Gray, Weisstein et al; edited the Wikipedia page on Pseudosphere to read infinite volume #1025 Correcting Math
Archimedes Plutonium
--- quoting
http://en.wikipedia.org/wiki/Pseudosphere ---
Tractricoid
The term is also used to refer to a certain surface called the
tractricoid: the result of revolving a tractrix about its asymptote.
It is a singular space (the equator is a singularity), but away from
the singularities, it has constant negative Gaussian curvature and
therefore is locally isometric to a hyperbolic plane.
The name "pseudosphere" comes about because it is a two-dimensional
surface of constant negative curvature just like a sphere with
positive Gauss curvature. Just as the sphere has at every point a
positively curved geometry of a dome the whole pseudosphere has at
every point the negatively curved geometry of a saddle.
Both its surface area and volume are infinite, on account of the
infinite extent of the shape along the axis of rotation. All
asymptotic graphs have infinite area under the curve. Oresme's proof
that the Harmonic Series 1/k goes to infinity is an elegant proof that
a concatenation of truncated cones fits into the pseudosphere poles
and thus the volume and surface area of pseudosphere are infinite.
(Proof given by Archimedes Plutonium in sci.math #1024 Correcting Math
post of 29Sept10).
--- end quoting http://en.wikipedia.org/wiki/Pseudosphere
I do not expect it to be there long, before another editor reverts it.
But the complaint filed is archived so that future
readers can see where someone tried to fix the false information given
by both Wikipedia and MathWorld on the subject of pseudosphere.
If you read MathWorld's account on pseudosphere, is also
highly ambiguous where in one part they say 1/2 volume and another
part they say equal volume. And unclear whether MathWorld is offering
a proof of the volume and area of pseudosphere, or whether MathWorld
is offering what Gray did in 2006 as a boundary conditions where the
pseudosphere can be, but not necessarily the full volume
in several parametrizations.
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
Dirk Van de moortel
9d56c6e8-6589-42bd...@i13g2000yqd.googlegroups.com
You get finite volume and area when you make the curve revolve about
its assymptote. When you make it revolve about the other axis, you
get infinite volume and area.
See the talk page of the wiki-article at
http://en.wikipedia.org/wiki/Talk:Pseudosphere
Dirk Vdm
Archimedes Plutonium
I am revolving it around the axis of the pseudosphere poles.
First, let me ask you about the history of the pseudosphere volume and
surface area.
Who if anyone, had proven what the volume and surface area of the
pseudosphere was?
Was it Beltrami?
Secondly, how do you explain your paradoxical result of finite volume
and finite surface area
when the Funnel is infinite volume and infinite surface area? Keeping
in mind that the Funnel
can be made as small and thin and tiny as you wish yet still be
infinite and yours pseudosphere claim can be made as large as you wish
and still be absurdly finite. So how do you explain that Dirk? The
only explanation that is feasible is that you made a big mistake.
And thank goodness that the person who wrote the Funnel entry of
MathWorld was not the same person that wrote the pseudosphere entry
into MathWorld.
I cannot spot your Calculus error at the moment and it must be some
assumption on your part,
making you imagine that it is finite.
So, please tell about the history of the pursuit of volume and area of
pseudosphere, was it Beltrami who started by using cylinders and the
convergent Series? Or is the volume and area
a completely recent or 20th century pursuit without Beltrami?
Hopefully some others in differential geometry will weigh in and show
you your mistake, for how in the world can you ever reconcile a Funnel
with a Pseudosphere, which a pseudosphere is what-- two funnels put
together.
A overall Theme: in math, when something feels funny, looks awry, such
as the precaution of
infinite stretch, then a mathematician ought to doubly look at and
reconsider a finite conclusion.
Also, I have posted a conjecture that basically says a asymptote
revolved on any axis always yields infinite volume and surface area.
So when that becomes a theorem of mathematics, I would not be arguing
with Dirk, but insisting that he find out the errors of his
computations.
Archimedes Plutonium
Well, perhaps indirectly I spotted Dirk's mistake in finite
conclusion. He is using Maple
and then when he has Maple do the calculation's, Maple assumes the
volume for the
enclosing sphere over the pseudosphere which gives a finite volume and
finite surface
area as only that portion inside the sphere of same radius.
