Re: Triangle with more than 180 degrees-

*** wrote:
Hero wrote:
> *** wrote:

...
> > > > Yes, once you have defined things. But when you want to think about
> > > > hyperbolic spaces it is best to look at some of the possible models.
> > > > There are four well known models for the hyperbolic plane. They are
> > > > explained in <http://en.wikipedia.org/wiki/Hyperbolic_geometry>.
> > >
> > > This is application of Euclidian geometry too.
> > > Models are not realisations, here of a hyperbolic space.
> >
> > They are realisations. Moreover, they make it possible to visualise how
> > things work in hyperbolic space, and such give insight.
>
> Call them, whatever You like. But they are all done with the geometry
> of Euclid, so all four models have their shortcomings, f.e.they are
> done in a disk of finite radius, but the plane should have infinite
> size.

You read wrong. The metric is defined in terms of Euclidean metric, but
in the new metric the planes *have* infinite size. Size is related to the
metric used.

Using models, one has the metric of the model and the metric of the
realisation, or whatever You call it. Except for a model, which is one-
to-one, we have to calculate, f.e. 3 cm on a map of England might be
21 miles, when traveling through the landscape.
In hyperbolic space one has even an absolute measure, different from
the geometry of Euclid. In the geometry of Euclid, on the other hand,
one has similarity, which allows us to reduce a distance of 21 miles
to 3 cm of a model, but keeping angles and the numerical relations
between lengths, which is impossible in hyperbolic geometry. So in
assuming the space, we live in, as a hyperbolic space, a model is
quite different from what we are used to in the geometry of Euclid.
Changing the size of a square in here will also change it's angles.

> > Have also a look at <http://en.wikipedia.org/wiki/Hyperbolic_space>.

> Again, these geometers start with curvature first. And with this and
> the models it's inside the geometry of Euclid and Euler.

Yes. So what? Should they start completely anew? Without any possibility
to visualise what you have? When you simply abolish the parallel postulate
and replace it by a postulate that there are more than one line in parallel
with a given line through a given point, you will not get anywhere. You
may have geometries with varying curvature or with constant curvature.

One can start with the geometry of Euclid and can look at surfaces of
negative curvature. I wrote about it: Think of hyperbolic paraboloid
or the hyperboloid of one ***, where there are straight geodesics
and not straight ones. In fact they are generated as ruled surfaces
from straight lines.
This is geometry based on Euclids definitions and axioms, just as
spherical geometry was done too. Before the advent of hyperbolic
geometry Euler and other worked out the curvatures of surfaces,
surfaces of negative Gauss-curvature were known under the name of
anticlastic surfaces and according to Dupin's indicatrix one could
classify differences of surfaces.

Now, as hyperbolic geometry is non-euclidian, and therefore it starts
with a different axiom, one can not work with a concept of curvature,
which is defined upon the geometry of Euclid. And what is shown to us
as straight lines and as circles is obviously different.

With friendly greetings
Hero

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