Talk:Pseudosphere

image
I think that an image of a pseudosphere would be desirable as the first image on this page. I had to go to tractrix to visualize it myself. McKay 05:06, 12 October 2006 (UTC)

Removed paragraph
I removed this paragraph, which has been in the article since 2005:


 * It also denotes the entire set of points of an infinite hyperbolic space which is one of the three models of Riemannian geometry. This can be viewed as the assemblage of continuous saddle shapes to infinity. The further outward from the symmetry axis, the more increasingly ruffled the manifold becomes. This makes it very hard to represent a pseudosphere in the Euclidean space of drawings. A trick mathematicians have come up with to represent it is called the Poincaré model of hyperbolic geometry. By increasingly shrinking the pseudosphere as it goes further out towards the cuspidal edge, it will fit into a circle, called the Poincaré disk; with the "edge" representing infinity. This is usually tessellated with equilateral triangles, or other polygons which become increasingly distorted towards the edges, such that some vertices are shared by more polygons than is normal under Euclidean geometry. (In normal flat space only six equilateral triangles, for instance, can share a vertex but on the Poincaré disk, some points can share eight triangles as the total of the angles in a narrow triangle of geodesic arcs is now less than 180°). Reverting the triangles back to their normal shape yields various bent sections of the pseudosphere. While smaller local sections will stretch out to saddle shapes, large sections that extend to the infinite edge, are illustrated in their expanded form by being bent until their opposite sides are joined, yielding the aforementioned "tractricoid" shape, which is also called a "Gabriel's Horn" (since it resembles a horn with the mouthpiece lying at infinity). Thus the tractricoid is really only a part of the whole pseudosphere. At any point the product of two principal radii of curvature is constant. Along lines of zero normal curvature geodesic torsion is constant by virtue of Beltrami-Enneper theorem.

Almost everything in this paragraph seems to be incorrect, confusing or of questionable relevance. -- BenRG (talk) 23:24, 26 April 2009 (UTC)
 * "It also denotes the entire set of points of an infinite hyperbolic space" – I can't remember ever seeing "pseudosphere" used to refer to the whole hyperbolic plane; at the very least it's unusual. The tractricoid has finite area. The universal cover of the tractricoid (where you reinterpret the circular cross sections as "spirals of constant radius") has infinite area but still doesn't include the whole hyperbolic plane.
 * "which is one of the three models of Riemannian geometry" – Riemannian geometry is not the geometry of surfaces of constant Gaussian curvature.
 * "This can be viewed as the assemblage of continuous saddle shapes to infinity." – maybe; is this helpful? The tractricoid doesn't look like a bunch of saddles.
 * "The further outward from the symmetry axis, the more increasingly ruffled the manifold becomes." – obviously not true of the tractricoid.
 * The Poincaré model is not an isometric embedding and has no apparent relevance to this article.
 * "By increasingly shrinking the pseudosphere as it goes further out towards the cuspidal edge, it will fit into ... the Poincaré disk" – well, you can map the tractricoid into the Poincaré disc of course (as shown here) but not onto.
 * The stuff about triangles doesn't seem relevant and seems mostly incorrect/confused anyway.
 * Gabriel's horn is not a tractricoid.
 * "At any point the product of two principal radii of curvature is constant" – true, but that's just saying that it has constant Gaussian curvature, which has already been mentioned.
 * The final sentence may be correct, I don't know.


 * Ben, can you tell who added that worthless garbled mess that should have been removed in 2005 when first installed? Can you tell who it was that included that mess? Because, if it was DirkVdm, the impression or opinion that DVdm leaves with me is one of a prankster, who puts up Wikipedia math pages just to get laughs out of how far one can enter a page of false mathematics. So that in my opinion, DVdm is a prankster and not anything of serious and truthful mathematics.
 * So the question is, does Wikipedia have some sort of rules that you guys can distinguish pranksters as editors and get their editing privileges removed. 216.254.227.56 (talk) 19:48, 3 October 2010 (UTC)Archimedes Plutonium


 * Firstly I would like to drawn your attention to No personal attacks your comments towards DVdm are counter to that policy, please try and approach wikipedia with a less confrontational attitude. This is not usenet and we have a strong Civility policy. Breaches of that policy can lead to blocks or bans.
 * No it was not DVdm who added that material, you can use the article history to find out who did which edit, the actual editor is no longer active on wikipedia.--Salix (talk): 21:44, 3 October 2010 (UTC)


