# 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:** 2
**Author:** ![matt\_mcl](https://avatars.discourse-cdn.com/v4/letter/m/7ab992/32.png) [@matt\_mcl](https://boards.straightdope.com/u/matt_mcl)\ **Post date:** [July 30, 2002, 9:51am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/21 "2002-07-30T09:51:53Z")
> [@](#): > > “Heh yeah baby, let’s go make some of our parallel lines intersect, catch my drift?” > > Worst-pickupline-ever! Well, except for, “Hey baby, want to go back to my place and try out some bilabial fricatives?” [sub]It’s hard to be a linguist. [/quote] ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [July 30, 2002, 9:54am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/22 "2002-07-30T09:54:32Z")
Euclid’s Fifth Postulate (which some seem to be confusing with Playfair’s Axiom) did not speak of intersection at all, but of angles: > [@](#): > > If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles. — _The Elements_ The postulate has been controversial since its inception. Proclus wrote, “This postulate ought even to be struck out of Postulates altogether; for it is a theorem.” Even Euclid himself was wary of it, not so much as invoking it until his 29th proposition. He 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. ---
**Author:** ![SPOOFE](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@SPOOFE](https://boards.straightdope.com/u/SPOOFE)\ **Post date:** [July 30, 2002, 10:22am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/23 "2002-07-30T10:22:43Z")
> [@](#): > > How do you know they don’t intersect? Did anyone go all the way to the end? How do you know they don’t intersect when you are not looking? How do you know scientist cannot invent a way to make them intersect? Well, it is completely possible and conceivable that, in the next thousand years or so, scientists come up with a series of wondrous technological breakthroughs that revolutionize physics and culminate in the greatest achievement of mankind’s history… they redefine the word “parallel”. ---
**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, 11:16am UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/24 "2002-07-30T11:16:53Z")
> [@](#): > > \*Originally posted by DrMatrix \* > \*\*No. They can’t. Unless you change the definition of parallel. > \*\* Ok, but many people (non-mathematicians?) define parallel as “running in the same direction.” This doesn’t seem so outrageous to me. \*\* > [@](#): > > Parallel lines do _not_ intersect. Parallel lines do not exist in positively curved space.\*\* What about the Equator and the Tropic of Cancer? I’m not saying you’re wrong, but why can’t we treat them as parallel lines? ---
**Author:** ![UrbanChic](https://avatars.discourse-cdn.com/v4/letter/u/9dc877/32.png) [@UrbanChic](https://boards.straightdope.com/u/UrbanChic)\ **Post date:** [July 30, 2002, 1:32pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/25 "2002-07-30T13:32:53Z")
Parallel lines _do_ intersect. They’re just very discreet. ---
**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, 1:45pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/26 "2002-07-30T13:45:13Z")
You know, **ultrafilter** I feel exactly the same way you do. In fact, I was just arguing this fact the other day. Funny to think that we both dealt with this around the same time. ---
**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, 2:11pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/27 "2002-07-30T14:11:52Z")
This is a great example of the need to agree on definitions - or at least precisely understand each other’s definitions and refer to them consistently - before you get started arguing. If - as ultrafilter, correctly AFAIK said - parallel lines by definition never intersect, then it is true that lines meeting Euclid’s conditions as cited by Libertarian will be parallel in the flat and not parallel in other geometries. If, on the other hand you wish to define “parallel” as lines that satisfy the properties in _Elements_ then lines such as these can intersect in non-Euclidian geometries. The reason the first definition is preferable is that that’s what mathematicians mean when they use the word so they own it. ---
**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, 2:24pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/28 "2002-07-30T14:24:23Z")
Dr. Matrix: You must’ve missed the discussion in Geometry class regarding non-Euclidian geometries. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [July 30, 2002, 2:28pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/29 "2002-07-30T14:28:13Z")
> [@](#): > > The reason the first definition is preferable is that that’s what mathematicians mean when they use the word so they own it. Euclid was not a mathematician? What was he, chopped liver? ---
**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, 2:36pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/30 "2002-07-30T14:36:46Z")
> [@](#): > > \*Originally posted by Libertarian \* > \*\*Euclid was not a mathematician? What was he, chopped liver? \*\* Euclid was a mathematician, but his definitions and postulates are no longer widely used, AFAIK. Modern geometry is quite different from what Euclid laid out in _The Elements_. Just for the record, the definition I’m using is the one favored today. Even in non-Euclidean geometries, it’s the same idea. There just aren’t any parallel lines in some non-Euclidean geometries. The point at infinity, found in projective geometry, is not an actual point, but a simplifying concept used to make it more elegant. ---
**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, 2:39pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/31 "2002-07-30T14:39:31Z")
Here’s more on [projective geometry](http://www.anth.org.uk/NCT/basics.htm). ---
