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Jan
28
accepted In neutral geometry, can a family of parallel lines leave holes in the plane?
Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
Thank you all. I've been over-thinking this one. @hardmath After you clarify your answer, I'll accept it.
Jan
28
revised Number system Problem
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Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
@hardmath Prop. 12 is true in neutral geometry and this implies that given a line $l$ and a point I could construct a parallel line through that point. Hence, I could construct a line parallel to $l$ through every point on the plane, but such lines might not be all perpendicular to one line. Indeed, they might not be parallel to each other.
Jan
28
revised In neutral geometry, can a family of parallel lines leave holes in the plane?
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Jan
28
awarded  Student
Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
A quick proof that two lines perpendicular to a given line are parallel: If they were not parallel, they would intersect forming a isosceles triangle with two 90 degree angles. the complimentary angle to either angle would also be 90 degrees. But by the exterior angle theorem, that exterior angle must be strictly greater than 90 degrees, which is a contradiction.
Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
"... in hyperbolic geometry the relation of parallelism between lines is not transitive" -- true indeed; nonetheless, the set of lines perpendicular to a given line will all be parallel to each other.
Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
There are parallel lines in neutral geometry as mentioned in the comments to the question. In the geometry of the sphere, two points do not determine a unique line (using the poles as the points). So it does violate a very basic axiom. In elliptical geometry, a line does not divide the plane into two disjoint sets (the two sides of the line). So that also does not satisfy all the axioms of neutral geometry. Euclidean and hyperbolic geometry are the only two models of neutral geometry that I am aware of.
Jan
28
answered Number system Problem
Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
Also, it is true in neutral geometry that for each line and each point not on the line, there exists a line passing through the point and not intersecting the given line, but that does not address my question, which is about the family of parallel lines perpendicular to a given line. Does one of those parallel lines pass through each point on the plane?
Jan
28
comment In neutral geometry, can a family of parallel lines leave holes in the plane?
I imagine there are quite a number of axiom systems for Euclidean geometry, but one of the axioms will usually be equivalent to "Given a line $l$ and point $P$, the line parallel to $l$ passing through $P$ is unique." Clairaut's axiom and the angle-sum postulate are examples. An axiom system for Euclidean geometry without that axiom is an axiom system for neutral geometry. Adding the axiom "Given a line $l$ and point $P$, there at least two lines parallel to $l$ passing through $P$" to neutral geometry gives you hyperbolic geometry.
Jan
28
asked In neutral geometry, can a family of parallel lines leave holes in the plane?
Jan
26
awarded  Teacher
Jan
26
awarded  Editor