about half these authors believes pseudosphere has infinite area,other half not #817 Correcting Math #286 Atom Totality

I know I am on the correct path of finding the natural boundary
between finite and infinite
with the pseudosphere and sphere, because the pseudosphere is infinite
in range, yet has
the same area as its associated sphere.

But notice this list of authors on a Google search for pseudosphere
area:

--- quoting Google hits on pseudosphere area ---

Pseudosphere -- from Wolfram MathWorld
Aug 13, 2010 ... The pseudosphere therefore has the same volume as the
sphere while having constant negative Gaussian curvature (rather than
the constant ...
mathworld.wolfram.com › ... › Surfaces › Surfaces of Revolution -
Cached - Similar

The pseudosphere « General Musings
Dec 19, 2009 ... Perhaps more interestingly, the Pseudosphere is an
object that is infinite in extent, yet has both a finite surface area
and encloses a ...
danielcolquitt.wordpress.com/2009/12/19/the-pseudosphere/ - Cached

pseudosphere
As a result, a sphere has a closed surface and a finite area, while a
pseudosphere has an open surface and an infinite area. In fact,
although both the ...
http://www.daviddarling.info/encyclo...udosphere.html - Cached -
Similar

Pseudosphere - Wikipedia, the free encyclopedia
In its general interpretation, a pseudosphere of radius R is any
surface of ... Both its surface area and volume are finite, despite
the infinite extent of ...
en.wikipedia.org/wiki/Pseudosphere - Cached - Similar

The Pseudosphere
The representation on the pseudosphere gives us the opportunity to
give some geometric meaning to the fundamental constant k in the
theorem on area. ...
math.uncc.edu/~droyster/math3181/notes/.../node69.html - Cached -
Similar

Rudy's Blog » Blog Archive » Pseudospheres
Aug 28, 2009 ... And I think this surface is something like a
pseudosphere. ... But if the circle is on a sphere, the area of the
cap inside it is LESS than ...
http://www.rudyrucker.com/blog/2009/...pseudospheres/ - Cached - Similar

Pseudosphe Definition from Answers.com
pseudosphere ( ′südə′sfir ) ( mathematics ) The pseudospherical
surface ... For a given edge radius R, the area is 4Ï€R2 just as it is
for the sphere, ...
www.answers.com/topic/pseudosphere-1 - Cached

Chaos on the pseudosphere - Elsevier
by NL Balazs - 1986 - Cited by 234 - Related articles
Classical motion on compact surfaces of constant negative curvature
The pseudosphere has an infinite area, and a point moving freely on it
will escape to ...
linkinghub.elsevier.com/retrieve/pii/0370157386901596

Pressures for a One-Component Plasma on a Pseudosphere
by R Fantoni - 2003 - Cited by 6 - Related articles
the case of a pseudosphere, let us consider a large disk of area A,
filled with a 2D OCP. For compressing it infinitesimally, changing the
area by ...
www.springerlink.com/index/HX58350656N42QVP.pdf - Similar

The Math Book: From Pythagoras to the 57th Dimension, 250 ... - Google
Books Result
Clifford A. Pickover - 2009 - Mathematics - 528 pages
Thus, a sphere is a closed surface with a finite area, while a
pseudosphere is an open surface with infinite area. British science
writer David Darling ...
books.google.com/books?isbn=1402757964...

--- end quoting Google hits ---

Notice in that list, about roughly 1/2 say the area is infinite.

Remember I opened the chapters of the math book "Correcting Math" by
correcting
the Euclid Infinitude of Primes proof where roughly half the authors
thought it was indirect
and not direct method. I am probably staring at what could be a
geometry misconception
akin to the Euclid Infinitude of Primes misconception.

So I have the hunch that I am in the proper place in all of
mathematics to eke out this
boundary of finite versus infinite with the pseudosphere and its
associated sphere.

And it ties in with my famous conjectu Euclidean Geometry =
Elliptic unioned Hyperbolic
geometries. And Physics would write it as Eucl Geom = broken symmetry
of Elliptic unioned
Hyperbolic geometries.

I am making very slow progress, for I am not convinced myself that at
10^500 is a special
number as the boundary, for as of yet I find nothing special of the
pseudosphere versus the
sphere of 10^500.

