I started to ask this question in physics. Then after considering the
problem a bit I realized it is perhaps better considered a math
problem.

Suppose I am located outside of the earth's atmosphere and that I have
in my hands two perfect laser pointers. Suppose also that there is no
dust or matter in front of me in space, such that the beams extend
infinitely.

I am holding my laser pointers such that their beams intersect
immediately in front of me. I then begin to uncross the beams. As I
uncross the beams, their point of intersection will move away from me
at an ever increasing rate toward a point in the infinite distance.

It seems to me that because the beams are infinite in length, they can
never become completely uncrossed. At the point at which I finally
hold the beams parallel, they will in fact be intersecting at a point
in the infinite distance. Is this correct? How can the beams be at
once parallel and intersecting?

Your premise is flawed. Realistically, you cannot have infinitely long
laser beams. Light only travels at 186,000 Miles Per Second, which is
admittedly very fast, but extremely slow considering the (infinite)
distances we are talking about.

However, assuming for a moment that these beams already exist, and
already extend to infinity, we still run into problems. As soon as you
begin to move the pointers, your beams are no longer straight. The
light already emitted from the pointers is going to keep moving
straight away from where it was emitted, but if you were to look at
this scene from above, you'd see the beam curving toward the new
direction. If you wiggled the pointer back and forth, you'd have a
beam that looked something like a Sine Wave. If it helps, imagine the
beams are actually Silly String. :)

So, while your intersection will continue outwards forever, you don't
actually have parallel beams, and thus, there's no conflict.

Please, let me know if you need any clarification of this answer.

---

Request for Answer Clarification by gts-ga on 21 Jul 2002 17:13 PDT

googlebrain-ga,

Thank you for your reply to my question. Unfortunately I'm not very
satisfied with it.

First let me say that my premise was not flawed as you suggest; I
realize perfectly well that the perfect infinite laser beams in my
question can exist only in abstract theory. It was in fact for that
very reason that I decided after writing the question that I should
post it here in the math forum rather than in the physics forum as I
had originally intended. Mathematically, the laser beams represent
intersecting rays of infinite length. I thought this much would be
clear to any mathematician. I'm sorry I should have been more
explicit.

You continued, "However, assuming for a moment that these beams
already exist, and already extend to infinity, we still run into
problems. As soon as you begin to move the pointers, your beams are no
longer straight."

Of lasers in space/time this would be true, but not so for
mathematical rays on a Euclidean plane.

Feel free to try again if you like, with the understanding that these
laser beams are mathematical entities rather than physical entities.

I realize also, by the way, that my conclusion is nonsensical.
Obviously two rays (or two lines) cannot be at once parallel and
intersecting. My reasoning must be wrong. What I am looking for is an
elegant explanation of the proper reasoning.

My inclination is to conclude simply that Euclidean geometry breaks
down at infinity, analogous to the way in which Newtonian physics
breaks down as one approaches the speed of light. If this is so then I
would like to see an explanation of why Euclidean geometry breaks down
at infinity.

Thank you again for your effort, and thanks also to chrisccc_3k-ga,
rmarkd-ga and ula-ga, whose comments all seem to point in the right
direction.

---

Clarification of Answer by googlebrain-ga on 21 Jul 2002 18:43 PDT

Lets try this again.

A: Your lines become uncrossed as soon as they become parallel. 

This is Euclid's "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."

Euclid doesn't defend this position at all. There is no need. It is
one of the 'givens' in Euclidian Geometry. If they meet somewhere in
the infinite distance, then they aren't parallel.

Euclid's "The Elements" Table Of Contents
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

This answer may be unsatisfactory, but it is the best that Euclidian
Geometry can provide. It is, however, unfair to say that Euclidian
Geometry 'breaks down at infinity.' By asking about Parallel lines,
you are looking directly at an undefined area. Other systems of
mathematics may not have this particular problem, but they will have
problems of their own. There are no systems of mathematics that cover
all cases.

It's clear to me that googlebrain's answer and clarifications are
strictly correct. However I learned little from him that I did not
already know. The comments from other users were for me far more
illuminating, especially those by blader and ulu.

