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Since Coxeter’s “Introduction to Geometry“ is a classic, I think I can ask a question referring to it (2.ed, 1969). For me it is important, because it lies at the foundation of how Coxeter defines parallelism in affine geometry.

At the end of his axiomatic outline of ordered geometry, chapter 12, being the basis for affine geometry, Coxeter introduces the concept of rays from a point A being parallel (in opposite senses) to a line r (not going through A), in theorem 12.61: “For any point A and any line r, not through A, there are just two rays from A, in the plane Ar, which do not meet r and which separate all the rays from A that meet r from all the other rays that do not.” The proof is clear to me, but not corollary 12.62 derived from it. It states that for any point A and any line r, not through A, there is at least one line through A, in the plane Ar, which does not meet r. Coxeter does not give a proof, implying that it is obvious.

My understanding is that we do not know yet that the two rays, let’s call them p and q, found in theorem 12.61 are collinear. (This is known, as Coxeter explicitly says, only after introducing axiom 13.11 in affine geometry, which states that there is at most one line through A which does not meet r.) Therefore, we do not know yet that (together with A) p and q form a line which would not meet r. I don’t see how the corollary can be proved from 12.61.

However, Coxeter continues with theorem 12.63 stating that “the parallelism of a ray and a line is maintained when the beginning of the ray is changed by the subtraction or addition of a segment” from the line on which the ray lies. With the help of this theorem, 12.62 could be proved in my opinion: Any point on the line on which ray p of 12.61 lies can be taken as the origin of p, and therefore p can be extended to a line being parallel to r. The same can be done with q. So we have found at most two lines that are parallel to r. If p and q are collinear, they are identical and there is only one line left which is parallel to r.

This seems to be a nice solution, but it has two drawbacks. First, I cannot imagine that someone like Coxeter switches around the order of his propositions by accident. Second, I don’t get the proof of 12.63. It states for the case of addition of a segment to the ray, that the resulting ray “obviously does not meet r”. This is not obvious to me. Of course, from the drawing it seems evident, but not from the axiomatic system and the propositions proved so far. How do we know that the segment added to the ray does not meet r somewhere way back? Thanks for help!

For clarification: Coxeter introduces the notion of parallelism by referring to rays being parallel to a line. In fact, the rays p, q determined in theorem 12.61 are defined to be parallel to the line r. So the condition is not only that they do not meet r, but in addition they separate all the rays that meet r from all the others that don’t. Furthermore, there is a sense (dt.: Richtungssinn) in this parallelism associated not only with the ray, but with the line, too, when looking at its parallelism with a ray. Then, as a consequence of theorem 12.63, the concept of parallelism is extended to two lines being parallel, but still associated with a sense for each of the lines. Only when Coxeter introduces his axiom 13.11 of affine geometry, he comes to the conclusion that “any two lines in a plane that do not meet are parallel”. (BTW: Only at this point, where the collinearity of the rays p and q is shown, the sense associated to parallelism can be abandoned.)

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Well, "more than 1" does not imply "at most 2". Coxeter there is still allowing for hyperbolic geometry as well. So you might think here about a branch of a hyperbola as a representation of the given "line", and take the 2 asymptotes as the 2 parallel lines you're aware of. But then consider all the lines in the angle range in between those asymptotes. All those would not intersect that branch of the given hyperbola neither. Thus those are said to be parallel as well. In fact: "parallel" does not mean to be asymptotic in some sense, it simply means not to intersect!

--- rk

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  • $\begingroup$ Thanks a lot for the hints. I certainly agree with your first sentence. Your reference to hyperbolic geometry is interesting. Please see the additional paragraph at the end to my original question, since it was too long for a comment here. This said, I am still not convinced by Coxeter’s proofs of 12.62 and 12.63. $\endgroup$ – Roland Salz Dec 16 '18 at 13:10
  • $\begingroup$ Two more hours of thinking about the case of hyperbolic geometry you mentioned helped me to find the necessary hint in Coxeter which I had overlooked and which clears up both proofs. It was right in front of 12.62, referring to hyperbolic geometry! Thanks again! $\endgroup$ – Roland Salz Dec 16 '18 at 16:09

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