Xref: utzoo talk.religion.newage:1325 alt.flame:1089
Path: utzoo!hoptoad!amdcad!ames!lll-lcc!rutgers!ucla-cs!troly
From: troly@CS.UCLA.EDU
Newsgroups: talk.religion.newage,alt.flame
Subject: Re: Platygaeanism
Keywords: platygaeanism, Riemannian geometry
Message-ID: <10064@shemp.UCLA.EDU>
Date: 28 Dec 87 03:11:12 GMT
References: <27455COK@PSUVMA> <4249@bellcore.bellcore.com> <1359@quad1.quad.com> <9959@shemp.UCLA.EDU> <1471@cartan.Berkeley.EDU>
Sender: root@CS.UCLA.EDU
Reply-To: troly@CS.UCLA.EDU (Bret Jolly)
Followup-To: Alt.flame
Organization: LA Platygaean Society
Lines: 86


In article <1471@cartan.Berkeley.EDU> ir353@sdcc6.ucsd.edu (Matthew Grayson)
writes:
>That does it. You guys have blown your cover. A projective plane cannot be
>given a flat metric. Not intrinsicly, anyway. If you want it to be
>extrinsically flat, then it must be embedded (I'm assuming you don't
>claim that the Earth intersects itself) in a higher dimensional positively
>curved manifold. If you insist that the earth's surface be flat in the
>metric sense, then your only possible non-singular structures for compact
>complete 2-manifolds are tori and Klein bottles.

This is quite true (courtesy of the Gauss-Bonnet theorem). We could have
a projective plane which is pretty flat except in one locality (the
Bermuda triangle? :-)), but I agree that this is not morally flat.
However I don't completely reject it out of hand for that reason. The
LA Platygaean Society considers all sorts of possibilities, indeed some
of our members are round-earthers.  But the appeal of the projective
plane lies in something apart from Riemannian geometry. If there is
one thing we know from round-earth based physics it is the power of
duality. Consider coordinates and conjugate momenta in Hamiltonian
mechanics, Legendre transforms in Thermodynamics, not to mention
the Heisenberg uncertainty principle in quantum mechanics. Like Yin
and Yang, yes, there is a Tao of physics! And might not this duality
spring from some geometrical basis of the world we live in? Projective
geometry embodies duality from the most basic level.  But this is not
metric, the theory would eventually have to explain our perception
of physical distance in terms of fundamental incidence relations. 
There is good reason to consider distance to be an epiphenomenon,
even in conventional round-earth worldviews.  Topological or incidence
relations are much more fundamental. Personally I am not (currently)
a projective planer but I think the theory deserves the same level
of critical consideration that I give to platygaean theories. 

If the ancient Greeks had taken a modern view and Euclid had written
his Elements on topology and projective geometry the subsequent history
of science would have been much different. But the Greeks could not
even imagine geometry aside from that imbedded in Euclidean 3-space.

> Since no-one has travelled the earth and come back reversed, we can
> conclude that the torus is the only possibility.

WAITAMINNIT! There are plenty of non-orientation reversing paths "around"
a Klein bottle earth.

> Very good. Please be kind enough to tell us where the
>non-trivial loops are. What path on the earth's surface does not bound a disk.

This seems very difficult experimentally.  I'd need to mount many worldspanning
expeditions, a heck of a lot of thread, and a way of pulling it taut without
breaking it :^).

>What's that ? I'm getting topological? Oh. Well, suppose that every loop
>CAN be contracted, then the surface is a sphere, but then.. oh dear... oh my..

Exercise: Prove the analogous result for a 3-sphere :^) :^) :^).

Recommended reading for all, no matter what shape you think the earth is,
THE SHAPE OF SPACE by Jeffrey R. Weeks. (Pub. Marcel Dekker). Mind-stretching!

>	BTW. A projective plane has a non-contractible loop. Where is it?

I found it! (No spoiler).

> {Matt quoting me}
>>(Correct me Phil, if I am misrepresenting you.) But a manifold need not
>>be embedded in *any* euclidean space.
{he replies}
>	But it can be, even isometrically ( see John Nash's embedding theorem).
I said *need* not be ...

>OK, flat earthers, is the surface of the earth intrinsically flat,
>extrinsically flat, or both. What 3-manifold is it embedded in, and what
>metric does the 3-manifold have. Let's see a model!!
If I figure this out I'll try to write my thesis on it :^).

This discussion has drifted far from the new age issues I originally addressed
so I move that followups be directed to alt.flame. Unfo