But what I am insisting that Dirk and Weisstein and others do, is
consider that the Funnel
is infinite in volume and area, which immediately implies the
Pseudosphere has to be
infinite in volume and area.
--- quoting what Dirk and I said to the Talk portion of Wikipedia on
pseudosphere and
which Dirk referred to in his prior post ---
Wikipedia and MathWorld are correct.
Take a tractrix with parametric equation (see for example this, 102b,
where R=12 and where I have swapped x(t) and y(t)), defined for t in
[ 0, +inf [
Make it revolve around the assymptote, which is the x-axis and
calculate Ax and Vx in Solid_of_revolution#Parametric_form (note that
I just added these sourced expressions to that article, and note that
I doubled both integrals for symmetry)
with which my version of Maple 13 produces the results of the article:
You probably calculated the volume and the area when you make the
thing revolve around the y-axis (i.e. not about the assymptote). In
that case indeed you get infinity for both volume and area. You just
made a silly mistake. DVdm (talk) 21:15, 30 September 2010 (UTC)
Well, honestly, I have not spotted your error in Calculus, perhaps
your use of the computer in the derivation is such that it only
computes the portion of the pseudosphere enclosed in the sphere of
same radius and does not include any of the extensions of the two
poles of the pseudosphere, so then anyone can see that volume is 1/2
of sphere and area is same as sphere. So your calculus is only
including a cutaway portion of the pseudosphere.
And be thankful that you were not the writer of the MathWorld funnel
entry, where the correct conclusion was infinite volume and infinite
area. So Dirk has to answer how in the world can the fattest largest
pseudosphere have finite volume and area when the tiniest and thinnest
of funnels can be packed inside that psuedosphere which has infinite
volume and area. Is Dirk's face turning red?
Here is that website of MathWorld:
--- quoting MathWorld on funnel ---
http://mathworld.wolfram.com/Funnel.html
The Gaussian curvature can be given implicitly as
(14)
Both the surface area and volume of the solid are infinite.
--- end quoting MathWorld on funnel ---
So, Dirk, can you assist the people at MathWorld in correcting their
error filled pseudosphere page?
216.16.54.108 (talk) 23:56, 30 September 2010 (UTC)Archimedes
Plutonium
--- end quoting of what Dirk and I said in the Talk portion of
Wikipedia that Dirk
referred to in previous post ---
I repeat my question, anyone know of a history of the volume and area
of pseudosphere
calculations? Because I find it difficult to believe that noone from
Beltrami until
recently worked out the volume and area. I find it unpalatable to
accept that Dirk was
the first to "offer an opinion of the volume and area of
pseudosphere."
Archimedes Plutonium
sci.math, sci.physics, sci.logic, sci.edu
Sep 29, 5:58 pm
Date: Sep 29, 2010 6:58 PM
Author: plutonium....@gmail.com
Subject: will Wikipedia correct its false pseudosphere claims?? Will
MathWorld? #1026 Correcting Math
http://en.wikipedia.org/wiki/Pseudosphere#Tractricoid
Tractricoid
The term is also used to refer to a certain surface called the
tractricoid: the result of revolving a tractrix about its asymptote.
It is a singular space (the equator is a singularity), but away from
the singularities, it has constant negative Gaussian curvature and
therefore is locally isometric to a hyperbolic plane.
The name "pseudosphere" comes about because it is a two-dimensional
surface of constant negative curvature just like a sphere with
positive Gauss curvature. Just as the sphere has at every point a
positively curved geometry of a dome the whole pseudosphere has at
every point the negatively curved geometry of a saddle.
No proof has ever been given as to the volume and surface area of
pseudosphere that is citable as a source of reference. What
MathWorld
talks of volume and surface area are only parametrics on the
pseudosphere, not a proof, and even that is garbled where MathWorld
says it is 1/2 sphere volume and later says it is 1 sphere volume.
So
Wikipedia should do the prudent thing, not say anything until a
reliable reference is cited. And it is obvious that noone in math
has
to prove that a funnel that goes to infinity has infinite volume and
infinite area. And since the funnel can be made to fit inside the
pseudosphere hemispheres that the pseudosphere is infinite volume
and
area.
