Understanding Non-Euclidean Geometry

Date: 05/26/98 at 20:57:40
From: Yvona Nestorowicz
Subject: Solid geometry

This question is more a question of ideas. Can more than one line 
really connect two points? Also, what is a definition of a straight 
line which is true not only on a coordinate plane or a flat plane, but 
on any surface? Thanks.

Date: 05/27/98 at 03:08:17
From: Doctor Pat
Subject: Re: Solid geometry

Yvona,

Wow, I'm not sure I can give you a great answer, but since it is a 
question of "ideas," here are my ideas.  

Can more than one line connect two points?  It depends somewhat on the 
surface. If you visualize the room you are in as a box and pick a 
point on one wall and another point on the opposite wall, a "straight" 
line might go along the wall to the corner, across that wall to the 
next corner, then along that wall to the second point. It might 
actually slope up or down a little as it goes around the wall to make 
up for one point being higher on a wall than the other. You couldn't 
move through the center of the room because that is not part of the 
surface. Remember the surface of the box is just the outside 
"wrapper." The inside of the box would be a 3-dimensional space that 
is purely imaginary to us 2-dimensional people. It would be like 3-D 
people taking a short cut through the fourth dimension.  

Okay, so that might be a "straight" line, but what about a line 
leaving point A, going down to the floor, across the floor and up the 
other wall to point B. That is "straight" also, as is one going up 
over the ceiling. I can even imagine a "straight" line that touches 
all six walls as it makes its way to Point B. Can you see that?  

Now a little about point B. We can define a straight line for any 
surface, but it may not work for all surfaces. A common idea is that a 
straight line is the one that is the "shortest" distance between two 
points. This is okay for our regular plane geometry, and works for 
most points on a sphere, but then we get to the north and south pole 
and realize that any "straight" line between them is the same length. 
By the same example, two points in the exact center of opposite faces 
of a cube will have four straight lines that are the "shortest."  

Scientists who work with gravity and black holes and exotic stuff like 
that frequently use the path that light travels as their "straight" 
line through the "curved" space around dense matter.  

I guess I've come back to the point that I don't know if I can answer 
your question. Have you got any ideas how you could define straight 
lines? 

Would they work on a basketball?  How about on a torus (doughnut) or a  
cube's surface? What about on a doughnut with a another doughnut 
intersecting it (like a figure 8 solid)? So many wonderful ideas.  

I hope some of this helps you channel your thinking about this, 
because I think your thoughts are what is wanted here.

Good luck,

-Doctor Pat,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/