Given the simplicity of the problem I had hoped to see the correct
line of reasoning without use of trigonometry or calculus or
non-euclidean geometries. It seems however that one of more of these
branches of mathematics are necessary for any complete answer to the
problem.

what if instead of laser beams finitely limited in velocity, ideal
infinite euclidean lines were used, and again uncrossed (or an attempt
made) in a similar manner?

"what if instead of laser beams finitely limited in velocity, ideal
infinite euclidean lines were used, and again uncrossed (or an attempt
made) in a similar manner?"

You run into many a funny thing dealing with infinities. It seems like
you're trying to look at infinity through finite glasses. The simple
answer is when the lines are parallel, the lines would uncross
infinitely quick "at infinity". Remember, infinity is not a point, and
you can't treat it like you would x=5 or y=10^100 (there's a name for
10^100, but I forget what it was. Maybe a google search will help...
;-> )

Think about it this way, Say you have one of those infinite lines and
turn it very slightly over the course of a second. "At infinity" the
"infinity point" of the line has been displaced an infinite distance.
Wha? Infinite distance traversed over finite time? Yeah, again, that's
the problem when dealing with infinities.

hth,
-Mark

Refering to chrisccc_3k-ga alternative question, Euclidean geometry
by definition, has parallel lines that don't intersect, hence they can
be uncrossed.  There are Non-Euclidean geometries where parallel lines
do intersect.  Take the earth's surface, where lines are all great
circle routes.  There, all distinct lines intersect in two points. 
Scientists are still wondering about the shape/geometry of our
universe.  Googlebrain did a great job showing the difference between
the ideal world and the real world.  Rmarkd also showed the problems
of dealing with the word "infinity".  In mathematics, there are many
"infinities".
http://mathworld.wolfram.com/ParallelPostulate.html
http://www.sciencefriday.com/pages/2002/Jul/hour2_071202.html 
http://www.amazon.com/exec/obidos/ASIN/0691096570/sciencefriday 
http://mathworld.wolfram.com/Infinity.html

Why of course, 10^100 = 10 duotrigintillion

What else would it be? :)

http://mathforum.org/library/drmath/view/57575.html

Appropriately enough, 10^100 is also known as a googol.  10^googol is
a googolplex.  How about that?

How about reversing the question to get the answer?

If I had two laser pointers and held them parallel to each other,
would the beams intersect?

No.

How much would you have the move one pointer towards the other for the
beams to intersect?

An infinitely tiny amount.

Dear gts:

Thank you for your very interesting question! I think I may have the
answer for you.

"It seems to me that because the beams are infinite in length, they
can
never become completely uncrossed."

Intuitively, this make some sense. But it breaks down if we analyze it
with formal mathematics.

The length (l) from the point between the two laser beam's origins and
the point where they intersect depends on the angle made between the
laser beams and the line connecting the two origins (angle theta), and
the distance (d) between the two laser pointers originally.

A crude ASCII drawing of the situation follows:

                     /|\
                    / | \
                   /  |  \
                  /   |   \ 
                 /    |(l) \ 
                /theta|     \
Laser Pointer 1 ------------- Laser Pointer 2
                     (d)

So, from basic trigonometry:

l = tan(theta)*d/2

The Easier Explanation: 

As the limit of theta approaches 90 degrees, l approaches infinity.
This is true. However, the key to this is that there is a difference
between "infinity" and "undefined." I think this is where you are
getting the parallel-yet-intersected concept form.

The key is that when theta equals 90 degrees, tan(theta) (and
therefore l)  is undefined. What this means is not that l equals
infinity, but l doesn't exist, so when it is parallel, it is
mathematically provable that they two in fact do not intersect at all.

The "infinite" length you speak of only exists as you get "infinitely
close" to the 90 degrees, or parallel. It (l) does not at all exist
when it IS parallel. This is why your initial presmise of "At the
point at which I finally hold the beams parallel, they will in fact be
intersecting at a point
in the infinite distance." The point, in fact, does not exist. As a
final example:

This is a false statement:
"At the point at which I finally hold the beams parallel, they will in
fact be intersecting at a point in the infinite distance."

