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.
Scientists believe that elliptical space cannot exist because it contradicts the fundamental principles of geometry and physics. In elliptical space, parallel lines would eventually intersect, and the angles of a triangle would add up to more than 180 degrees, which goes against the laws of Euclidean geometry. Additionally, the concept of gravity in elliptical space would not follow the inverse square law, making it incompatible with Einstein's theory of general relativity.
While it is possible to imagine a universe with elliptical space, that does not necessarily mean that it can actually exist. The principles of mathematics and physics are based on empirical evidence and observations, not just imagination. Therefore, even if we can imagine a universe with elliptical space, it does not mean that it is a plausible or scientifically valid concept.
No, there has never been any evidence or observations that support the existence of elliptical space. In fact, all of our current understanding and observations of the universe align with the principles of Euclidean geometry and general relativity, which do not allow for elliptical space. Until there is empirical evidence that suggests otherwise, scientists will continue to reject the idea of elliptical space.
While it may be interesting to explore and consider different theoretical concepts, there is currently no practical benefit to considering the possibility of elliptical space. This is because there is no evidence to support its existence, and it goes against our current understanding of the laws of physics. Scientists must focus on theories and concepts that are supported by evidence and can be tested through experiments.
It is always possible that our understanding of physics could change in the future, but it is highly unlikely that it would change to allow for elliptical space. The principles of Euclidean geometry and general relativity have been extensively tested and confirmed through experiments and observations. In order for our understanding of physics to change, there would need to be substantial evidence and observations that contradict these principles, which has not been the case for elliptical space.