When I was in college a million years ago, one of my math classes featured a brief discussion of non-Euclidean geometry. Our teacher suggested that if we could measure triangles of interstellar proportions, the sum of the angles might not be 180 degrees. Maybe all our math is micro-math, if we but knew it.
I want to see two parallel lines meet.
Joke from a Dashiel Hammett novel (I think Red Harvest):
Blonde - “I can put two and two together.”
Detective - “Yeah, but sometimes the answer’s four and sometimes it’s twenty two.”
The mailbag item being referenced is
If the universe was curved, how would you know the lines were straight?
Wouldn’t it just seem like two lines (which were “straight”) seemed to converge and were therefore curved?
That would depend on your definition of
parallel lines.
In Euclidean geometry, parallel lines are
lines in the same plane which do not
intersect.
It’s equivalent to saying lines that lie in
the same plane, for which a third line
can be constructed which is perpendicular to
both.
Or, lines in the same plane which are the same distance apart at three distinct points.
If we use either of those as the definition
(and use a lax concept for what ‘in the same
plane’ means), then parallel lines could meet
in curved space.
…AND the lines could be ‘straight’ in the
sense that the segment connecting any two
points in the line was entirely contained
by the line.
I’m confuzzled by this. (Not an uncommon occurrence.)
I could understand it if you used -two- points, but not three points along the same lines. (Sort of a Mercator-projection thing extrapolated to infinity, or has gravity and the origination point got something to do with this?)
Well, it’s been a long, LONG time since I took that course. Don’t know about 3 points. The model I found most useful was to imagine that a line is actually a great circle on a sphere. These are circles whose center is also the center of the sphere. That works for all the axioms except “that one.” If the circle is really, really big, then anything measurable would appear to be Euclidean. My problem with relating this to curved space, tho, is that space is 3-dimensional (4 if you count time but I don’t think that matters here) and the surface of a sphere is sort of 2-dimensional.
Nonetheless, any three points (even in n-dimensional space) that are not on the same line determine a unique plane. Hence any three non-colinear points can set up a triangle, the degrees in the angle of which can be measured to determine whether they are equal to, greater than, or less than 180 degrees. If the triangle is big enough, and the measurement accurate enough, that would tell us whether the universe is (respectively) flat, positively curved, or negatively curved.
-Three points, because a line is always going
to have two points at which the distance to another line is equal.
-The curved space argument is that there are FOUR physical dimensions. We are three dimensional beings living in a 4-d universe. So the surface of a sphere is analogous - we can tell things about a surface, because we
are in three dimensions, that ‘dwellers in the
surface’ could not perceive.
So you’re supposed to think of space as a curved surface, and we live within that 3-d surface, unable to see the curve. Just bump
everything up by one dimension.
Thanks. I’m a graphical thinker which makes me too literal for this, and if I think about spheres and great circles, I’m probably imagining ants crawling around on an orange. Trying to think about 4 physical dimensions hurts my brain. Because really, there’s east and west, north and south, up and down, and what else is there? How could there be a point in space that you couldn’t describe using 3 coordinates, provided that you have specified where the origin is and where the axes are? OK, polar coordinates are coming back to me, but they describe the same thing. This is not at all what you are talking about, right?
I do remember one exercise in which we had to start with axioms and prove theorems, but “line” and “point,” and so forth, were replaced by nonsense words. This prevented us from trying to visualize what we were doing and was more useful as a logic exercise; but probably no good at all in trying to figure out if space is curved, or whatever.
I also remember a Twilight Zone episode … oh, never mind.
You might check out: How many dimensions are there?
The fact that we can’t imagine such dimensions visually doesn’t mean they don’t exist. Deep down inside, we can’t visually imagine subatomic activity either (we think of an electron as a point zipping around a nucleus, for instance) but that doesn’t mean it ain’t real.
The analogy of ant crawling on orange helps us to draw analogies.
Einstein predicted that large mass (gravity) causes a bend (distortion) in space. This has been tested, measuring the position of a star, and then the same star when the sun was in between (like, during a solar eclipse). The position of the star seemed to change when the sun was in the way; the mass (gravity) of the sun was bending the space, so that the light from the star didn’t come “straight” to earth. [/oversimplification]
I thought that experiment demonstrated that
gravity bent light, not space itself.
I know gravity is believed to do that too,
but I thought that solar eclipse/star
position thing was about light, not space
itself.
Light follows the shape of space. IOW, if space is curved, the light will also curve.
It’s possible, I hear, that light (electromagnetic energy) is conducted by three-dimensional space, similar to (but not exactly like) electricity being conducted by copper or aluminum or any other non-insulator.
It’s part of a theory that there may, indeed, truly be another universe that we cannot see because light cannot travel across the “bulk,” as it’s called, that separates this universe from the other. However, gravity may be able to cross this non-conducting “bulk” and the effects of this etra-universal gravity may account for the “missing mass” or “dark matter” phenomenon.
I posted this theory on this topic last month on another thread.
Feel free to correct me at any time. But don’t be surprised if I try to correct you.
See what happens, boys and girls, when you leave out ONE zero?
Feel free to correct me at any time. But don’t be surprised if I try to correct you.
OK, I’m starting to get the more-than-three-dimensions thing. It’s kind of like Magic Eye, tho, if I think about it too hard I lose it.
Is it like this: If I put my fingertip on the tabletop in front of me, I can describe its exact location by specifying latitude, longitude, distance from (for example) the center of the Earth, and a point in time; but if there is another dimension I don’t know about then I could actually be describing two or more points.
Then, if space is curved, what appears to me to be a straight line is actually curved if I could view it in terms of that extra dimension?
You’ve got it Southern.
My brane… er… brain hurts!
Posted by tanstaafl
I don’t know about anyone else, but this cracked me up. 
Does that make me overeducated and weird? Probably.
Where else can you read good topology puns, if not for the SDMB?
“If you prick me, do I not…leak?” --Lt. Commander Data
Saltire, you’re overeducated and weird.
Now I have to find my college topology textbook because I absolutely do not remember what a brane is. And that was one of my favorite classes. My mind is going … I can feel it …
my mother said this would happen
That’s not quite correct. We are four dimensional beings, but we only perceive three dimensions. Or, more accurately, we perceive all four dimensions but put one of the dimensions into a special “time” category, and don’t realize that it is a dimension. Even once we factor in the fourth dimension, the universe is still curved.