Why can't elliptical space exist?

[acesuv]

I'm reading a book called Euclid's Window, and in passing the author says that elliptical space cannot exist (something analogous to the surface of a sphere). However, curved and flat spaces can exist.

Why is that?

 

[Bacle2]

Do you know the author's actual definitions? If not, I assume s/he is referring to the fact that projective spaces for n>1 are not embeddable in lower Euclidean spaces.

 

[micromass]

I'm reading a book called Euclid's Window, and in passing the author says that elliptical space cannot exist (something analogous to the surface of a sphere). However, curved and flat spaces can exist.

Why is that?

Can you give the exact quote?

 

[acesuv]

Do you know the author's actual definitions? If not, I assume s/he is referring to the fact that projective spaces for n>1 are not embeddable in lower Euclidean spaces.

Can you give the exact quote?

"Like Poincare, Reinmann gave his own interpretations of the terms point, line and plane. As the plane, he chose the surface of the sphere. His points, like Poincare's, were positions, in the manner of Descartes considered to be pairs of numbers, or coordinates (essentially the point's latitude and longitude).Reinmann's lines were the great circles, the geodesics on the sphere.

As in Poincare's model, it must be confirmed that Reinmann's admits consistent interpretations of the postulates. Now might be a good time to recall that it had been proved that elliptic space could not exist. Sure enough, Reinmann's model did turn out to have a few little problems. It was one thing to create a new space based on a new version of the parallel postulate; Reinmann's space was inconsistent with the existing versions of other postulates as well..."

Euclid's Window, PG 138-139

 

[pasmith]

"Like Poincare, Reinmann gave his own interpretations of the terms point, line and plane. As the plane, he chose the surface of the sphere. His points, like Poincare's, were positions, in the manner of Descartes considered to be pairs of numbers, or coordinates (essentially the point's latitude and longitude).Reinmann's lines were the great circles, the geodesics on the sphere.

As in Poincare's model, it must be confirmed that Reinmann's admits consistent interpretations of the postulates. Now might be a good time to recall that it had been proved that elliptic space could not exist. Sure enough, Reinmann's model did turn out to have a few little problems. It was one thing to create a new space based on a new version of the parallel postulate; Reinmann's space was inconsistent with the existing versions of other postulates as well..."

Euclid's Window, PG 138-139

I think the author is saying that Reinmann's model (or possibly Poincare's) doesn't admit of the possibility of elliptic spaces, and noting that this is a defect in that model.

 

[lavinia]

When I Learned plane geometry, two postulates were: two lines in a plane can intersect in no more than one point: a line in a plane separates the plane into two half planes.

On a sphere, two lines(great circles) always intersect in two points(opposite poles). One can try to fix this by identifying all opposite poles to make lines intersect in exactly one point. But then the line will not separate the plane into two half planes.

 

[HallsofIvy]

I'm reading a book called Euclid's Window, and in passing the author says that elliptical space cannot exist (something analogous to the surface of a sphere). However, curved and flat spaces can exist.

Why is that?

You need a bit more than that. I suspect the author was talking specifically about Eucidean and non-Euclidean geometries. The distinction between Euclidean and non-Euclidean geometry is the "parallel postulate", typically given today as "Playfair's axiom", "There exist exactly one line through a given point parallel to a given line" (equivalent to Euclid's original postulate). "Non-Euclidean" geometries deny that axiom.

And, since it says "there exist exactly one line", there are basically two ways to deny that:
1) (Hyperbolic geometry) "There exist more than one line through a given point parallel to a given line."
2) (Elliptic geometry) "The exist no line through a given point parallel to a given line."

We can construct many models for "hyperbolic geometry" which obey all the other postulates but when we attempt to do that for "elliptic geometry" we hit a snag- we find that any geometry obeying the "elliptic" parallel postulate also violates another postulate of Euclidean geometry- that two lines intersect in no more than one point. So there cannot exist an elliptic geometry that violates only Euclid's parallel postulate.

But if we are willing to allow violation of other postulates as well there exist many useful elliptic geometries. For example "spherical geometry", in which "points" are points on a given sphere and "lines" are great circles, has the property that, not only are there no parallel lines (all lines intersect) but, in fact, all lines intersect twice!

 

[lavinia]

We can construct many models for "hyperbolic geometry" which obey all the other postulates but when we attempt to do that for "elliptic geometry" we hit a snag- we find that any geometry obeying the "elliptic" parallel postulate also violates another postulate of Euclidean geometry- that two lines intersect in no more than one point. So there cannot exist an elliptic geometry that violates only Euclid's parallel postulate.

The other possibility is that two lines intersect in exactly one point but that a line does not separate a plane into two disjoint half planes.

 

[mathwonk]

Since Euclid's postulates were incomplete as given, it is a little tricky for me at least, to talk precisely about logical equivalences and other fine distinctions among them and other versions. The 5th postulate said essentially there is at most one line through P, parallel to a given line. To get at least one, he first proves the exterior angle theorem, which uses in all likelihood, the SAS theorem which Euclid's own postulates do not allow to be proven.

His postulate #2 also does not make clear for instance, when a finite segment is extended "arbitrarily" to a line, whether that line is allowed to double back on itself, as in the case of geodesics on a sphere. Without this, e.g., on a sphere, I believe the exterior angle theorem is false. Which is why one does not get parallel lines there.

Euclid also never postulated that two lines meet in at most one point, although he claims it is clear, in a proof.

So Euclid's own postulates allow a good bit of fun with variations and speculation.