**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, 2:55pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/32 "2002-07-30T14:55:24Z")
> [@](#): > > \*Originally posted by lucwarm \* > **Ok, but many people (non-mathematicians?) define parallel as “running in the same direction.” This doesn’t seem so outrageous to me.** 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. \*\* > [@](#): > > What about the Equator and the Tropic of Cancer? I’m not saying you’re wrong, but why can’t we treat them as parallel lines? \*\* 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. ---
**Author:** ![armedmonkey](https://avatars.discourse-cdn.com/v4/letter/a/d6d6ee/32.png) [@armedmonkey](https://boards.straightdope.com/u/armedmonkey)\ **Post date:** [July 30, 2002, 3:31pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/33 "2002-07-30T15:31:11Z")
Sure parrallel lines don’t have a point of intersection all the way out to infinity … but what about infinity + 1? Or infinity squared? Did you ever think of that? Huh, well did you? Jeez, you don’t have to look at me like that On a more serious, yet extremely picky note, I would like to point out that that “parrallel lines” and “lines that never intersect” are not equivalent. It is entirely possible to have two non-parralell lines that never intersect. The lines have to be in the same plane to be parrallell, ya freakin’ two dimmenssional thinkers. ---
**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, 3:44pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/34 "2002-07-30T15:44:55Z")
Just thought I’d chime in to support **ultrafilter** and **Dr. Matrix**. In modern mathematical usage, “parallel” is used to refer to lines which lie in the same plane, but do not intersect. Full stop. (On preview: points for **Beeblebrox** for anticipating the “same plane” thing. Smartass 🙂 ) > [@](#): > > _Originally posted by Libertarian_ > \*\* > > > [@](#): > > > > Nicolai Lobachevsky, founder of non-Euclidean geometry in which Euclid’s Fifth Postulate does not hold > > “Parallel lines meet each other at a point of infinity” > > Riemann geometry is not the only non-Euclidean geometry. > \*\* The points at infinity are not, strictly speaking, points of Lobachevskian (aka hyperbolic) space. They are a separate set of points, very closely related but not part of the Lobachevskian plane. > [@](#): > > The postulate has been controversial since its inception. Proclus wrote, “This postulate ought even to be struck out of Postulates altogether; for it is a theorem.” Even Euclid himself was wary of it, not so much as invoking it until his 29th proposition. Proclus’s comment is very wrong in hindsight, since Euclid’s fifth postulate is quite emphatically _not_ a theorem. After all, it doesn’t follow from the other postulates of Euclidean geometry; if it did, Lobachevskian geometry wouldn’t exist. ---
**Author:** ![gobear](https://avatars.discourse-cdn.com/v4/letter/g/8dc957/32.png) [@gobear](https://boards.straightdope.com/u/gobear)\ **Post date:** [July 30, 2002, 3:56pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/35 "2002-07-30T15:56:02Z")
> [@](#): > > Dr. Matrix: You must’ve missed the discussion in Geometry class regarding non-Euclidian geometries. 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. In any event, in 3-dimensional Euclidean space, parallel lines never intersect, period. [Here](http://www.math.lsa.umich.edu/~mathsch/courses/Infinity/Geometry/Lesson1.shtml), by the way, is an art-major- friendly explanation of projective geometry. ---
**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, 4:15pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/36 "2002-07-30T16:15:34Z")
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. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [July 30, 2002, 4:19pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/37 "2002-07-30T16:19:32Z")
I consider Math Geek to be an authority on the matter, having seen his analysis of matters mathematical in other threads. I therefore concede the point to DrMatrix, et al. From now on, since they have been redefined, I will not use the parallel lines postulate as an example of nonuniversal axioms. ---
**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, 4:21pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/38 "2002-07-30T16:21:39Z")
Alright **Monty** , put up. In exactly which “viable system of mathematics” do some parallel lines intersect? I’m quite curious. ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [July 30, 2002, 4:29pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/39 "2002-07-30T16:29:15Z")
To summarize some incredibly geeky and intelligent thoughts here I think the answer can be summarized as: 1. As a mathematical concept parallel lines _by definition_ never intersect. 2. In the real world/universe there is no such thing as a perfectly straight line. Sooner or later any two lines will intersect even if they have to cross much of the universe to do it (space itself is curved). Hence, truly parallel lines don’t exist. ---
**Author:** ![lieu](https://avatars.discourse-cdn.com/v4/letter/l/c2a13f/32.png) [@lieu](https://boards.straightdope.com/u/lieu)\ **Post date:** [July 30, 2002, 4:33pm UTC](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484/40 "2002-07-30T16:33:20Z")
Back in the 80’s, parallel lines used to meet in my sinus cavity. [Previous page](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484.md?page=1) [Next page](https://boards.straightdope.com/t/listen-closely-people-parallel-lines-never-intersect/121484.md?page=3)