Xref: utzoo talk.religion.newage:1279 alt.flame:999
Path: utzoo!utgpu!water!watmath!clyde!rutgers!ucla-cs!zen!ucbvax!cartan!jell-o!grayson
From: grayson@jell-o (Matthew Grayson)
Newsgroups: talk.religion.newage,alt.flame
Subject: Re: Platygaeanism
Summary: Learn some geometry....
Keywords: platygaeanism, Riemannian geometry
Message-ID: <1471@cartan.Berkeley.EDU>
Date: 17 Dec 87 23:05:18 GMT
References: <27455COK@PSUVMA> <4249@bellcore.bellcore.com> <1359@quad1.quad.com> <9959@shemp.UCLA.EDU>
Sender: nobody@cartan.Berkeley.EDU
Reply-To: ir353@sdcc6.ucsd.edu (Matthew Grayson)
Organization: UC San Diego Math Department
Lines: 61

In article <9959@shemp.UCLA.EDU> troly@CS.UCLA.EDU (Bret Jolly) writes:
.... more flat earth stuff
>a summary of the major theories someday when I have time, but I would like
>to note that many of them consider the earth's surface to be a compact
>2-dimensional manifold, just as the round earthers do. But a compact manifold
>(without boundary) can still be flat.  Round-earthers are always confusing
>topological and metrical arguments.  They also confuse extrinsic and intrinsic
>geometry.
>For example, Phil Wayne, who apparently is a round-earther boldly toying
>with platygaeanism, suggests a projective plane as the surface of the earth.
>This is one of the major compact models (actually it leads to a whole class
>of models).

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. Since no-one has travelled
the earth and come back reversed, we can conclude that the torus is the
only possibility. 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.
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..
you're back with a round earth, which may have zero extrinsic curvature,
but then there's that positively curved 3-manifold again. What's your choice?
	BTW. A projective plane has a non-contractible loop. Where is it?

> But Phil's description involves *twisting* and *connecting* the
>edges in 4 dimensions.  At least I think that is what he is trying to say.
>(Correct me Phil, if I am misrepresenting you.) But a manifold need not
>be embedded in *any* euclidean space.
	But it can be, even isometrically ( see John Nash's embedding theorem).
>Things get worse when round-earthers
>1) Automatically embed any manifold you talk about in some euclidean space,
>and
>2) then proceed to borrow the *metric* of that euclidean space without even
>realizing what they're doing.
	It's called the induced metric, and some people even realize it.
Like it or not, the surface of the earth which seems flat is embedded in
some 3-manifold which seems even flatter, namely an open neighborhood of
the surface, so maybe embedding the earth in euclidean space is  not such 
a bad idea, or is the fact that the
volume of space near the earth's surface seems to be metrically flat
insufficient to conclude that it is (hee hee). Well then, suppose that it's 
not euclidean 3-space. It's some flat 3-manifold.  We're back to non-trivial 
loops. Where are they?
	I thought this theory was supposed to be simpler....
>Well, I think I'll sit back and watch the discussion for a while.  Hopefully
>some intelligent knowledgeable people like Miriam Nadel will soon join in.
>
>>                 /
>Bret Jolly (bo'-ret tro ly) Mathemagus  LA platygaean society
>             .

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!!

	Matt