So maybe we really have not ironed out the concept of pseudosphere
area. Maybe it is not equal in terms of "absolute value" to the sphere
area, with a change in sign. Maybe no-one
has factored in the cutoff portion and the reason that Wikipedia lists
the "theoretical pseudosphere". And that is a rather funny
circumstance to have anything in mathematics called a "theoretical
pseudosphere" as if there is a "practical pseudosphere" to compare
with.
I suppose these same blokes would call some numbers as "theoretical
numbers" versus
a natural-number?

You know, it is hard for me to resolve a problem and when the math
community is divided
on issues, it is doubly hard to resolve the problem.

So maybe this search, this hunt of mine for why 10^500 is special for
the pseudosphere involves the area, and not the circumference of the
defect cutoff.

And if I am correct that 10^500 is the start of infinity, then all
pseudospheres poles go to
10^500. Meaning that we assume they converge at that point of 10^500.
So that my hunt
was not one of "specialness at 10^500" but rather convenience at
10^500. That when we say a
pseudosphere area equals the absolute value of the associated sphere
area, we have pre-agreed that the poles end at 10^500.

Although I do, firmly believe there is something special of the number
10^500 with the pseudosphere and will not be restful until I find it.


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

I am really, really really glad I diverted two posts to history of
science for I see a immediate
solution to my snagged up on the pseudosphere defect cutout. I had
imposed a embedding sanction on the pseudosphere.

Funny how modern day politics sometimes enters into my science
achievements "sanctions".

I had imposed a sanction that the pseudosphere had to be embedded
inside the sphere.

But let me say, suppose I lifted that imposed sanction. What if I
said, give me a radius of a sphere and then a pseudosphere with the
same radius but then allow me to draw the pseudosphere with a
logarithmic curve that has a cutoff but whose surface area exactly
matches the sphere's surface area. So I elminate these pseudospheres
that supposedely stretch endless out. The pseudosphere I am left with
will not fit inside the sphere embedded
inside the sphere but will poke out of the sphere in order to have
equal surface area. And we
will not count the area of the "stubb cutaway of the pseudosphere".

Now correct me if wrong, but I am under the vision that the tractrix
curve, a logarithmic curve
is a variable curve, just as there are many types of logarithmic
spirals, there are many types
of tractrix curves.

So now, the condition I want to satisfy is no longer a embedding of
the pseudosphere with its
defect cutout, but rather I want an equal surface area of the
pseudosphere with its cutout and
with a given sphere.

And now I can begin to look at 10^500 and wonder what is special about
this number and sphere and pseudosphere? Is it at 10^500 that we reach
a special relationship of a equal surface area that we cannot have
reached with a smaller radius or diameter?

All of a sudden this hunt has become very exciting, indeed.

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

Archimedes Plutonium wrote:
I am really, really really glad I diverted two posts to history of
science for I see a immediate
solution to my snagged up on the pseudosphere defect cutout. I had
imposed a embedding sanction on the pseudosphere.

Funny how modern day politics sometimes enters into my science
achievements "sanctions".

I had imposed a sanction that the pseudosphere had to be embedded
inside the sphere.

But let me say, suppose I lifted that imposed sanction. What if I
said, give me a radius of a sphere and then a pseudosphere with the
same radius but then allow me to draw the pseudosphere with a
logarithmic curve that has a cutoff but whose surface area exactly
matches the sphere's surface area. So I elminate these pseudospheres
that supposedely stretch endless out. The pseudosphere I am left with
will not fit inside the sphere embedded
inside the sphere but will poke out of the sphere in order to have
equal surface area. And we
will not count the area of the "stubb cutaway of the pseudosphere".

Now correct me if wrong, but I am under the vision that the tractrix
curve, a logarithmic curve
is a variable curve, just as there are many types of logarithmic
spirals, there are many types
of tractrix curves.

So now, the condition I want to satisfy is no longer a embedding of
the pseudosphere with its
defect cutout, but rather I want an equal surface area of the
pseudosphere with its cutout and
with a given sphere.

And now I can begin to look at 10^500 and wonder what is special about
this number and sphere and pseudosphere? Is it at 10^500 that we reach
a special relationship of a equal surface area that we cannot have
reached with a smaller radius or diameter?

All of a sudden this hunt has become very exciting, indeed.


Now grant me the concession that the tractrix logarithmic curve is a
variable curve so I have a large long list of choices of a log-curve
to use for the tractrix and thus the pseudosphere which is a revolving
of the tractrix about its axis. And also, grant me the lifting of the
embedding of the pseudosphere inside the sphere of equal radius.

