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