Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 David Bernier Posts: 3,892 Registered: 12/13/04
Re: Can the Lobachevsky plane be embedded into R^3 ?
Posted: Mar 15, 2007 10:55 AM

Hero wrote:
> Denis Feldmann wrote:
>> Hero a écrit :
>>

>>> Thomas Mautsch wrote:
>>>> Hero wrote:
>>>>> On 5 Mrz., 22:07, k...@msiu.ru wrote:
>>>>>> Can the Lobachevsky plane be embedded into R^3 ?
>>> ......
>>> ......

>>>>> Lobachevsky is contradicting Euclid's fifth postulate.

>>>> This is completely irrelevant nonsense!
>>> You can read in Wiki:
>
>> Do you realize at what level the answer was posted ? Thanks for
>> reminding us all basic truths, and insisting on your misconceptions...
>> It was *still* irrelevant (nonsense, I would not have been so harsh on
>> first posting, but now...)
>>

>>> ,,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

>> Somehow, I had missed that. Who is that guy again?
>
> Here is that guy Lobachevsky:
> Geometrical researches on the theory of parallels
> http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAN2339
>
> Lower level of completely irrelevant nonsense...
>

>> Was Euclid wrong,
>> after all?

>

This is what I think: I suggest that you think about geometry on
a spherical surface. Given points A and B on a sphere, if A !=B, then
the shortest path on the sphere from A to B is an arc of
circle whose center = center of sphere. So geodesics,
"shortest-route" curves, are formed by great circles, those circles
with center = center of sphere. In the Euclidean plane,
every straight line (geodesic of the plane) has parallel
companions, i.e. lines distinct from it, but which never
intersect it.

Going back to a spherical surface, can two geodesics fail to
intersect? The answer is no. Note that the geodesics
on the sphere are circles in space, not straight lines.

As far as I know, what I have discussed with geodesics
of a sphere is close to an example of an embedding.

Let's say: in an embedding of some Riemannian surface
in R^n, the geodesics of the surface often are not
made up of straight line segments of R^n.

David Bernier

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