So, now, what is special about the number 10^500 and not the number 1,
or 2 or 3 etc etc until we reach 10^500 with these spheres and
pseudospheres?

Well, if my mind internal vision is not playing games and tricks on
me, I can envision where
at 1, there is no pseudosphere of equal area as the unit sphere, and
where that pseudosphere
embeds the unit sphere inside itself, nor at 2, at 3, etc etc, until
we reach 10^500.

At 10^500, there is a tractrix curve when revolved on its axis that
creates a pseudosphere of
Radius 10^500 and which is equal in area to the sphere of Radius
10^500 but which the pseudosphere can accomodate this sphere to be
embedded inside the pseudosphere. Maybe
my mind is playing tricks and games on me, and that is why I harken
that we need working
handheld models in such endeavors.

If my mind is not playing tricks on me, then I discovered the natural
boundary in pure mathematics of where finite numbers versus infinite
numbers meets, and it is 10^500.

And that makes sense in art and esthetics as well as science, because
the pseudosphere is
seen as going to infinity, but here we put an actual number to where
infinity starts.

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

So I think I am close to finding my solution as to why 10^500 would be
a pure math
natural boundary between finite and infinite numbers.

We see the problem having already existed ever since the discovery by
Beltrami
of the pseudosphere in 1868, that the infinite stretch of the
pseudosphere and why that
surface area should equal the associated sphere of equal Radius. Why
would any infinite
stretch still be a finite area?

This sounds like a repeat of the troubles and problems with math
professors not knowing
that Euclid's Infinitude of Primes proof was actually direct method
and not indirect. And so
we have probably every mathematician lined up and backing or
supporting the idea that
infinite stretch of a pseudosphere is not harmful and that it is a
finite area.

But I think they are all wrong.

What they should have done and said was that we can represent a
cutaway pseudosphere that
has equal surface area to a given sphere of radius R.

And now for my solution to 10^500 and why that number is so special.
As we increase the radius of the sphere and pseudosphere from 1 to 2
to 3 to 4 all the up, we begin to see a pattern that the pseudosphere
of equal surface area, not counting the cutaway of the poles
on the pseudosphere, we see as we increase the radius that the
pseudosphere gains in size
relative to the sphere as we go higher and higher. Some of this gain
is due to the fact that
the tractrix curve that generates the pseudosphere is a pliable or
variable curve, similar to the
fact that there are a range of logarithmic spirals and not all log
spirals are fixed to one type.
So that the pseudosphere tractrix curve has a range of types.

Now we keep the areas the same for the sphere and associated
pseudosphere and as we
grow bigger in size, the pseudosphere grows bigger relative to the
sphere, to the point where
at 10^500 the associated sphere fits embedded into the pseudosphere.
So at radius 1, the pseudosphere fits mostly inside the sphere, but at
radius 10^500, the sphere fits mostly
inside the pseudosphere. Where we have a reversal of roles.

Now it is hard for me to believe this because looking at that
pseudosphere as shown in the picture of Wikipedia, it is very hard to
imagine a sphere of radius 10^500 fitting mostly inside
that pseudosphere of radius 10^500. It is difficult to envision how
the area can be the same yet the sphere fitting mostly inside that
puny pseudosphere. And because it is so hard to imagine, I do not
believe
I am out of the woods yet on this topic, and that my imagination has
played tricks on me.
So I need further concrete evidence before I can accept this myself.

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

Archimedes Plutonium wrote:
So I think I am close to finding my solution as to why 10^500 would be
a pure math
natural boundary between finite and infinite numbers.

We see the problem having already existed ever since the discovery by
Beltrami
of the pseudosphere in 1868, that the infinite stretch of the
pseudosphere and why that
surface area should equal the associated sphere of equal Radius. Why
would any infinite
stretch still be a finite area?

This sounds like a repeat of the troubles and problems with math
professors not knowing
that Euclid's Infinitude of Primes proof was actually direct method
and not indirect. And so
we have probably every mathematician lined up and backing or
supporting the idea that
infinite stretch of a pseudosphere is not harmful and that it is a
finite area.

But I think they are all wrong.

What they should have done and said was that we can represent a
cutaway pseudosphere that
has equal surface area to a given sphere of radius R.

