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From: Archimedes Plutonium on 4 Feb 2010 16:47

Diving into a new subject for me. Call it Pedestal-Geometry. Another
name maybe
Anti-Calculus Geometry. Another name maybe Surface Contact Geometry.

This subject harbors alot of false ideas and misconceptions.

Take for example two perfect spheres. No problem here because the only
contact
or pedestal would be a "point intersection". But, now, changing the
problem just a
little bit by one perfect sphere in contact with a oblate sphere. What
is the maximum
contact here? Is it still a "point contact" or is there a "line
segment contact" or
possibly even a "area contact?"

Now take two identical infinite cylinders and ignore the ends. What is
the maximum contact?
It would be a infinite-line contact. But now twist one of the
cylinders starting to make a
X type of a pattern. Starting with a tiny twist. With the tiny twist
is the contact a point or line
segment? Of course when the X pattern is 4 right angles then the
contact is a point.

So here we begin to see how our intuition plays tricks on us. That we
begin to not know
what the truth is.

Now getting back to the oblate sphere. Suppose we have two very oblate
spheres, then does
the maximum contact become a "area contact"? So does area contact
occur in mathematics
only when there is "oblateness involved"? So that the only time we
have area-contact is when
we have a elliptic or hyperbolic arc that is to some extent flattened
with oblateness?

In other words, area contact occurrs only when there is some degree of
flatness? Is that true?

The question I am really in pursuit of, is the maximum contact of
Elliptic geometry with
Hyperbolic geometry. Given a "perfect sphere" with no oblateness, and
given the analog
"perfect pseudosphere with no oblateness, what is the maximum contact?
Is it a line-segment
contact?

Now I wish I had a perfect model such as the circles to determine that
60 by 60 by 60
degree arc on globe was the maximum reverse concavity for triangles. I
wish I had
the perfect model to tell me the maximum contact of Elliptic with
Hyperbolic analog
geometry. Maybe there is a perfect model, though I am suspicious of
this one. Take those
plastic pales that one can nest inside one another. Take two of them
and consider the
outer one as Hyperbolic geometry and the inner nested one as Elliptic
geometry. And if
you have had experience with these pales, they often become stuck and
hard to pull
apart. So, the question is, is that a example of Elliptic and
Hyperbolic geometry contact
where the contact is a "surface area" such as a band width of area, or
is the contact
really just a line contact?

Now I do not know where these ideas are going to go. Can one call a
oblate sphere still a
model of Elliptic Geometry? With its oblateness, has it become
Euclidean geometry? And
if so, is the only "area contact" occurring when there is Euclidean
geometry involved?

Is the maximum contact between Elliptic and Hyperbolic geometry that
of a line segment
contact? Or is it like that of two perfect spheres-- a single point
contact?

As you can see, this is not an easy subject but froth with
misconceptions. One could take
an algebra approach and manipulate the equations of a oblate sphere
with oblate sphere
and find out what the intersection maximum is.

The reason I am on this subject is that I want to see if the maximum
contact with Elliptic
and Hyperbolic geometry is a 10% of the surface area of the sphere.

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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