 * I'm the one who added that paragraph, and put it back in 2007. (Though some things were changed; I didn't say "assemblage" of saddle shapes. Also the last two sentences I didn't write). If it was 2005 when I first put it up, it may have been when I just began contributing and before I registered, so would be under an AOL IP#.
 * The information was taken straight from Rudy Rucker's The Fourth Dimension, and can be seen in the Non-Euclid link, which takes directly from the book, even down to the images. I may not have cited the book, but if I didn't; then that was because I was new and didn't know the ropes yet. (And perhaps because the Non-Euclid site was used as the source).
 * So I believe Rucker is a reliable source, and he suggested that calling a tractricoid the whole pseudosphere was in error. Pseudosphere is the name of the whole object, having constant negative curvature; not just what is essentially a portion of it, created by the Poincaré method of shrinking the surface to fit in a finite space. (He shows how the tractricoid is formed by shrinking a segment of the "infinite" extension, and then joining opposite edges of the cut-out. This was also deleted from the article. How could all of this be omitted when the Poincaré disk is the next most common illustration of the pseudosphere besides the tractricoid?)
 * And the tractricoid appears to be itself derived from the Gabriel's Horn, so that is why Rucker treats it as only part of the pseudosphere.
 * Though it is true that he (and those derived from him, such as Non-Euclid) seems to be the only one saying this, while everyone else defines it as a tractricoid.
 * However, one Scientific American article also described a "hyperbolic manifold" in which every point had the geometry of a saddle; though it didn't call it a pseudosphere. http://cosmos.phy.tufts.edu/~zirbel/ast21/sciam/IsSpaceFinite.pdf scroll down to p. 5 (I actually came here tonight thinking I had added a link to the article online, and was looking for it; but I guess I never did. Now I just found it again on Google instead).
 * Also, "the Riemann models of the universe" are always held to be the flat plane, the sphere and the saddle, representing negative space. Rucker holds the pseudosphere as being the extension of the saddle to infinity (that was what I originally put, rather than "assemblage").
 * So I guess the debate here is over the validity of his claim. If he's held to be a valid source, I say it should be put back. Eric B (talk) 02:15, 11 November 2010 (UTC)

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.


 * 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 [
 * $$x(t) = t- R \, \tanh{ \frac{t}{R} }$$
 * $$y(t) = R \, \operatorname{sech} { \frac{t}{R} }$$
 * Make it revolve around the assymptote, which is the x-axis and calculate $$A_x$$ and $$V_x$$ in Solid_of_revolution (note that I just added these sourced expressions to that article, and note that I doubled both integrals for symmetry)
 * $$V_{x} = 2 \, \int_0^\infty \, \pi \, y^2 \, \frac{dx}{dt} \, dt$$
 * $$A_{x} = 2 \, \int_0^\infty 2 \pi y \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt,$$
 * with which my version of Maple 13 produces the results of the article:
 * $$V_{x} = \frac{2}3{} \pi R^3 \,$$
 * $$A_{x} = 4 \pi R^2 \, $$
 * 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)


 * For what it's worth, the arguments made by Archimedes Plutonium (posting from IP 216.16.54.*) can be found on the sci.math newsgroup here, here, and here. Even if any of it were correct, it would probably still be considered original research. — Loadmaster (talk) 04:11, 1 October 2010 (UTC)


 * Indeed. And the comments below are off-topic on this article's talk page, so I have collapsed them. DVdm (talk) 08:55, 1 October 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


 * Can someone help in assisting the insertion of the reference


 * http://mathworld.wolfram.com/Funnel.html


 * Help in assisting that reference to MathWorld be inserted since the **TRUE volume and area** of pseudosphere is infinite, and for Wikipedia and MathWorld to stop spreading math falsehoods. 216.16.56.151 (talk) 20:12, 2 October 2010 (UTC)Archimedes Plutonium —Preceding unsigned comment added by 216.16.54.168 (talk)
 * Note the formula for Gaussian curvature of the Funnel, this depends on the parameter and hence is non-constant. Definition of a pseudosphere is that the Gaussian curvature is a negative constant.--Salix (talk): 04:34, 3 October 2010 (UTC)