So is Wikipedia all about sources citing and forget the truth or
falsity of what they print, or is Wikipedia focused more on truth
and
falsity, than sources?
http://en.wikipedia.org/wiki/Talk:Pseudosphere
Talk:Pseudosphere
From Wikipedia, the free encyclopedia
[edit] incorrect volume and surface area, from a mistaken source
Currently Wikipedia has a falsehood about the area and volume of a
pseudosphere. It was cobbed from the MathWorld website where noone
refers to a proof of volume nor area. Gray is referenced but Gray's
analysis is only a parametric on the surface area and volume.
What is needed is an actual proof of surface area and volume and not
a
reciting source like MathWorld. It is likely that noone has ever
proven what the volume and surface area of a pseudosphere is.
Until a proof is cited, Wikipedia, in prudence should not list any
volume and area result, especially the ambiguous MathWorld reference
wherein one sentence says it is 1/2 sphere and another says it is
equal volume to the sphere.
Also, what is "edge radius" in the Wikipedia entry?
Usually the editors of Wikipedia act and behave like automatons with
rules for their convenience, and their obnoxious behaviour of
placing
"sources" above truth content. The obnoxious behaviour of valuing a
"source" as higher value, than the question of whether what is
printed
by Wikipedia is true or false.
Even a High School student has the commonsense that a funnel, no
matter how thin, how tiny the cone, that if the end of the funnel
goes
to infinity that the volume and area of the funnel is infinite. Does
there need to be a proof that any funnel, when stretching to
infinity
has infinite volume, and thus a psuedosphere is **two curved funnels
set back to back** and each of these funnels of a pseudosphere since
they stretch to infinity, hence have infinite volume and infinite
area.
So can Wikipedia use a average commonsense High School student who
knows that a funnel that is infinitely long has infinite volume and
hence the Pseudosphere has infinite volume.
Can we use a High School kid as a source for volume of Pseudosphere,
rather than Weisstein's MathWorld which lacks commonsense for their
synopsis of Pseudosphere volume?
I mean, honestly, is Wikipedia stuck more on references rather than
honest to goodness truth?
Archimedes Plutonium, who published about half a dozen proofs that
the
pseudosphere has infinite volume and surface area in sci.math.
So is Wikipedia, lazier or less lazy as MathWorld in correcting its
false entries??
216.16.54.214 (talk) 22:38, 29 September 2010 (UTC) Archimedes
Plutonium, who hates to see false encyclopedia claims over
pseudosphere volume.
Is "edge radius" a deception foisted by Dirk and Wikipedia?
Why cannot Weisstein and MathWorld recognize their page on Funnel
contradicts their page on pseudosphere volume and area??
--- quoting MathWorld on funnel ---
http://mathworld.wolfram.com/Funnel.html
The Gaussian curvature can be given implicitly as
(14)
Both the surface area and volume of the solid are infinite.
--- end quoting MathWorld on funnel ---
How many days of countdown before Wikipedia and MathWorld correct
their
huge blunder?
This is 3rd Day Countdown, and let us see how long it takes them to
correct their
mistake that misleads those in math education.
Dirk Van de moortel
See below.
>>
>> Secondly, how do you explain your paradoxical result of finite volume
>> and finite surface area
>> when the Funnel is infinite volume and infinite surface area? Keeping
>> in mind that the Funnel
>> can be made as small and thin and tiny as you wish yet still be
>> infinite and yours pseudosphere claim can be made as large as you wish
>> and still be absurdly finite. So how do you explain that Dirk? The
>> only explanation that is feasible is that you made a big mistake.
Or that you confuse a the tractrix with the curve that produces
Mathworld's funnel.
See below.
[snip part that be read at http://en.wikipedia.org/wiki/Talk:Pseudosphere ]
> Wikipedia that Dirk
> referred to in previous post ---
>
> I repeat my question, anyone know of a history of the volume and area
> of pseudosphere
> calculations? Because I find it difficult to believe that noone from
> Beltrami until
> recently worked out the volume and area. I find it unpalatable to
> accept that Dirk was
> the first to "offer an opinion of the volume and area of
> 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
By the way, http://books.google.be/books?id=kJa15IMxAoIC&pg=PA324
tells that Huygens knew in 1693 that the pseudosphere has finite area
and volume. That as a few centuries before Beltrami.
