found a Proof outline that Old-Poincare-Conjecture is contradictory to all of geometry #696 new book 2nd edition: New True Mathematics
Archimedes Plutonium
Call it the luck of the Irish... that he was working on 4Color
Mapping, Poincare Conjecture and Kepler Packing all simultaneously
at once so that allowed him to see a proof that the Old Poincare
Conjecture is inconsistent with the Axioms of Geometry.
Archimedes Plutonium wrote:
> New Poincare Conjecture: every closed loop on sphere or Euclidean
> geometry
> can at most shrink to the primitive set of 9 consecutive points as
> such:
>
> . . .
> . . .
> . . .
>
> 4Color Mapping Problem: 4 mutual adjacencies is the maximum possible.
>
> I want to prove those two are the same. Both spring from the wellhead
> of the Revised Betweenness Axiom: given any two points (numbers) A
> and B is always a C, unless A and B are consecutive.
>
> In other words, geometry and numbers have holes between points and
> numbers.
> So the Old Poincare Conjecture and Betweenness Axiom are bogus.
>
> In this diagram of countries L, M, B, J, O can we consider J as the
> center point of the 9 Point Poincare Cell:
>
> . . .
> . . .
> . . .
>
>
>
> LLLLLLLLLLL
> LMMMMMML
> LMMMMMML
> LLBBJJOLLLL
> LLBBJJOL
> LLOOOOL
> LLLLLLLLL
>
> The important proof is that the Betweenness Axiom allows us to have
> a triangle with two right angles:
>
> A
> ____________
>
>
> ____________
> B C
>
> As C moves closer and closer to B the triangle formed by ABC
> is a triangle with two right angles at B and C because of the old
> Betweenness Axiom is an infinite downward regression that becomes
> a Dedekind limit so that the angle at C is 90 degrees from the limit.
>
> Now I need a similar proof scheme that the Old Poincare Conjecture
> allows me to penetrate the J country from the L country. And thus,
> showing that the Old Poincare Conjecture allows the existence of
> Five-mutual adjacencies.
>
> I may not see it at this moment, but I sense it is there.
>
> Maybe I should try a different tactic. Show that the Old Poincare
> Conjecture allows for the existence of a triangle with less than
> 180degree
> interior angles.
>
Today I begin to find the function that tells me the Core HCP-Pure
in Euclidean stacking of spheres. It is near that of the function
f(x) = x^3 - [[tan(x)]] but have to adjust it so that is preliminary
function.
So how is it that the Old Poincare Conjecture is false and has
to be revised so that a closed loop shrunk cannot be a point but
has to be a Poincare Array of points of at least 9 points as such:
. . .
. . .
. . .
This is because Geometry has holes in between numbers
and that the Betweenness Axiom must be as such--
between any two A and B is a C, except if A and B are consecutive
Reals.
So the adjustment to the Poincare Conjecture of Old has to
be this NINE Point Array.
Now I finally am seeing how to prove it. What happens is
that the Poincare Conjecture is in Elliptic Geometry and thus
our sensations or illusions are still back in Euclidean Geometry
of envisaging a sphere surface. It is as if we are asked for
a answer in metric distance of millimeters when the respondent
can only do inches. Or when a document requires English
and the writer can only do French.
In Elliptic Geometry where the Old and New Poincare Conjecture
is based, then the Old Poincare Conjecture is inconsistent
with the Axioms of Geometry and in specific the axiom of
what is a parallel line in Elliptic Geometry.
In Elliptic Geometry all the lines therein intersect, so there are
no parallel lines. The Old Poincare Conjecture destroys that
axiom.
And to see how it destroys it, I use the Kepler Sphere Packing
program. In Elliptic geometry, and also in Hyperbolic geometry
there is no **nesting of upper layers into the hollows of below
layers**
The creation of hollows in packing comes about only in Euclidean
Geometry. And the entire Kepler Packing program is a Euclidean
geometry exercise. But when you go over to Elliptic and Hyperbolic
geometry, not those silly Euclidean-elliptic-imitations or
hyperbolic-imitations (those plastic fake flowers trying to appear
like real flowers).
Here I am talking of actually being in Elliptic geometry. In
Elliptic Geometry a Kepler style packing program is nothing
that one can visualize and the key feature is that there are
no hollows to stack as a hexagonal close packing.
So the conclusion will be that in Elliptic Geometry when you
shrink a closed loop, it will eventually shrink to not a point
but this Poincare Array of 9 consecutive points
. . .
. . .
. . .
The Old Poincare Conjecture was working with the Old
Betweeness axiom that allowed an infinite downward
regression. That is a false axiom and the patch that fixes
it is consecutive Reals. That patch forces the Old Poincare
Conjecture to shrink not to a point but to a 9-point-array.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Archimedes Plutonium
Archimedes Plutonium wrote:
>
> So the conclusion will be that in Elliptic Geometry when you
> shrink a closed loop, it will eventually shrink to not a point
> but this Poincare Array of 9 consecutive points
>
> . . .
> . . .
> . . .
>
> The Old Poincare Conjecture was working with the Old
> Betweeness axiom that allowed an infinite downward
> regression. That is a false axiom and the patch that fixes
> it is consecutive Reals. That patch forces the Old Poincare
> Conjecture to shrink not to a point but to a 9-point-array.
>
I have a very nice analogy here and using the Kepler Packing as a
brake
concept. Instead of viewing the Kepler Packing as to how many spheres
fit into a cube, consider it as a gigantic braking exercise. What is
the
maximum braking possible? In other words what is the most friction
rubbing possible between spheres in Euclidean geometry and the answer
involves the hollows where you fit upper layers into the hollows of
below layers.
But here is the rub (thanks for the pun) for Elliptic geometry that
the
Old Poincare Conjecture was based in. In Elliptic geometry, in packing
there are no hollows to speak of. And whatever can be construed as
a "hollow" in Elliptic geometry involves that 9 Point Array as shown
in previous post.
Braking as in physics is maximum in Euclidean geometry but braking in
either Elliptic or Hyperbolic is minimal.