Re: Triangle with more than 180 degrees-

*** wrote:
Hero wrote:

> In hyperbolic space one has even an absolute measure, different from
> the geometry of Euclid.

That is the point. There is no absolute measure. There is only the
measure you define.

> 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.

You are wrong here. Take any model of hyperbolic geometry, and *use*
the metric defined on that model. Let's have a look at Klein's model.
He defines it on a disc x_1^2 + x_2^2 < 1. Let me define:
delta(x,y) = 1 - x_1.y_1 - x_2.y_2.
Then he defines a distance function as:
d(x,y) = a.arcosh[delta(x,y)/sqrt(delta(x,x).delta(y.y))]
See the constant a in front. Changing that constant changes distances,
but does not necessarily change angles, once you have defined angles.

I'm afraid, You are applying Euclid's geometry. In this, one can
scale, change a constant to change distances. The essence of non-
similarity of hyperbolic geometry is a constant one can not change,
it's a natural constant - if our space is non-euclidian. But if it is,
the model has to be done with this geometry too. Changing the constant
would be a change to the space, we live in.
In Euclid's geometry scaling has a lot of invariants, the relation of
lengths, the angles, the relation of diameter to the circumference of
a circle,... but there is f.e.the relation between a closed surface
and the volume, it encloses: the size of the skin in relation to the
volume of the body changes with scaling, therefore a mouse has
different temperature regulation to an elephant.
In hyperbolic geometry are a lot more changes, when one maps points
onto a map or into a model.

> Changing the size of a square in here will also change it's angles.

Well, that entirely depends on how you define angles.

I do not believe You, unless You show this to us.


> 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.

They are different when you look at them from an Euclidean standpoint.

You mean, taking an hyperbolic point of view let straight lines in
both geometries look the same?

What you are bothering about is, I think, that when you live in a hyperbolic
space how it is possible that you experience only Euclidean space. The
point is that at a small scale all spaces are nearly Eucliden, the difference
is negligible. Your map of your favourite city will be Euclidean, and you
will find that it fits perfectly well with what you experience. Nevertheless
there are distortions, but they are negligible.

So in consequence hyperbolic geometry is negligible?
And with this mapping of my favourite city we are back to the
beginning. Here are triangles with a sum of angles of more than 180
degrees, which is true, as long as the edges of these triangles are
the following the curvature of earth's surface, ( rising about 2
meters above a direct distance of 10 km), of course in the geometry
of Euclid.

With friendly greetings
Hero

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