And now for my solution to 10^500 and why that number is so special.
As we increase the radius of the sphere and pseudosphere from 1 to 2
to 3 to 4 all the up, we begin to see a pattern that the pseudosphere
of equal surface area, not counting the cutaway of the poles
on the pseudosphere, we see as we increase the radius that the
pseudosphere gains in size
relative to the sphere as we go higher and higher. Some of this gain
is due to the fact that
the tractrix curve that generates the pseudosphere is a pliable or
variable curve, similar to the
fact that there are a range of logarithmic spirals and not all log
spirals are fixed to one type.
So that the pseudosphere tractrix curve has a range of types.

Now we keep the areas the same for the sphere and associated
pseudosphere and as we
grow bigger in size, the pseudosphere grows bigger relative to the
sphere, to the point where
at 10^500 the associated sphere fits embedded into the pseudosphere.
So at radius 1, the pseudosphere fits mostly inside the sphere, but at
radius 10^500, the sphere fits mostly
inside the pseudosphere. Where we have a reversal of roles.

Now it is hard for me to believe this because looking at that
pseudosphere as shown in the picture of Wikipedia, it is very hard to
imagine a sphere of radius 10^500 fitting mostly inside
that pseudosphere of radius 10^500. It is difficult to envision how
the area can be the same yet the sphere fitting mostly inside that
puny pseudosphere. And because it is so hard to imagine, I do not
believe
I am out of the woods yet on this topic, and that my imagination has
played tricks on me.
So I need further concrete evidence before I can accept this myself.


A long time ago I had a set of nesting funnels. So if I can get
together a
bunch of those plastic vases shaped like trumpets or those plastic
trumpets
used in the world cup soccer match earlier this year, I can play
around with
concrete items and seeing is believing.

Now I keep the area and radius the same, but the Gaussian curvature
can be different in value but opposite in sign. Now I think this is an
overall improvement in that the identical
Gaussian curvature does not allow for a snug fit of the sphere resting
on the saddle shape of
the pseudosphere.

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

Archimedes Plutonium wrote:


Now I keep the area and radius the same, but the Gaussian curvature
can be different in value but opposite in sign. Now I think this is an
overall improvement in that the identical
Gaussian curvature does not allow for a snug fit of the sphere resting
on the saddle shape of
the pseudosphere.


No, I made a mistake there, keep the Gaussian curvature the same, of 1/
R^2 and
(-)1/R^2 and keep the surface areas the same and keep the Radiuses the
same.
What we do is when we cut off the pseudosphere polar regions we splice
that amount
of area onto the curves of the pseudosphere.

Here is a picture of the pseudosphere if you do not know what they
look like:

http://en.wikipedia.org/wiki/Pseudosphere

So as we splice this extra area from the cutaways of the pseudosphere
poles,
we gradually keep increasing the interior volume of the pseudosphere
relative to the
sphere. At 10^500, the sphere should fit inside that hollow of the
pseudosphere centered
at its equator.

It is like taking two trumpets and putting them together and haveing a
baseball or softball
fit into that hollow.

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

Archimedes Plutonium wrote:
Archimedes Plutonium wrote:


Now I keep the area and radius the same, but the Gaussian curvature
can be different in value but opposite in sign. Now I think this is an
overall improvement in that the identical
Gaussian curvature does not allow for a snug fit of the sphere resting
on the saddle shape of
the pseudosphere.


No, I made a mistake there, keep the Gaussian curvature the same, of 1/
R^2 and
(-)1/R^2 and keep the surface areas the same and keep the Radiuses the
same.
What we do is when we cut off the pseudosphere polar regions we splice
that amount
of area onto the curves of the pseudosphere.

Here is a picture of the pseudosphere if you do not know what they
look like:

http://en.wikipedia.org/wiki/Pseudosphere

So as we splice this extra area from the cutaways of the pseudosphere
poles,
we gradually keep increasing the interior volume of the pseudosphere
relative to the
sphere. At 10^500, the sphere should fit inside that hollow of the
pseudosphere centered
at its equator.

It is like taking two trumpets and putting them together and haveing a
baseball or softball
fit into that hollow.


Alright, that did not take long at all.

We can generalize the pseudosphere as being two right circular cones.
And for brevity
we can consider only 1/2 the sphere and thus one cone.

We can generalize the cutaway of the infinite stretch of the
pseudosphere and then the tacking on of the missing area so the
pseudosphere area is equal to the sphere area, as
a exercise on the right circular cone of forming that cone into a
cylinder. A cylinder that
at the radius of 10^500 is able to contain inside itself the sphere of
radius 10^500.