This is a true statement:
"As I approach infinitely closely to the point where I finally hold
the beams parallel, they will be in fact be intersecting at a point in
the infinite distance."

Best Regards,
blader-ga

The other explanations are correct, but perhaps you are still looking
for more.

One concept that blader brought up was limits.  What happens if you
uncross the lines (two-ended lasers) a little more?  You get an
intersection behind you, at "negative infinity".  For the limit to
exist, it needs to be equal from both directions.  That is why the
value is undefined.
http://mathworld.wolfram.com/Limit.html

What you are talking about is a Non-Euclidean Geometry, one that has
changed parallel lines from not intersecting, to instersecting at a
"point at infinity".  These links provide good expanations about
"Projective Geometry".
http://www.math.lsa.umich.edu/~mathsch/courses/Infinity/Geometry/Lesson2.shtml
http://mathworld.wolfram.com/PointatInfinity.html
http://www.math.toronto.edu/mathnet/questionCorner/infinity.html
http://www.math.toronto.edu/mathnet/questionCorner/projective.html

"Given the simplicity of the problem I had hoped to see the correct
line of reasoning without use of trigonometry or calculus or
non-euclidean geometries. It seems however that one of more of these
branches of mathematics are necessary for any complete answer to the
problem."

It's those simple problems that help create new branches of
mathematics.

Your question is the dividing point between Euclidean and Projective
Geometry.  Asking what happens when the point gets further and further
out creates limit theory as a tool.  And then there's 1/0 = ?

Does this explanation help?
Imagine you've got the lasers on each rail of a train track.  You have
the lasers crossed in front of you and you lay track out to the point
where they intersect.  The intersection may get further and further
out, but so can the track.  You probably want to jump to "infinity"
here, but that would be cheating ;-)  You have to get there just by
going further.  But when that happens, the track follows.  As long as
the beams intersect, the beams are never parallel (only the rails). 
But when the lasers are on the rails (parallel), the beams never
intersect, like the rails, even at "infinity".

It is those "simple" questions that are the most interesting.

ulu,

Thanks for your additional insight.

> You probably want to jump to "infinity"
> here, but that would be cheating ;-)  

lol. :)

> It is those "simple" questions that are the most interesting.

I agree. I love mathematics but for practical reasons I never really
pursued the subject academically. (I was a business major. We can get
by with a $5 calculator from Walmart. :) My laser beam question is
entirely my own invention... just something I've often wondered about.

This has been fun and well worth the 2 bucks I paid for it. 

Stay tuned for my next math question, in which I'll offer proof that
humanity is very likely become extinct in the near future...

See here: https://answers.google.com/answers/main?cmd=threadview&id=43608
for another one of those simple yet interesting problems. =)

I think it's possible to answer your question precisely and numerically
when we consider angular speed with which you rotate the
two straight lines in question. Once you specify the initial
angle between two straight lines and the angular speeds at which
two lines rotate, it's easy to calculate at which moment in time
the two lines become parallel. Let's say the angular speed
of each line is constant. It's easy to write the formula which shows
that it takes finite time for the lines to become parallel (that is,
angle between them becomes 0) under these conditions.
   The time for the lines to become parallel is finite here.
The thing that may trouble you may be that intersection point
moves away with infinitely increasing speed within finite time.
There is no contradition here, though.
   If we specify different angular speed function, for example
non-constant angular speed exponentially slowing down as 
lines approach zero angle, then the time to parallelism would become
infinite indeed. Another example of conditions under which 
time to parallelism is infinite is such rotation of lines
when intersection point moves at constant speed.
   But assuming the simplest case, that is, when lines rotate
with constant angular speed in opposite directions, it takes 
finite time for them to become parallel. At no point in time the 
lines become "simultaneously parallel in intersecting" --
provided those are lines in Euclidian planar geometry. 
   Had we talked about non-Euclidian plane, then the lines
could indeed be parallel and intersecting at the same time.
But that would be different question.