent the existence of an explicit smooth proper
> isometric embedding of the hyperbolic plane into R^6 (Blanusa);
> so why should it contradict an embedding into R^3 ?!

Because in geometric space, which has the structure of R^3, Euclid's
fifth postulate is true.

With friendly greetings
Hero

Date Subject Author
3/5/07 ksp4@msiu.ru
3/6/07 Hero
3/11/07 Thomas Mautsch
3/12/07 Hero
3/12/07 Denis Feldmann
3/12/07 Hero
3/15/07 Hero
3/15/07 Rainer Rosenthal
3/15/07 Hero
3/15/07 Denis Feldmann
3/15/07 Rainer Rosenthal
3/15/07 Hero
3/15/07 David Bernier
3/16/07 Hero
3/15/07 Denis Feldmann
3/15/07 Rainer Rosenthal
3/15/07 Hero
3/15/07 Denis Feldmann
3/15/07 David Bernier
3/16/07 narasimham
3/17/07 narasimham
3/17/07 narasimham
3/17/07 narasimham
3/18/07 Thomas Mautsch
3/28/07 narasimham
3/28/07 JEMebius
3/28/07 Chan-Ho Suh