# Listen closely, people: parallel lines NEVER intersect! **URL:** \ **Category:** The BBQ Pit\ **Created:** [July 30, 2002, 4:13am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484 "2002-07-30T04:13:14Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![UncleBeer](https://avatars.discourse-cdn.com/v4/letter/u/977dab/32.png) [@UncleBeer](https://boards.straightdope.com/u/UncleBeer)\ **Post date:** [July 30, 2002, 4:34pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/41 "2002-07-30T16:34:10Z")
Now let’s talk about over-lapping concentric circles. ---
**Author:** ![hawthorne](https://avatars.discourse-cdn.com/v4/letter/h/c89c15/32.png) [@hawthorne](https://boards.straightdope.com/u/hawthorne)\ **Post date:** [July 30, 2002, 4:37pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/42 "2002-07-30T16:37:43Z")
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. ---
**Author:** ![Buck\_The\_Diver](https://avatars.discourse-cdn.com/v4/letter/b/ac8455/32.png) [@Buck\_The\_Diver](https://boards.straightdope.com/u/Buck_The_Diver)\ **Post date:** [July 30, 2002, 5:03pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/43 "2002-07-30T17:03:09Z")
Anybody wanna wager that they intersect in the [Time Cube](http://www.timecube.com) universe? ---
**Author:** ![Spritle](https://avatars.discourse-cdn.com/v4/letter/s/5daacb/32.png) [@Spritle](https://boards.straightdope.com/u/Spritle)\ **Post date:** [July 30, 2002, 5:33pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/44 "2002-07-30T17:33:39Z")
For the record, Parallel Lines was a damn fine album, albeit not Blondie at their recording best. ---
**Author:** ![Squish](https://avatars.discourse-cdn.com/v4/letter/s/8797f3/32.png) [@Squish](https://boards.straightdope.com/u/Squish)\ **Post date:** [July 30, 2002, 6:39pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/45 "2002-07-30T18:39:07Z")
> [@](#): > > \*Originally posted by matt\_mcl \* > **Well, except for, "Hey baby, want to go back to my place and try out some bilabial fricatives?"** Reminds me of that book that was so popular in my high school: _Hymen: the Story of the Bilabial Stop_. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 30, 2002, 6:39pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/46 "2002-07-30T18:39:56Z")
> [@](#): > > \*Originally posted by Monty \* > \*\*No cupie doll there, gobear. The OP said “parallel lines never intersect.” Some do. Some don’t. Don’t like it? Write your own viable system of mathematics and get back to me then. Dr. M can help you, I suppose. \*\* I’d also like to see some support for your claim that some parallel lines do intersect. ---
**Author:** ![Monty](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/monty/32/47_2.png) [@Monty](https://boards.straightdope.com/u/Monty)\ **Post date:** [July 30, 2002, 7:17pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/47 "2002-07-30T19:17:35Z")
> [@](#): > > Non-Euclidean Geometry > One of the postulates of Euclid’s geometry of the plane states that through a point outside a given line it is possible to draw only one line parallel to the given line, that is, one that will never meet the given line no matter how far the lines are extended in either direction. For many centuries mathematicians believed that this postulate could be proved on the basis of the remaining postulates, but all efforts to discover such a proof were fruitless. Then in the first part of the 19th century the German mathematician Carl Friedrich Gauss, the Russian mathematician Nikolay Ivanovich Lobachevsky, and the Hungarian mathematician János Bolyai independently demonstrated the possibility of constructing a consistent system of geometry in which Euclid’s postulate of the unique parallel was replaced by a postulate stating that through any point not on a given straight line an infinite number of parallels to the given line could be drawn. Later, about 1860, the German mathematician Georg Friedrich Bernhard Riemann showed that a geometry in which no parallel lines occurred was equally possible. > The details of these two types of non-Euclidean geometry are complex, but both systems can be demonstrated by means of simple models. The Bolyai-Lobachevsky geometry, often called hyperbolic non-Euclidean geometry, describes the geometry of a plane consisting only of the points on the inside of a circle in which all possible straight lines are chords of the circle. As can be seen from Fig. 10, an infinite number of parallels to line L can be drawn through the point P. Similarly Riemannian, or elliptic non-Euclidean geometry, is the geometry of the surface of a sphere in which all straight lines are great circles. Fig. 11 makes apparent the impossibility of drawing any pair of parallel lines on this surface. > > “Geometry,” Microsoft ® Encarta. Copyright © 1994 Microsoft Corporation. Copyright © 1994 Funk & Wagnall’s Corporation. 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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 30, 2002, 7:38pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/48 "2002-07-30T19:38:41Z")
**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. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [July 30, 2002, 8:06pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/49 "2002-07-30T20:06:04Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\* > > In the analytic Euclidean plane (as developed by Descartes), these two conditions are equivalent. But that’s not the only setting in which geometry is interesting. > > \*\* 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. \*\* 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. \*\* [/QUOTE] This would depend on your definition of a “line,” no? (I’m sure the definition you propose is reasonable, however.) ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [July 30, 2002, 9:03pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/50 "2002-07-30T21:03:01Z")
> [@](#): > > _Originally posted by Monty_ > **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.** 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. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [July 30, 2002, 9:58pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/51 "2002-07-30T21:58:21Z")
> [@](#): > > \*Originally posted by Math Geek \* > \*\* > Aside to **lucwarm** : > 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). ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [August 1, 2002, 4:48pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/52 "2002-08-01T16:48:50Z")
> [@](#): > > \*Originally posted by gobear \* > > Given that **Dr. Matrix** is a physicist, and therefore well-grounded in mathematics, I suspect **Monty** will need to put some Old Bay seasoning on his post, for he will be forced to eat 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). > [@](#): > > \*Originally posted by \*\*\*Libertarian \*\* > > [Euclid] actually defined parallel lines in Proposition I.27 > > > [@](#): > > > > If a straight line crossing two straight lines makes the alternate angles equal to one another, the straight lines will be parallel to one another. - _The Elements_ > > Obviously, on spherical surfaces, such lines would intersect. 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. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [August 1, 2002, 6:56pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/53 "2002-08-01T18:56:34Z")
What about two identical lines (in 2d Euclidean space)? Do they intersect? Are they // ? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 1, 2002, 6:59pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/54 "2002-08-01T18:59:50Z")
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. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [August 1, 2002, 11:45pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/55 "2002-08-01T23:45:22Z")
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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 1, 2002, 11:56pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/56 "2002-08-01T23:56:31Z")
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. ---
**Author:** ![QueenAl](https://avatars.discourse-cdn.com/v4/letter/q/d26b3c/32.png) [@QueenAl](https://boards.straightdope.com/u/QueenAl)\ **Post date:** [August 2, 2002, 12:28am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/57 "2002-08-02T00:28:43Z")
> [@](#): > > **parallel** : adjective (of lines, planes, surfaces, or objects) side by side and having the same distance continuously between them. (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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 2, 2002, 12:31am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/58 "2002-08-02T00:31:08Z")
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”. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 2, 2002, 2:19am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/59 "2002-08-02T02:19:34Z")
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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 2, 2002, 3:49am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/60 "2002-08-02T03:49:17Z")
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. [Previous page](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484.md?page=2) [Next page](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484.md?page=4)