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Topic: easiest proof of Jordan Curve Theorem as a corollary of Moebius
theorem #1982 Correcting Math

Replies: 7   Last Post: Aug 11, 2014 3:19 PM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
easiest proof of Jordan Curve Theorem as a corollary of Moebius
theorem #1982 Correcting Math

Posted: Aug 9, 2014 4:29 PM

Last night I wrote:

On Saturday, August 9, 2014 2:14:22 AM UTC-5, Archimedes Plutonium wrote:
> Alright, I think I found a far far easier proof of the Jordan curve theorem.
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> Now Wikipedia describes this theorem as saying, draw a closed curve in the plane and that closed curve divides the plane into two regions of all the points inside the closed curve and all the points exterior to the closed curve. So that if we take any point inside and any point exterior to the closed curve and followed a path containing those two points we end up crossing the closed curve.
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Now if you look at that Wikipedia page we see the Math community is unhappy over the Jordan proof where many say it is not a proof at all but a huge assumption carried forward disguised as a proof.

And where Veblen complains about the Jordan proof and offers his own as being the first solid proof.

Well, I am going to say that neither Jordan's or Veblen's offering are proofs but rather are flawed.

I think I am lucky in finding a second proof in mathematics wherein the only means of proof is via a Corollary of a larger more general proof. Here I speak of the Moebius theorem that 4 mutual adjacencies is the maximum. Within that easy proof of Moebius theorem the Jordan Curve Theorem falls out as a simple easy to prove Corollary.

Remember when I did the Beal with FLT as corollary, and said that the structure of Logic was such, that many proofs in mathematics are proofs only when their "more general statements" are proven first and the underlying minor statement FLT for Beal and Jordan Curve for Moebius can thence be proven.

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> Now the date of Jordan Curve theorem seems to be about 1887, while the Moebius theorem is far far older.
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> The Moebius theorem says that 4 mutual adjacencies is the maximum adjacencies of closed curves can have. It is proven simply by the Euler formula of
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> V - E + F = 2 where V stands for vertices, E for edges and F for faces.
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Now here is a picture of 4 maximum mutual adjacencies, for a 5th can not be realized because the B region is blocked from any further adjacencies. This is proven by the Moebius theorem which uses Euler's formula.

MMMMMMMO ?
MMMMMMMO ?
MMMMBBJJJO ?
MMMMBBJJJO ?
MMMMOOOO

So, now, let us take that B region and call it the Jordan Closed Curve loop and use the same Euler formula to prove Jordan Curve Theorem.

Notice now that B is surrounded by a "greater closed loop of M, O, J.

So that to prove that any interior point of B with a path to any exterior point of B, must cross the closed loop that forms B, but also, we can say the path crosses the closed loop formed by M, O, J if the exterior point is beyond M, O, J interiors.

So, if the Jordan Curve theorem were not true, then we have the error that there is a 5th mutual adjacency for B, M, O, J as that path from a interior point of B goes out and reaches a exterior point of B, M, O, J without crossing any of the closed loops of B,M,O,J.

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> Now, if we examine 4 mutual adjacencies can we not picture the Jordan Curve closed loop being that region R that cannot receive a 5th mutual adjacency? And can we not draw a path from any interior point of R to any point exterior to R and it would have to cross the closed loop?
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> So, the Jordan Curve theorem is a direct result of the Moebius theorem proven by the Euler formula.
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So, I waited a long time to find another proof in math where it comes directly from proving a more general statement first. Many would not consider the Moebius Theorem as a more general statement of the Jordan Curve Theorem, but often, statements and their generality fools many in mathematics. For in a sense, the Jordan Curve Theorem is a simplistic version of the Moebius Theorem where it asks for many paths leading out to concentric shaped closed loops around a central closed loop.

--

Recently I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of front-page-hogs since they crowd out everyone else reading "mobile" instead of "desktop". Many posters in sci.physics and sci.math are not doing science but seeing whether they can dominate the front page of the newsgroup, and PAU is free of hogs, mockers and hatemongers.