The discussion centers on the impossibility of elliptical space existing, particularly in relation to Euclidean and non-Euclidean geometries. It highlights that while curved and flat spaces can exist, elliptic geometry cannot be constructed without violating other Euclidean postulates. The conversation references Riemann's and Poincaré's interpretations of geometric concepts, specifically the parallel postulate, and emphasizes that any geometry adhering to the elliptic parallel postulate also contradicts the principle that two lines can intersect at most once. The conclusion drawn is that elliptic geometries, such as spherical geometry, require the violation of additional postulates, making them inconsistent with traditional Euclidean frameworks.
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acesuv said: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 said: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.
"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.micromass said:Can you give the exact quote?
acesuv said:"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
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.acesuv said: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?
HallsofIvy said: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.