So now, these formulas:

right circular cone: pi*r*s where s is slant-height = sqrt(r^2 + h^2)

sphere area = 4*pi*r^2

cylinder: 2*pi*r*h

I needed to convince myself that as the radius increases, the adding
on or splicing more
area segments onto the pseudosphere (cone) increases relative to the
sphere, so that as the radius
increases, at some moment in the radius, the cone has transformed into
a cylinder that is able to contain the sphere. It is the slant height
that provides that ever increasing area for
the cone to transform into a cylinder.

So is 10^500 unique to the above process? Well, remember it is the
cone that is modeling the
pseudosphere so that 10^500 is not going to be indicative of the cone
transformed into cylinder.


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

Archimedes Plutonium wrote:
(snipped)

Alright, that did not take long at all.

We can generalize the pseudosphere as being two right circular cones.
And for brevity
we can consider only 1/2 the sphere and thus one cone.

We can generalize the cutaway of the infinite stretch of the
pseudosphere and then the tacking on of the missing area so the
pseudosphere area is equal to the sphere area, as
a exercise on the right circular cone of forming that cone into a
cylinder. A cylinder that
at the radius of 10^500 is able to contain inside itself the sphere of
radius 10^500.

So now, these formulas:

right circular cone: pi*r*s where s is slant-height = sqrt(r^2 + h^2)

sphere area = 4*pi*r^2

cylinder: 2*pi*r*h

I needed to convince myself that as the radius increases, the adding
on or splicing more
area segments onto the pseudosphere (cone) increases relative to the
sphere, so that as the radius
increases, at some moment in the radius, the cone has transformed into
a cylinder that is able to contain the sphere. It is the slant height
that provides that ever increasing area for
the cone to transform into a cylinder.

So is 10^500 unique to the above process? Well, remember it is the
cone that is modeling the
pseudosphere so that 10^500 is not going to be indicative of the cone
transformed into cylinder.


Alright, I am happy with what I wrought above. I can begin to see that
patching the
original pseudosphere as the radius increases would alter the
conditions of where the
pseudosphere is embedded inside the sphere to where the sphere is
embedded inside
the pseudosphere.

But I think I could have arrived at that situation in a more easier
and elegant fashion.

Something I have not hand held and constructed. I am thinking of
enclosing a sphere
inside a cube and then cutting the cube into equal 8 sections of its
corners and then
removing the sphere and leaving that 8 sections of residue. Then
rearranging those
8 sections so that they form what I call a circular-pseudosphere. It
has a circular curve
rather than based on a logarithmic curve that the true pseudosphere
has.

The beauty of this circular-pseudosphere is that it has the identical
same surface area
as the sphere itself. So there is no patching or adding a patch of the
infinite stretch of the
pseudosphere that escaped the sphere and rambled out to infinity.

So here we see the difference of that infinite stretch of a log-
pseudosphere and the circular-pseudosphere.

Now how does the number 10^500 play a role with the log-pseudosphere
and the circular-
pseudosphere? It is where the **integer value only** is large enough
to make the log curve
be the same as the circular curve. In other words, dealing only with
integers and not fractions
that at 10^500 we pass a boundary where the integer intercepts the
circular with the log curves.

Also, I want to begin to discuss the fact that with Conic Sections of
classical mathematics we
were able to produce all the curves-- straight lines, circles,
parabola and hyperbolas.

But notice that the Pseudosphere is the Conic Sections in Hyperbolic
geometry and whereas the Conic Sections is the frustrum or apexes
meeting at the two cones apex, here in Hyperbolic geometry the
pseudosphere is the reverse of the Conic sections where the cones
do not meet at the apexes but where they meet at the bases.

And a conic section is like a knife cut or a plane that cuts into the
Conic Section. In Hyperbolic Geometry we replace that with a
hyperbolic cut and thus we produce all the curves possible-- straight
line, circle, parabola and hyperbola.

Now I should also be able to do the Conic Sections in Elliptic
Geometry and the cuts should be elliptic cuts to produce all the
curves.

So here we begin to see the force of Quantum Mechanics dualism on
mathematics that
we have the inverses or reverses in all the three geometries.

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

Alright, it is about time I performed this experiment. I did the 2D
experiment a long time ago
since it is easy to cut out a circle inside a square and rearrange the
4 residue triangles that
forms a 4 pointed star. But now I need to do the 3D experiment.

