Before I came to the Internet in 1993, I had the persistant nagging
idea that if you unioned the Riem geometry to Loba geometry that you
end up with Euclidean geometry. I think I am nearer to that goal.

My beliefs:
(1) I believe Physics is everything and that mathematics is a minor
subset of physics. Physics creates math as well as every other thing in
the Universe.
(2) I believe Quantum Duality is the superstructure of mathematics and
duality has not been represented in mathematics. One of that duality
superstructure is that geometry has two forms -- Eucl and the other
dual is Riem + Loba. I think of Eucl as the wave form and Riem as the
particle form.
(3) I believe that these two geometries of Eucl and of Riem + Loba have
"native numbers" which are the points of those geometries.

This is what guides me and spurs me on.

For years I have looked for the very best model of Lobachevsky
geometry, such as saddle shaped, such as the shape of a trumpet. But I
am going to settle with a shape of what I call the residue of when you
cut out a circle that is inscribed inside of a square. Those 4 corners.

So how am I going to make the n-adics a field? I am going to claim that
the p-adics and n-adics form a circle and this circle winds around to
infinity. And it matters not whether the adic is prime or composite,
they wind around and come back to the starting point of 0.

I am going to define imaginary numbers as Doubly Infinites. Reals are
infinite rightward strings with a finite leftward portion, and Adics
are the reverse of Reals infinite leftward string with finite rightward
portion. Doubly Infinites are as the name says, strings both infinite
rightwards and leftwards.

I am going to use Doubly Infinites to obtain those points of the
inscribed square with its circle cut out. Those 4 portions when the
circle is cut out.

Now in 10-adics we have a number of ....9999999_10_ but we also have a
number of Doubly Infinites of ...99999_10_111111.... and of
.....99999_10_444444....

What I am getting at, is to obtain the points of those 4 regions of
Loba geometry when the inscribed circle is cut out of the square. I
want every point of those 4 regions represented by a Doubly Infinite.
And these DI are imaginary numbers to the Adics.

Now with the 2-adics I am going to focus on the square from 1 to 3. So
just the square from 1 to 3. For the 3-adics I focus on the square from
2 to 4 etc etc, and for the 10-adics I focus on the square from 9 to
11. Each adics spins a circle out to infinity but I am interested only
in the circle spun by each adic whether prime or composite confined to
a square of a radius of 1.

This is why I need n-adics as well as prime-adics. Because as I go from
one to the next number, I go through every number and not just jump
from prime to prime.

Now each adic whether prime or composite is inscribed inside a square
from its radius of 1, even though we know they all spin and turn to
infinity we are interested only in the turnings inside this unit radius
square. And here we apply the Doubly-Infinites as imaginaries to
capture those 4 Loba regions in each square.

These Doubly Infinites which are the points of those 4 Loba regions in
each square when added or union to the p-adics and n-adics encompasses
the entire Euclidean plane and thus there is a 1-1 correspondence
between All-adics because they have these Doubly-Infinites as imaginary
number add-ons. And since All Adics are 1-1 correspondence with the
Real Plane and the Reals are a field then All-Adics are a field.

Now what specific Doubly-Infinites do I need for say the 10-adics to
obtain those 4 Loba regions? I wish the answer was as simple as Complex
where you merely tack onto the Reals just one number of sqrt-1. I wish
it were that simple, but it is not that simple because here we need an
infinite number of Doubly Infinites as imaginary. For every 10-adic
number has a Doubly Infinite associated to that particular 10-adic
number.

And where n-adics stand alone do not form a field because they lack
inverses or other requirements, then the Doubly-Infinite imaginary
number steps up and fulfills the requirement.

Pictured geometrically is this. The Reals are the Euclidean plane. The
All Adics by themselves are spirals at each point, a spiral at 1,
another spiral at 2, etc etc. Think of the Euclidean plane filled up
with circles of radius 1 at each whole number. When we inscribe those
circles inside a square then the 4 regions of each square are
Doubly-Infinites as imaginary tack-ons to the adics.

A huge amount of work has to be done.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

In reference to this belief:
(2) I believe Quantum Duality is the superstructure of mathematics and
duality has not been represented in mathematics. One of that duality
superstructure is that geometry has two forms -- Eucl and the other
dual is Riem + Loba. I think of Eucl as the wave form and Riem as the
particle form.

Can someone answer a question here. We know about periodic functions
such as the trigonometry functions of sine, cosine, etc etc. Can Riem
geometry or Loba geometry accomodate periodic functions in the way that
Euclidean geometry can accomodate periodic functions or is there an
inherent rejection or inability to have periodic functions in Riem and
Loba geometries? Are periodic functions native to only Euclidean
geometry?

