Re: Triangle with more than 180 degrees-

In article <4a9cd241-edd1-42f0-8c00-2338bbf4e397@xxxxxxxxxxxxxxxxxxxxxxxxxxx> Hero <Hero.van.Jindelt@xxxxxx> writes:
*** wrote:
....
> So the space of euclid is given and we can connect points in this.
> In the disc geometry we are not allowed to leave the disc.

And how about 4-D space? Are you allowed to leave 3-D space for new points
to connect?

I'm afraid, You have to find a forth spatial dimension first.

Did you ever read Flatland?

Any discussion of geometry should be done within the realm of
discourse. Obviously, if you go outside that realm other things can
happen. I have defined Euclidean and hyperbolic geometries on the
inside of a disc. The inside of the disc is what we are talking about,
and the definition of straight lines.

The inside of what?

The inside of a disc.

You wrote (some postings back):
Consider the following:
take a function f: [0,1) -> [0,oo) that is monotonously increasing
(x > y -> f(x) > f(y)), has f(0) = 0 and when x goes to 1, f(x)
grows without bound.
Now consider the inside of the unit circle in Euclidean 2D space.

You considered 2D space and with Euclid this is always inside 3D
space, for which he defined his geometry, of course treating 2D planes
and other forms of surfaces as well.
One has to be careful, if someone is using the wording 'euclidian', as
this might differ from the elements of Euclid.
So sorry, we can not contain the geometry of Euclid in a finite disc.
The geometer is living inside the realm of discourse.

But if we restrict the realm of discourse to the inside of a disc, the
geometer is living inside that disc. As an observer, you can do things
outside that disc, but still the geometer is living inside it.

But even here, we can do something different. Take a hyperbolic geometry
inside that disc, with a definition of straight lines, and of distance.
Consider any function f(r) with:
(1) when r_1 > r_2, f(r_1) > f(r_2).
(2) f(0) = 0.
(3) when r goes to 1, f(r) increases without bounds.
This is an invertible function, say the inverse of f is g.
Now the following definitions (using polar coordinates in the 2-D plane):
d((r_x, phi_x), (r_y, phi_y)) = d'((g(r_x), phi_x), (g(r_y), phi_y))
where d' is the distance fnction on the hyperbolic disc, and d is the
distance function on the plane. Further, we call a line
a.r.cos(phi) + b.r.sin(phi) = c
straight if
a.g(r).cos(phi) + b.g(r).sin(phi) = c
is straight in the hyperbolic disc (by the definition provided). This
gives a hyperbolic geometry to the 2-D plane. And not one of them, but
a whole lot. In the same way you can define a whole lot of different
Euclidean geometries on the plane.

Doing that for elliptic geometries is problematical, because for them you
need bounds.

The whole still hinges on: "what are straight lines"? Once you have
defined what straight lines actually are, you get something that either
is Euclidean or hyperbolic.

What makes straight lines so problematic?

The lack of definition.

I only see attempts to make geometry and the universe finite,
bordered, bounded, limited, enclosed, contained and having ends, one
should stay inside.

But for the geometer who is living inside such a disc, the geometry as
I described is would look infinite.

So i want to thank You for this discussion, in which You gave me the
opportunity to promote some old truth, like a triangle can not have a
sum of angles exceeding 180 degrees, when it's edges are not curved,
but lying equal to the points on them.

You are still reasoning from Euclidean geometry only, refusing to consider
definitions that lead to something different.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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