To do this I used the end of a toothpaste box and cut out a cube
looking portion. Of course it
had just one end of the cube since I do not need the full cube but
only 1/2 cube to visualize.
I thence cut the cube in 1/2 and that portion I cut into 4 pieces. Now
each of those 4 pieces has 3 tip ends and it is those tip ends that
the sphere is tangent.

For the sphere enclosed inside the cube, I did not have available. So
I got a firm hard orange
and with a sharp knife I cut the orange into the 8 equal pieces. Each
piece is a elliptic geometry triangle. So that 4 will rest on the top
4 cube cutaways and the other 4 resting
below in the 4 cube cutaways.

Now I needed to perform this experiment because I can no longer
accurately visualize it in
my mind's eye.

Now some are going to say that this figure is in no way a
pseudosphere. I am going to
call it, or name it the "circular pseudosphere" because the curves are
not logarithmic as
the true pseudosphere is, but instead are circular curves derived from
the sphere itself.

Mathematics probably has a name for this figure, but I am unaware of
what it is called.

I see it as a Circular Pseudosphere with spines on it, as there are 4
spines. And the equator
is not a circular disc as in the true pseudosphere but is a
discontinuous region connected by
4 points of the top layer with 4 points of the bottom layer.

Now why do I bother with this figure? It is important because it is a
primitive pseudosphere
on the way to becoming a full fledged pseudosphere and its main
feature that is desirous is
the fact that the surface area of this primitive pseudosphere matches
the surface area of
the sphere itself. All that needs to be done to transform this
primitive pseudosphere is to
make the curves logarithmic instead of circular. And this primitive
pseudosphere fits totally
inside the sphere itself from which it sprang from.

Now the mission project of this experiment is that in the course of
transforming this circular
pseudosphere into a logarithmic pseudosphere that the radius of 1 or 2
or 3 etc etc are not
special, but at 10^500 this transformation is very special.

It is at the number 10^500 radius that the transformation of the
circular pseudosphere into
a full fledged logarithmic pseudosphere that the sphere of radius
10^500 is able to be contained inside the hollow of the 10^500
logarithmic pseudosphere.

And this is where the natural boundary between Finite Numbers versus
Infinite Numbers
occurs in mathematics. It exists in the fabric of mathematics itself
and is not artificially imposed. It is where the pseudosphere no
longer nests inside the associated sphere of
equal radius but where the sphere nests inside the associated
pseudosphere of equal
radius. It happens not at 10^499 but uniquely starts to happen at
10^500 and continues
to nest inside thereafter.

Why does this crossing of embeddedness happen? And the answer is that
the surface area
of the pseudosphere has to be equal to the surface area of the
associated sphere. This stretch area of the pseudosphere is collected
and patched onto the pseudosphere cutaway,
and the reason the Circular Pseudosphere has holes and gaps in it. So
as this area is patched
onto the cutaway pseudosphere and because this area increases as the
radius increases (see
the slant height of the cone reference), since this patch area
increases with the radius
increase, means there is a cross-over at some large number radius
where the sphere fits inside the hollow of the logarithmic
pseudosphere cutaway. I conjecture this radius number
is 10^500 or thereabouts because in Physics the end of the
StrongNuclear force is
253! for element of atomic number 100, which is 10^500 Coulomb
Interactions. The end of Physics is the end of mathematics as well, or
the start of infinity.

In New-Mathematics, the better definition of Infinity is not that of
"endlessness" but rather, a
far more commonsense definition is that infinity means the end of
Physics. Physics such as
renormalization in QED, has the burden of getting rid of infinities
and that is the situation throughout physics as to get rid of
infinities.

And the sphere versus pseudosphere is the geometry analogy of finite
number versus infinite
number in algebra.

So, I am glad I finally performed that experiment to know visually
what happens.

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

Archimedes Plutonium wrote:
Alright, it is about time I performed this experiment. I did the 2D
experiment a long time ago
since it is easy to cut out a circle inside a square and rearrange the
4 residue triangles that
forms a 4 pointed star. But now I need to do the 3D experiment.

To do this I used the end of a toothpaste box and cut out a cube
looking portion. Of course it
had just one end of the cube since I do not need the full cube but
only 1/2 cube to visualize.
I thence cut the cube in 1/2 and that portion I cut into 4 pieces. Now
each of those 4 pieces has 3 tip ends and it is those tip ends that
the sphere is tangent.


Now I may run into trouble here visualizing this. So I need to do this
experiment
more refined.



