Euclidean/Non-Euclidean Geometry

Date: 02/21/2003 at 08:01:03
From: Sue
Subject: Euclidean/Non Euclidean Geometry

Consider the following geometry called S:

Undefined terms: point, line, incidence
Use Logic Rules 0-11
Axioms: 
  I) Each pair of lines in S has precisely one point in common.
 II) Each point in S is incident with precisely 2 lines.
III) There exist precisely four distinct lines in S.

Prove the theorems:
  i) There exist precisely 6 points in S.
 ii) There exist precisely 3 points on each line.
iii) Is this system categorical? Justify answer.
 iv) Can it be proved that precisely one of the following properties 
     hold in S:
        a) the elliptic parallel property
        b) the Euclidean parallel property
        c) the hyperbolic parallel property
If so, which one? Prove answer.

I'm not sure if I'm supposed to start off with lines already existing 
or not. And if I have 4 lines, how do I show there are 6 points if 
each pair of lines only has 1 point in common? I have no idea what it 
means to be categorical or how to prove it or the parallel properties.

Date: 02/21/2003 at 09:25:46
From: Doctor Edwin
Subject: Re: Euclidean/Non Euclidean Geometry

Hi, Sue. 

You don't have to know anything about geometry to prove #1.

How? It's a classic handshake problem. You have 4 people (lines) and 
each one shakes hands with (shares exactly one point with) each other 
person (line).

Line 1 hits lines 2, 3, and 4.
Line 2 also hits lines 3 and 4. 
Line 3 also hits line 4.

How many "hits" are there?

#2 is also possible to do without any special knowledge of non-
Euclidean spaces. If each point is where two lines intersect, and if 
every line intersects every other line exactly once, then each line 
has as many points on it as there are other lines for it to intersect. 
Does that make sense?

As for the parallel properties: I'm guessing that if you are studying 
non-Euclidean spaces, you are familiar with Euclid's 5th postulate 
and the variants that make non-Euclidean geometries.

Euclidean geometry says that there is exactly one line parallel to 
line AC though point B (assuming that point B is not on line AC). 
That's the Euclidean parallel postulate.

The hyperbolic parallel postulate says that there is more than one 
such parallel line. Obviously we are no longer talking about the nice 
straight space we like to picture when we're doing Euclidean 
geometry.

The elliptic parallel postulate says that there are NO such lines. 
That is, that there is no such thing as a parallel line.

Now, which of these three describe S? Does S have any parallel lines? 
If so, can there be more than one line through a given point that are 
all parallel to some other line?

I hope this helps. Write back if you're still stuck.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/21/2003 at 20:13:53
From: Sue
Subject: Euclidean/Non Euclidean Geometry

Dr. Edwin,

Please tell me if this makes sense.

For i)There exist precisely 6 points in S.
Proof: 
1) By Axiom III, there exist 4 distinct lines: 1, 2, 3, 4 in S.
2) By Axiom I, each pair of lines has 1 point in common.
3) Lines 1 and 2 have a point, A, in common.
4) Lines 1 and 3 have a point, B, in common.
5) Lines 1 and 4 have a point, C, in common.
6) Lines 2 and 3 have a point, D, in common.
7) Lines 2 and 4 have a point, E, in common.
8) Lines 3 and 4 have a point, F, in common.
9) By sets 3-8, there exist 6 points in S.

For ii) There exist precisely 3 points on each line.
Proof:
1) By Axiom III, there exist 4 distinct lines: 1,2,3,4 in S.
2) By theorem i, line 1 has points A,B,C.
3) By theorem i, line 2 has points A,D,E.
4) By theorem i, line 3 has points B,C,D.
5) By theorem i, line 4 has points C,E,F.
6) Thus, each line in S has precisely 3 points on it.

iii) Is the system categorical?
Justification: This system is categorical because all models of this 
system would have to contain 4 distinct lines where each pair of 
lines has precisely 1 point in common and each point is incident with 
2 lines. All models would be isomorphic to each other making the 
system categorical.

I hope I did that correctly. I'm still not too sure of how I should 
go about writing a proof for the parallel properties section.
I am pretty sure that the answer is the elliptic property because 
according to the Axioms each pair of lines has a point in common. So 
if every line in the system intersects with another line at some 
point, there is no way there can be a parallel line. How do I write 
the proof? 

Thank you again, Dr. Edwin!

Date: 02/25/2003 at 10:52:11
From: Doctor Edwin
Subject: Re: Euclidean/Non Euclidean Geometry

Hi, Sue.

Everything you did looks good (although I don't know what
"categorical" means in that context so I can only guess that you 
got that part right). As for the proof of the elliptic parallel 
property, I would say you've proved it. If you want to make it more 
formal, you'll need to start with a more formal statement of the 
various parallel postulates. The ones I gave you were just summaries 
of what I remembered plus what I looked up on the Web. You probably 
have formal statements of them in your class notes or text. Once 
you've got those it will be more clear how to state the proof more 
formally.

Good luck and write back if you need more help. 

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/25/2003 at 16:14:56
From: Sue
Subject: Thank you (Euclidean/Non Euclidean Geometry)

Thank you, Dr. Edwin, you were a big big help!