Re: The sphere and hyperbolic geometry

Ken Quirici wrote:
On Dec 24, 10:08 pm, "I.M. Soloveichik" <ims...@xxxxxxxxxxxxx> wrote:
Euclidean geometry, hyperbolic and elliptic geometry are categorical which means there is only one of them---except there can be several diferent representations or models. But in any of these geometries 2 points determine a line, so this also holds in any model you come up with. Maybe your geometry satisfies only some of the axioms needed to categorize it.

izzy

1. by categorical, do you mean for example that the surface of a 3-
dim. sphere can have only one of the types of
geometry defined on it, but this type may have several different
models? I have no reason to argue against this.

2. My geometry - the surface of the sphere, all points and all circles
- may indeed satisfy only some of the
axioms listed for Euclidean geometry, and claimed (by Mathworld at
least) to hold - the first four, specifically
are at issue. And 2 points determining a line is definitely NOT
satisfied by my model - in fact two points determine
an infinite number of lines (circles) in my model. So you may be right

The model I've heard about for elliptic geometry (which gives
a 2-dimensional "elliptic plane") starts with the surface
of a sphere and identifies antipodal points, those that
lie on a line through the center of the sphere.

That way, the Northern hemisphere plus the equator
has a copy of every point. Also, from the North Pole,
the farthest points are all on the equator, and at
the same distance.

Two points on the equator which are antipodal are
identified. So from the North Pole to 0 lat.,
0 longitude, there seem to be two lines:

- follow a meridian from North Pole to (0,0).
- follow a meridian from North Pole to 180 longitude,
zero latitude.

I don't know whether professional geometers would
accept the "identify antipodal points" model ...

David Bernier


thst it is not a valid hyperbolic geometry in the
'officisl' sense. However as I said before Mathworld lists the first
axiom as - between any two points there is a
straight line - it doesn't specify there is only one. But it's the
third that baffles me - any line segment can
be the radius of a circle whose center is one of the endpoints of the
line segment. How the heck does a circle
or arc of a circle act as the radius of a circle?

I suspect that the first 4 axioms of Euclidean goemetry can be
satisfied by many systems that don't at first
glance appear to satisfy them - it all depends on how you define lines
and parallelism (and given the third
axiom, 'circles' in some official sense).

Isn't it true that the sphere with great circles is an Elliptic
geometry - there are no parallel lines? Every line
intersects every other line. Mathworld explicitly makes this claim,
that the sphere with great circles is an
Elliptic geometry - but it seems to violate the 'infinite extension'
axiom, for one.

--
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