For the sphere enclosed inside the cube, I did not have available. So
I got a firm hard orange
and with a sharp knife I cut the orange into the 8 equal pieces. Each
piece is a elliptic geometry triangle. So that 4 will rest on the top
4 cube cutaways and the other 4 resting
below in the 4 cube cutaways.


I need to see how the edges of those sphere cutaway rests in the cube
cutaways.


Now I needed to perform this experiment because I can no longer
accurately visualize it in
my mind's eye.

Now some are going to say that this figure is in no way a
pseudosphere. I am going to
call it, or name it the "circular pseudosphere" because the curves are
not logarithmic as
the true pseudosphere is, but instead are circular curves derived from
the sphere itself.

Mathematics probably has a name for this figure, but I am unaware of
what it is called.

I see it as a Circular Pseudosphere with spines on it, as there are 4
spines. And the equator
is not a circular disc as in the true pseudosphere but is a
discontinuous region connected by
4 points of the top layer with 4 points of the bottom layer.

Now why do I bother with this figure? It is important because it is a
primitive pseudosphere
on the way to becoming a full fledged pseudosphere and its main
feature that is desirous is
the fact that the surface area of this primitive pseudosphere matches
the surface area of
the sphere itself. All that needs to be done to transform this
primitive pseudosphere is to
make the curves logarithmic instead of circular. And this primitive
pseudosphere fits totally
inside the sphere itself from which it sprang from.


I probably am wrong with that statement of fitting inside. I know in
the
2D experiment of cutting out the circle inside the square that the
residue
totally fits inside the circle, but here in 3D that is likely not to
be the case.

Can I cut the cube into 8 equal pieces and some how rearrange them to
fit inside the sphere? The answer is no. But the question for whether
I can
cut the sphere into 8 equal triangles and then invert them and to see
if they
fit inside the sphere is perhaps a whole another question.

And I am not sure in my mind's eye whether that is possible or
impossible.
If the best rearrangement of the inverted sphere is for the 3 points
of the
1/8 cubelets matches the 3 points of the 1/8 spherelets, then I cannot
pack
the inverted sphere inside the original sphere. And I suspect that is
the answer
but I need to verify with a hand held model.



Now the mission project of this experiment is that in the course of
transforming this circular
pseudosphere into a logarithmic pseudosphere that the radius of 1 or 2
or 3 etc etc are not
special, but at 10^500 this transformation is very special.

It is at the number 10^500 radius that the transformation of the
circular pseudosphere into
a full fledged logarithmic pseudosphere that the sphere of radius
10^500 is able to be contained inside the hollow of the 10^500
logarithmic pseudosphere.

And this is where the natural boundary between Finite Numbers versus
Infinite Numbers
occurs in mathematics. It exists in the fabric of mathematics itself
and is not artificially imposed. It is where the pseudosphere no
longer nests inside the associated sphere of
equal radius but where the sphere nests inside the associated
pseudosphere of equal
radius. It happens not at 10^499 but uniquely starts to happen at
10^500 and continues
to nest inside thereafter.

Why does this crossing of embeddedness happen? And the answer is that
the surface area
of the pseudosphere has to be equal to the surface area of the
associated sphere. This stretch area of the pseudosphere is collected
and patched onto the pseudosphere cutaway,
and the reason the Circular Pseudosphere has holes and gaps in it. So
as this area is patched
onto the cutaway pseudosphere and because this area increases as the
radius increases (see
the slant height of the cone reference), since this patch area
increases with the radius
increase, means there is a cross-over at some large number radius
where the sphere fits inside the hollow of the logarithmic
pseudosphere cutaway. I conjecture this radius number
is 10^500 or thereabouts because in Physics the end of the
StrongNuclear force is
253! for element of atomic number 100, which is 10^500 Coulomb
Interactions. The end of Physics is the end of mathematics as well, or
the start of infinity.


And also, in making the pseudosphere of logarithmic curve, probably
some
added area is needed to make it so that no gaps or holes except for
the cutaway
poles, exists.



In New-Mathematics, the better definition of Infinity is not that of
"endlessness" but rather, a
far more commonsense definition is that infinity means the end of
Physics. Physics such as
renormalization in QED, has the burden of getting rid of infinities
and that is the situation throughout physics as to get rid of
infinities.

And the sphere versus pseudosphere is the geometry analogy of finite
number versus infinite
number in algebra.

So, I am glad I finally performed that experiment to know visually
what happens.


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