I can't put the remainder in Wiki since it is off-topic there.
Take this halve funnel producing curve
y = sqrt( exp( -2 x/R ) ) with R > 0 and x in [ 0, +inf [
which should correspond to Mathworld's funnel
z = 1/2 r ln(x^2+y^2)
( http://mathworld.wolfram.com/Funnel.html )
For R >= 1 it lies within the R-pseudosphere.
For R < 1 it intersects it.
This curve is clearly not a tractrix, which is needed to produce
a pseudosphere.
Rotating the funnel about the x-axis and doubling produces
Volume = pi R (try it - it is trivial)
Aera = 2 pi ( sqrt(R^2+1) + R^2 arctanh(1/sqrt(R^2+1)) )
which are clearly far from infinity.
It all depends on the curve.
It looks like Mathworld is wrong about the infinities for the funnel.
Mathworld has many mistakes
Try emailing Eric Weisstein about this error.
I did that on 2 different occasions. He never replied.
Dirk Vdm
Dirk Van de moortel
5prpo.14516$Fk6....@newsfe13.ams2
> By the way, http://books.google.be/books?id=kJa15IMxAoIC&pg=PA324
> tells that Huygens knew in 1693 that the pseudosphere has finite area
> and volume. That as a few centuries before Beltrami.
>
> I can't put the remainder in Wiki since it is off-topic there.
>
> Take this halve funnel producing curve
> y = sqrt( exp( -2 x/R ) ) with R > 0 and x in [ 0, +inf [
Duh, silly me.
Make that just
y = exp( -x/R )
of course :-)
Dirk Vdm
Archimedes Plutonium
Dirk Van de moortel wrote:
(snipped)
>
> By the way, http://books.google.be/books?id=kJa15IMxAoIC&pg=PA324
> tells that Huygens knew in 1693 that the pseudosphere has finite area
> and volume. That as a few centuries before Beltrami.
>
Mind providing a synopsis of what Huygens did to convince himself of
finite volume and finite area
AP
Dirk Van de moortel
Perhaps, see http://books.google.be/books?id=ogz5FjmiqlQC&pg=PA314
[ with area (2 pi) of half a pseudosphere of radius 1, consistent with the
(4 pi R^2) value ]
There's some synopsis at the bottom of the page. If that's not what you
need, you'll have to hit the libary and search for Huygens' work.
Dirk Vdm
Archimedes Plutonium
Tractricoid
The term is also used to refer to a certain surface called the
tractricoid: the result of revolving a tractrix about its asymptote.
It is a singular space (the equator is a singularity), but away from
the singularities, it has constant negative Gaussian curvature and
therefore is locally isometric to a hyperbolic plane.
The name "pseudosphere" comes about because it is a two-dimensional
surface of constant negative curvature just like a sphere with
positive Gauss curvature. Just as the sphere has at every point a
positively curved geometry of a dome the whole pseudosphere has at
every point the negatively curved geometry of a saddle.
As early as 1639 Christian Huygens reported that the volume and the
surface area of the pseudosphere are finite, and although no proof was
ever given, and what ensued was a mistaken belief that the
pseudosphere was finite in volume and surface area. The proof that the
pseudosphere is infinite in both volume and area was given when it is
noted that the Funnel is infinite in volume and area and hence the
pseudosphere is two funnels put together.Cite error: Invalid <ref>
tag; refs with no name must have content; see the help page[2] </
ref>[3]
--- end quoting my edit of Wikipedia ---
Sorry about that for the reference to MathWorld is really this:
http://mathworld.wolfram.com/Funnel.html
Where MathWorld got that correct by saying a Funnel is infinite in
area and volume and anyone, even a teenager can recognize that a
pseudosphere is two funnels put together.
For some reason, Dirk and the editors of Wikipedia and the author of
the MathWorld pseudosphere are totally blind when it comes to
recognizing that a pseudosphere is two
funnels.
Can someone inform Eric Weisstein that a pseudosphere is two funnels
and to correct his
MathWorld error? Perhaps the error is because Dirk is the only
mistaken chap who is vigorously promoting error clad mathematics.
AP