Who proved pseudosphere has what volume? Oresme or Gray or Weisstein? #1024 Correcting Math
Archimedes Plutonium
Now looking at this website of MathWorld:
http://mathworld.wolfram.com/HarmonicSeries.html
We see how Oresme proved that the Series 1/n diverges by means of
grouping the terms
as in
1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) +. . .
So that each successive grouping is at least 0.5 or greater. That is
the whole
of the proof.
Now let us do the same for the pole of a pseudosphere that comes
poking out of the
enclosing sphere with its exit-circle. Let us mold a truncated cone
whose volume is
1 and will fit like a bathtub plug into the exit circle hole of the
pseudosphere pole and
the fit is only tight at the narrow end so that there is some gaps
along the side of the
plug for it is not a contiguous intersection of truncated cone with
the sides of the
pseudosphere pole. Now we mold another rubber truncated cone with
volume 1/2
and whose narrowest end is contiguous with the next stretch of the
pole. We thence
have begun a concatenation of truncated cones to fit inside the
pseudosphere pole.
Now we mold two more bathtub plugs whose volume is 1/3 and 1/4 and
fit
them after
the 1/2 plug. And there is nothing to stop us from doing this to
infinity. So the volume
of the pseudosphere pole is infinite volume.
So one really has to wonder how in the world anyone came up with a
cylinder modelling
that yielded a finite volume using the convergent Series 1 + 1/2 +
1/4
+ . . . It sort of reminds me of Statistics work, where field
researchers go out and look for what they would like the
conclusions to be and ignore anything that is opposed to what their
end conclusion is wanted to be. Sure we can take a pseudosphere pole
and mold cylinders of volumes 1 then 1/2, then
1/4 and say that they are upper bounds of the entire pole, and then
falsely conclude the pseudosphere is finite. Just like in the Hotel
Paradox of false accounting of $27 + $2 + where is the missing $1 to
make $30, whereas the true accounting is $25 +$3 + $2 = $30. So by
using the convergent Series for the pseudosphere is a false
accounting. Just like if someone told you to take a thermometer and
go
over there and measure the distance between A and B, and then take
the
Convergent Series and go and measure the volume of a pseudosphere.
So Oresme's proof that the harmonic Series 1/n is divergent is an
elegant proof. And so elegant that it easily is the same proof that
the pole of a pseudosphere is infinite in volume.
Now where does the math community get the idea that
the volume of pseudosphere is finite?
--- quoting in parts from MathWorld ---
http://mathworld.wolfram.com/Pseudosphere.html
(Gray et al. 2006, p. 477).
In the first parametrization, the coefficients of the first
fundamental form are
.
.
.
The pseudosphere therefore has the same volume as the sphere while
having constant negative Gaussian curvature (rather than the constant
positive curvature of the sphere), leading to the name "pseudo-
sphere."
--- end quoting in parts ---
Now, looking at Wikipedia,they quote Weisstein's MathWolrd
--- quoting Wikipedia ---
http://en.wikipedia.org/wiki/Pseudosphere
Both its surface area and volume are finite, despite the infinite
extent of the shape along the axis of rotation. For a given edge
radius R, the area is 4(pi)R^2 just as it is for the sphere, while the
volume is 2(pi)R^3/3 and therefore half that of a sphere of that
radius.[2]
2. Weisstein, Eric W., "Pseudosphere" from MathWorld.
--- end quoting Wikipedia ---
In Weisstein's MathWorld on pseudosphere, it says the volume is two
things, once it said 1/2 sphere and above it
says it is the sphere volume.
In Wikipedia it talks about "edge radius" whatever that means.
So who actually proved what the volume and surface area
of pseudosphere as talked about in MathWorld, was it Gray in 2006 or
is that Weisstein's own contribution to the
volume and area?
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