Dik T. Winter wrote:
> In article <1193933470.367670.303770@[EMAIL PROTECTED]
>
a_plutonium <a_plutonium@[EMAIL PROTECTED]
> writes:
> > Dik T. Winter wrote:
> ...
> > > And again you fail to see that in my system the distance between
Frankfurt
> > > and Zurich is *not* addition, but subtraction. And there you
indeed do
> > > have two choices, and subtraction obviously is *not* commutative.
> > >
> > > > And there is noway of removing those choices and thus never a
ring or
> > > > field.
> > >
> > > You just can't read or understand my system.
> ...
> > Wrong Dik, it is you who is unable to understand your mistakes. So
you
> > defined the minor arc as addition and the major arc as subtraction.
But
> > now what do you do for podal and antipodal points where minor arc
equals
> > major arc? Do you roll the dice or flip a coin to see which direction
or
> > clockwise or counterclockwise.
>
> Indeed, you do not understand my system, because that is *not* the way I
> defined it. One of the arcs is A - B, the other is B - A. A + B is
> neither.
>
You never considered that your model is all wrong, perhaps you never
found
anyone to discuss it with.
I found over 200,000 search hits talking on commutativity on a sphere
and the very first
one says it is noncommutative in general.
JSTOR: Random Walk on a Sphere and on a Riemannian Manifold - Oct 31
[ 317 ] RANDOM WALK ON A SPHERE AND ON A RIEMANNIAN MANIFOLD BY P. H.
ROBERTS ...... It is not known whether any commutative manifold exists
which is not ...
links.jstor.org/sici?sici=0080-4614(19600331)252%3A1012%3C317%3ARWOASA
%3E2.0.CO%3B2-5 - Similar pages - Note this
Here, Dik, the first hit in the first paragraphs of the abstract, by
P.H.Roberts and H.D. Ursell
in a Royal Society journal of Math and Physics state:
" It is shown that commutability does not hold in general, but that it
does hold in completely
harmonic spaces and in some others."
Dik, those guys don't often make statements like that if they were not
sure
that a sphere is not Commutative in general.
The interesting question, Dik, is what has to be done to your circle
in order to
make it Commutative, in order to make it Commutative in general. And
what has
to be done is to cut away your circle so that there is no choice of
direction nor
is there a choice of major or minor arc. So when you remove 1/2 of
your circle you
you no longer have a circle but a curved Euclidean line segment and
thus you can
establish your Ring and Field.
And that begs the question of a sphere. How to make it a Field and
Ring, and I suspect
the answer is cutting away 1/2 and what I suspect the minimum deletion
is a hemisphere.
So you have to cut away a hemisphere of a globe in order to make it a
Field or Ring.
In those 200,000 hits where they talk about establi****ng a Field or
Ring over a sphere,
what they are doing is merely cutting away alot of the sphere. And it
is no different in
the perspective that Earth is so large that we see land that looks
mostly flat and so
if you set up a Model on a small patch of a circle or sphere, then you
can get away with
the illusion that you have a Field or Ring on that small patch. And
that is what yours
has come to, Dik, a small patch of the circle that seems to obey your
definitions, but
not the circle in full.
> Again, to repeat the definition, take a circle with radius R and
midpoint
> M. Define an origin on the circle, say O and a direction from O. Let's
> have A and B two points on the circle. Let alpha be the angle between
> the line from A to M and the line from O to M in the direction from O to
A
> according to the preferred direction. Similar for beta and B. (Also
The Ring over Reals has no preferred direction. The Field over Reals
has no preferred direction. So that should have warned you that your
system
is doomed to failure.
> choose alpha and beta such that they are in [0, 2.pi).) To add A and B
we
> calculate: gamma = alpha + beta mod 2.pi. Take a line from O to M and
go
> in the preferred direction along an angle gamma; that is the point
> C = A + B. For multiplication we calculate:
> delta = fraction((A * B) / (4.pi^2)) * 2.pi,
> where fraction yields the fractional part of a number that is in [0,1).
> This delta gives D = A * B.
>
> Now, what about E = A - B? It would be the number that, when it is
added
> to B yields A. Or in angles, it angle would be the angle from the line
> through B and M and the line through A and M in the preferred direction.
> In arcs, it is the arc from B to A in the preferred direction. Similar
> for B - A.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland,
+31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
If you deleted half of your circle, then you can get away with your
above because there
never is two different number answers to a multiplication or addition.
If Earth were a hemisphere
and not a sphere then the distance between Frankfurt and Zurich would
be only 300 km
and not a second answer of 39,700 km.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
|
