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!
You can read in Wiki: ,,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.