Now let’s talk about over-lapping concentric circles.
And as the Good Knight departs, a small amount of steam may observed departing this peasant’s nostrils. Must be from battles with the Spaniards.
Anybody wanna wager that they intersect in the Time Cube universe?
For the record, Parallel Lines was a damn fine album, albeit not Blondie at their recording best.
Reminds me of that book that was so popular in my high school: Hymen: the Story of the Bilabial Stop.
I’d also like to see some support for your claim that some parallel lines do intersect.
I’ll thank both Math Geek and ultrafilter to get off their tushes next time and look it up in the closest encyclopedia. That way you can look at the figures referred to in the quoted portion above.
Monty: parallelism isn’t a transitive relation in Bolyai-Lobachevskian geometry. Take a line l and a point P in such a geometry. All the lines through P which are parallel to l are not parallel to each other, because they intersect at point P. Sure, it’s different from the way it would be in Euclidean geometry, but that’s no surprise.
Agree. Never said otherwise. The point is that it doesn’t seem outrageous to me to define “parallel” using either condition and excluding the other.
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On the surface of a sphere, the only lines are circles with the same radius as the sphere. So the Tropic of Cancer isn’t a line, in spherical geometry. **
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This would depend on your definition of a “line,” no? (I’m sure the definition you propose is reasonable, however.)
I don’t need to look up the article, as I have already seen diagrams of the type described many, many times. However, as ultrafilter has already explained, you are interpreting that article incorrectly. The situation described contains lines which are parallel, and lines which intersect, but it contains no pairs of lines which are both parallel and intersecting.
So no, you won’t thank me for getting off my tush since you are, in fact, wrong.
Aside to lucwarm:
It’s worth thinking about the reason why smaller circles, like the Tropic of Cancer, aren’t considered lines in spherical geometry. Imagine two topologists standing at two different points on the earth with the same longitude but different latitudes: one at the equator and one on the tropic of Cancer. Suppose that at the same time the both roll marbles along their respective lines of latitude, at the same speed and in the same direction (say, west). Ignoring those pesky mountains and oceans and stuff, the marble on the equator will stay on the equator but the marble on the Tropic of Cancer will not. Instead the second marble will follow a circular path the same length as the equator which starts and stops on the Tropic of Cancer and is tangent to the Tropic of Capricorn at its far point. If the second topologist wanted his marble to stay on the Tropic of Cancer, he’d have to continually nudge it away from the equator.
So in a sense, the Tropic of Cancer isn’t heading in the same direction as the equator at all; instead it’s continually curving away towards the pole. The fact that it can do this and maintain the same distance from the equator at all times is one of the things which makes spherical geometry different from Euclidean.
Agree. I would add that I’ve thought about your explanation and it seems to me that in a spherical system, the shortest distance between two points would be along a line (as you define it).
Thanks for the support, but . . . Let me clear something up here. I studied computer science and math (including Euclidian and non-Euclidian geometries). I am a computer programmer/analyst, not a physicist. Physics is an interest of mine and I have studied it on my own without formal training (unless you count a year of high school (Newtonian) physics).
Euclid studied plane geometry. I haven’t read The Elements, but I wonder if the above was intended to be a definition of parallel or a property that parallel lines enjoy? Not that it matters, because, as ultrafilter pointed out, Euclid’s definitions are not the same as those used today. I’m just wondering.
What about two identical lines (in 2d Euclidean space)? Do they intersect? Are they // ?
In the case of two identical lines, they are parallel, and they do not intersect. Remember, one of the axioms is that two lines intersect in at most one point. So since they have all their points in common (i.e., they’re coincident), they don’t intersect.
I’ll speak as one who has been confused in the past. I think that I get screwed up by the distinction between Euclid’s postulates, in which the parallel line not intersecting bit is a postulate upon which all else is built upon, but not proven itself and not the “definition” of parallel lines … and what modern mathematicians (as well represented by Dopers here) say is the current use and defintion of the word. This is a big distinction: is it a postulate or is it a definition?
I had previously accepted the former rather than the latter, and as such had thought of non-Euclidian geometries as systems that arise from accepting a different postulate about parallel lines, and ones which may also be valid for describing some aspects of reality. Certainly some books about math for the nonmathematician (eg “The history of Pi”) promote that view. If it is a postulate then a different reality can be supposed by not accepting it and following out the logical consequences of a different postulate. If it is a defintion then what ultrafilter says.
The parallel postulate doesn’t say that parallel lines don’t intersect. It says that given a line l and a point P not on line l, that there is some number of lines through P parallel to l. Two lines are defined to be parallel if they don’t intersect.
(From the New Oxford dictionary of English).
This is not really an issue of mathematics, but language. Parallel lines must have the same distance continuously between them. Thus, if two parallel lines eventually meet, then they not consistently have the same distance between them, and they are not parallel lines. Likewise, if a straight line then takes a curve, it’s no longer a straight line. It’s just words.
In non-Euclidean geometry, lines which are apparantly parallel can intersect, like the lines of longitude, but by definition they aren’t parallel at all - they just appear so.
Sorry, QueenAl, but the dictionary is not a good reference for mathematical definitions. While that condition is equivalent to parallelism in the Euclidean plane, the definition we use is “they don’t intersect”.
When I took geometry at U-Va. an age ago, Dr. Faulkner’s lecture notes set for projective geometry "the following axiom system.
A projective plane consists of points, lines, and incidence satisfying:
(P1) Every pair of points is incident to a unique line.
(P2) Every pair of lines is incident to a unique point.
(P3) Thre are 4 points, no 3 of which are incident to the same line."
IOW, I don’t see anything in the axioms about the point at infinity, as a simplifying concept or otherwise.
Sure, the idea of the point (and line) at infinity is tucked back in your head somewhere, and comes into play when you introduce some other stuff to the mix. But it’s not strictly there, at least not given this particular axiomatic treatment.
For that matter, neither are parallels. But to the extent that the term is used in projective geometry (and it is), they meet at a single point.
In order for (P2) to hold, you need the point at infinity. And given that, you need the line at infinity for (P1) to hold. But those are qualitatively different, as you can’t construct them.