Let me tell you how I arrived at this insight. I have a vase, in fact,
alot of vases near my bed where I sleep and I awoke staring at a
particular vase that was negative curvature or hyperbolic geometry.
That is to say it was trumpet shaped at the lip and the the base
so that as I lay in bed looking at the vase my mind was saying,
"let me try to squeeze in or fit into that negative curvature one
of the circles of latitude of a nearby globe."
Then my mind goes to work. It says that every latitude circle of a
globe would create a vase of negative curvature.
So then the question becomes can I define any given sphere as merely
a deconstructed set of all the circles from 0 to 1 radius, restricting
the globe to unit radius.
So in other words, I define a sphere as a set of all circles of radius
of all the Reals from 0 to 1. So that the North Pole would be a circle
of 0 radius and the Equator a circle of 1 radius and where there is a
circle of 0.5000.... radius etc etc. So this sphere is the set of all
circles from 0 to 1. Correction-- Hemisphere.
Now, what I want to do is construct a Pseudosphere of negative curvature
from that same pile or set of circles that were lines of latitude.
Now, here, the mind seems to go puzzled. Puzzled and confused because
to build the hemisphere from that set of latitudes, it is puzzling how
those latitudes can thence build a pseudosphere. But it is only a mind
illusion like an optical illusion. We can build a trumpet shape from
those same set of latitudes.
What I have described above is a new world definition of a sphere and
a pseudosphere because we build both from the very same set of circles.
And what I have described above is a way to prove that the Reals are a
mirror image of the AP-adics.
Now the largest circle in building the sphere is a great-circle of
radius 1. And the largest circle in building the Pseudosphere is
9999...9999radix99
So what I am going to show and prove in the second edition of this
textbook is that I can tie together Hyperbolic geometry with Elliptic
geometry in this sphere and pseudosphere building. Building both from
one set of circles. Building them in one way, I end up with a
hemisphere, and building them in another way I end up with a 1/2
Pseudosphere. That tells me the Reals are connected to the AP-adics.
The new definition of a unit-sphere is that of a construction of all the
circles whose radii have Real value from 0 to 1 and the new definition
of pseudosphere is the construction of all the circles from those same
radii excluding the 0 and 1.
I remember when I studied geometry in the 1970s that there was a
distinction between analytic geometry and synthetic geometry. The above
construction is a bit of both for there is analysis and there is
synthesis in the construction.
The above is a very new definition in mathematics for the sphere because
it conjoins the lines of longitude with latitude as never before done.
In the past, we thought of the lines of latitude as unim****tant or
nonessential. Our ideas of a sphere relied on great-circles or
longitudes and latitudes were thought of as garbage-extras. But what my
new definition brings to light is that lines of latitude are just as
essential to Elliptic geometry as are lines of longitude. My new
definition of a sphere tells us that a sphere is dependent on lines
of latitude.
So I should look forward to the 2nd edition of this textbook.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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