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Scratch the above. Working with just 2-adics which is already a field,
let me see what Doubly Infinites to append to 2-adics to make a Riem +
Loba geometry that equals Eucl geom.

Here I envision a circle in the Real plane and that 2-adics are the
points of the circle centered at zero. So if we have a truncated Real
plane of say only 1 unit long then the 2-adics occupy points of the
circle of radius 1 but leave the residue of those 4 regions of a circle
inscribed in a square.

So the 2-adics are already a field and we have no concerns about field
requirements. The concern we have is to fill in those 4 regions when
you cut out the inscribed circle from the square. And to assign
Doubly-Infinites to those points of the 4 regions.

Now if we look at only quadrant 1 of a unit circle of radius 1 the arc
of the circle from 0,1 to 1,0. I am saying that the 2-adics will cover
those points from the arc 0,1 to 1,0 but the points in the region
outside the arc bounded by the unit square is a Lobachevskian geometry
region to the point of 1,1. A Loba triangle which has the arc side as a
Loba line segment even though its other two sides is still Eucl lines
of the square.

Dik, or Chris, can you tell me whether 2-adics yield any of those
points in that arc segment? If not, then I am rather safe to proceed.

If not, then I propose that this arc line segment of Lobachevskian
geometry has Doubly Infinites as its points. And thus what I propose to
do is tack onto the 2-adics an imaginary number of a Doubly Infinite to
be this Lobachevskian line segment.

That would be tremendous progress towards the equation
Eucl geometry = Riem geom union or + Loba geom

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

I may end up agreeing with Dik that the n-adics are impossible to
wrangle into a field. I may end up with the idea that only 2-adics is
all that is needed and the rest are just meaningless equivalent forms
such as 1/2 is 2/4 is 3/6.

A.P.

In article .com writes:
Scratch the above.

Indeed, it makes no sense. But the same applies to this article.

Dik, or Chris, can you tell me whether 2-adics yield any of those
points in that arc segment? If not, then I am rather safe to proceed.

How can I tell? It makes no sense to me.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter wrote:
In article .com writes:
Scratch the above.

Indeed, it makes no sense. But the same applies to this article.

Dik, or Chris, can you tell me whether 2-adics yield any of those
points in that arc segment? If not, then I am rather safe to proceed.

How can I tell? It makes no sense to me.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Okay, we have a circle of radius 1 centered in the Real plane at zero.
We could do this with radius 2, radius 3 etc etc. We inscribe the
circle inside a square. That leaves 4 regions that are triangular
shaped if we cut out the circle. We focus on just the quadrant 1 of
this triangular shaped region.

Now we take the 2-adics and form a x,y axis of 2-adics, just like the
x,y axis of Reals to form the Real plane. We ask whether any 2-adic
points fall within that triangular shaped region when the circle is cut
out of the square. In Quadrant 1, the triangular region is 0,1 and 1,1
and 1,0 with two straight line segments and the curved segment that
used to be the circle.

Someone said adics forms a treelike structure and I am asking whether
adics are points within these circles of ever increasing radius and
always fail to fall into one of these 4 triangular shaped regions.

So the radius is 1 and the distance from zero to 1,1 is sqrt2 or 1.4...
So the points from where the circle was to 1,1, whether any of those
points arise in 2-adics when the x and y axis are 2-adics

Basically I want to find out if the 2-adics fall into the circle and
miss falling into the triangle-shaped regions.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Dik Winter wrote in another thread:
It is exactly that. They offer idempotents, and so they also offer
zero
divisors, and so they can not be part of a field. (This is not true
when
n is a power of a prime, that is just a change of notation, so I
exclude
them.) So the p-adics offer something the n-adics do not offer: the
ability
to be part of a field.

A.P. writes: Thanks for that information but I want even more of a
thorough answer. I want to know, Dik, if idempotents are the only thing
that n-adics offers that the p-adics cannot offer. Is there anything
else that the n-adics can create for which the p-adics are powerless to
create? Because if the n-adics can only offer idempotents, then I
should have abandoned them a long time ago. And the moral to the story
is that sometimes we fail to ask the best question first and do
heedless work.

So, Dik, are idempotents the only difference between n-adics and
p-adics.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Basically a point like 0.99 or -0.99. Are they possible in 2-adics? For
they fall into that triangular shaped region of unit circle inscribed
inside of square.

A.P.

Dik, how many of the points on the unit circle itself can the 2-adics
generate? Is it a finite set or an infinite set? And what of the points
inside the unit circle?

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

No, I would have trouble here with 0.99 in that it would be in the
triangular shaped region but also it would be on the y-axis before 1.

So how can I overcome that obstacle? Can I make the 2-adics the x-axis
and the 3-adics the y-axis? And then ask whether a point exists with
the coordinates of 0.99,0.99 ?

A.P.