Where’d Monty go? I wanted him to stick his foot down his throat and gargle some more.
Have I mentioned I love geek fights? (And I’m with ultrafilter, Math Geek, and friends)
What I love is that Medea’s Child, posting on page 2, is the only person besides me to use any synonym for “geek”. This place is great.
ultrafilter,
What is your definition of intersect? The intersection of L and L would be L. I don’t have the axioms handy, but shouldn’t the axiom be that two distinct lines intersect in at most one point?
The distinction between ordinary points and points at infinity does not exist in the axioms of projective geometry. Since all provable statements within projective geometry are derived from the axioms, projective geometry cannot make any distinction between ordinary points and points at infinity. The Euclidian plane together with a line at infinity form a model for the axioms. The distinction exists only in the particular model, not in the geometry.
ultrafilter
I know some terms in mathematics differ in meaning to their real world usage, but I didn’t realise that was the case with this term. If all ‘parallel’ lines need to do is not intersect, that means that any lines anywhere in the world could be called parallel just because they don’t meet. Is that right? Very strange. I thought parallel lines would have to be, well, parallel.
Consider this for a definition of parallel:
Two distinct lines l and m are parallel if and only if for every point P that lies on (is incident with) l, P does not lie on m.
This guarantees non-intersection of parallel lines.
DrMatrix: Yes, it should be two non-distinct lines etc. etc. In fact, let us take the following premeses as axioms of “incidence geometry”
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For any two points P and Q, there exists a unique line l such that P and Q both lie on l.
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Every line contains at least two distinct points.
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There exist at least three points such that no line contains (is incident with) all three.
As a result the statement “For any two lines that are not parallel, there is a unique point that lies on both lines” follows as a theorem. (Proof: use contradiction by axiom 1)
DrMatrix
It was an inference, actually, which was what he needed both his definition and his postulate for. I spoke sloppiliy when I said that he “defined” parallel lines in Proposition I.27.
He actually defined them in Definition 23:
“Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” — The Elements
At any rate, I’ve conceded the point to you already, thanks to the compelling arguments made here (particularly by Math Geek), and I accept that your understanding of parallel lines is the correct one.
Yes, as long as (a) the lines lie in the same plane and (b) the lines are both infinite and not just line segments.
MathGeek - OK, thanks. It seems odd to me that they would conrtinue to call them parallel lines, but I guess a lot of terminology in most disciplines seems strange to outsiders. shrugs
Math Geek
Oddly, despite what has been said about Euclid’s obsolescence, your definition seems to reflect his exactly.
Libertarian, there has been some confusion in this thread about what Euclid’s definition of “parallel” was, starting with your post on the first page which incorrectly identified Proposition I.27 as the definition of parallel lines. But yes, you are correct that Euclid’s definition of “parallel” as stated in Definition I.23 does agree with what we’ve been discussing here.
To which I answer with a resounding “So?”, because not only is that rather off-topic, it doesn’t change the fact that Euclid’s construction of geometry is not the construction used in modern mathematics.
I think you’re stuck inside a particular model of the projective plane. But even if it’s the normal one, it’s not the projective plane; it’s just an instance. We could say just as accurately that a 3-dimensional vector space V over a field is a projective plane, with ‘point’ being a 1-dimensional subspace of V, ‘line’ being a 2-dim’l subspace, and ‘incident’ being ‘containing’ or ‘contained in’. Now, where’s that point at infinity?
RTFirefly: You may be right. I need some time to think about it.
A pit thread on geometry.
A freakin’ pit thread on the variations in definitions, postulates, and theorems in classical and modern mathematical world models.
You guys are so esoterically weird!
I love this place.
Tris
“You can tell whether a man is clever by his answers. You can tell whether a man is wise by his questions.” ~ Mahfouz Naguib ~
Off topic? Off topic? I was responding to YOUR post! I was responding about parallel lines! In a thread about … parallel lines!
[…shaking head in disbelief…]
Isn’t it enough that I conceded your point on the first page, recognizing you as an authority on matters mathematical? You feel the need to kick sand in my face as well? Stalin on a bayonette!
Libertarian: upon second reading, my last reply to you does seem to be needlessly harsh. There was no need to be snippy to you; I apologize.
As far as being an “authority” goes: I do appreciate the compliment, but if you’re not careful you’re going to give me airs 
I will claim to know something about hyperbolic geometry, as that’s central to the topic of my thesis. But beyond that the rest of the “math crowd” are at least as much an authority as I am: ultrafilter, DrMatrix, RTFirefly, and so forth.
Thanks. That’s good information to have. I’ll rely on their expertise as well. And thanks for the gracious apology, Math Geek.
Ain’t it the greatest? Where else would this happen?
As I understand it (at least, from my notes for my axiomatic geometry class), in projective geometry, all we can state are the following:
1)Axioms 1-3 in my post above hold.
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Axiom 2 is strengthened to state that for any line, there are at least three points on that line.
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The “Elliptic Parallel Property” (any two distinct lines intersect) holds.
I think this is logically equivalent to what RTFirefly said above. Furthermore, for a line l, we define an equivalence class of l, [l], as the set consisting of l and all lines parallel to l. Moreover, we define [l] to be a point at infinity (mostly for historical reasons), and say that [l] lies not only on l, but also every line parallel to l. This satisfies (3) above. Of course, the “line at infinity” is just the unique line defined by the points at infinity.
As an example, consider the following model:
Take the set {A,B,C,D,E,F,G} and define a point to be any one of those letters. A line is considered to be any of the sets {A,C,G} {A,B,D} {A,F,E} {G,E,D} {G,F,B} {D,F,C} or {C,E,B}. {C,E,B} is considered to be the line at infinity. C, E, and B are all points at infinity, despite the fact that there is nothing “infinite” about this model.
Oh, and I agree with RTFirefly and Triskadecamus. It was exactly this sort of thing that got me to finally register.
Qir nha
What makes the points C, E, B points at infinity? I don’t see anything special about them.
Oh yeah, welcome to the Straight Dope. Stick around.
It’s really more a matter of notation than anything else. However, if we talk about the projective plane as an extension, or perhaps a completion, of the affine plane (any model of incidence geometry (whiose axioms I gave in my first post) having the Euclidean parallel property – for any line l, and any point P not on l, then there exists a unique line m through P, parallel to l) then it makes some sense for those points to be the ones at infinity. Consider a four-point model consisting of {A,D,F,G}, with lines consisting of any two points, since my second axiom is not strengthened in the affine plane. So, my lines are {A,D} {A,F} {A,G} {D,F} {D,G} and {F,G}.
This gives us a model with the Euclidean parallel property. (I’ll let you prove this yourself.) Then, based on our definition of equivalence classes (which we decided to call points at infinity). We also defined [l] to lie on [l] and any line parallel to [l]. So, if {A,D} is l, then B=[l]. Moreover, {G,F} is the line parallel to {A,D}. If you look at my example above, you’ll see that indeed, B lies on {G,F}. Similarly, C lies on both {A,G} and {D,F} (again, both of which are parallel in my four-point model), and E lies on both {D,G} and {A,F}. Naturally, that makes {B,C,E} the line at infinity.
Of course, we could have picked some other four-point model, which would have changed our points at infinity.
And I’m definitely sticking around. This place is great. (Two years of lurking was enough to tell me that much.)