 * The halve pseudosphere is the surface of revolution of the tractrix with parametric equation
 * $$( x(t), y(t) ) = \left( t - R \, \tanh{ \frac{t}{R} }, R \, \operatorname{sech}\,{\frac{t}{R}} \right), \quad \quad 0 <= t < \infty$$
 * whereas MathWorld's halve "funnel" is obtained by revolving the curve parametrized by
 * $$( x(t), y(t) ) = \left( t, e^{ -\frac{t}{R} } \right), \quad \quad 0 <= t < \infty$$
 * which also produces finite area and volume. MathWorld is wrong about that, but that is not our problem, since we do not mention any of this in.
 * So, wp:TPG this part of the discussion is off-topic here and please stop repeating this here. See 4th level warnings on your user talk page(s) User talk:216.16.56.151 DVdm (talk) 09:42, 3 October 2010 (UTC)

Pseudospherical surface redirects wrongly to this page
Pseudospherical surface redirects to this page not all Pseudospherical surfaces are pseudospheres (i know it is hard to get your head around)

But there are many more "Pseudospherical surfaces" or "surfaces of constant negative curvature": (see for example http://virtualmathmuseum.org/Surface/gallery_o.html#PseudosphericalSurfaces )

Three surfaces of revolution:

Pseudospherical Surfaces of the parabolic type (this is the tracioid or pseudosphere)

Pseudospherical Surfaces of the hyperbolic type

Pseudospherical Surfaces of the elliptic type (called the conic type at the virtual math museum)

and also other surfaces that are not surfaces of revolution what to do now? (PS I am not an expert in this field, nor on wikipedia) (I made a similar comment on https://en.wikipedia.org/wiki/Talk:Sine-Gordon_equation#Surfaces_of_constant_negative_curvature_not_always_a_pseudosphere i am still a beginner sorry) WillemienH (talk) 10:19, 16 September 2014 (UTC)

Multiple major errors
The first sentence, which is the entire introductory section, reads as follows:

"In geometry, the term pseudosphere is used to describe various surfaces with constant negative Gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid."

The first named section is titled Theoretical pseudosphere, and its first sentence is as follows:

"In its general interpretation, a pseudosphere of radius R is any surface of curvature −1/R2 (precisely, a complete, simply connected surface of that curvature), by analogy with the sphere of radius R, which is a surface of curvature 1/R2. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.[1]"

These sentences contain a number of errors.

1. The word "pseudosphere" has no "general interpretation". It means a specific surface in 3-dimensional space. (In fact, it refers to case of the tractricoid of constant negative curvature -1, not just any negative number −1/R2.)

2. First of all, the pseudosphere is a surface, not a "theoretical" surface. It is not at all clear why the word "theoretical" was used, but it is just confusing here, since it has no meaning.

3. The pseudosphere is a specific surface, not maybe one surface and maybe another one. It is emphatically not a hyperboloid, and any use of the word to mean a hyperboloid is a mistake.

This word should not be used in either the first sentence or in the first (or any!) section title.

The phrase "precisely, a complete, simply connected surface of that curvature" contains three errors.

4. The pseudosphere is not complete. Complete means that every geodesic can be extended indefinitely, but the pseudosphere has a circle of singularities (the circle in its plane of symmetry) past which geodesics cannot be extended.

5. The pseudosphere is not simply connected. For a space to be simply connected means that every closed curve in the space can be continuously shrunk to a point in the space. But any closed curve on the pseudosphere that goes around its axis of revolution cannot be shrunk to a point on the surface.

(6. Any explanatory statement beginning with the word "precisely" but which contains nothing but errors has also made an error by using the word "precisely".)

I hope someone will fix this article.Daqu (talk) 19:02, 16 June 2015 (UTC)

Problematic illustrations
The second illustration on the page has a caption that reads:

"The pseudosphere and its relation to three other models of hyperbolic geometry"

Nope, wrong. The illustration does show the pseudosphere and three other models of hyperbolic geometry.

But it most certainly does not clarify the relation of the the pseudosphere to any of those models. 2601:200:C000:1A0:687A:28A3:D31E:9855 (talk) 05:04, 5 May 2021 (UTC)


 * I think it does a pretty good job, but the linked interactive version at https://timhutton.github.io/PseudosphereGeodesics/ is better still. –jacobolus (t) 22:58, 22 August 2023 (UTC)

The illustration whose caption is "The pseudosphere and its relation to three other models of hyperbolic geometry" is terminally confusing with all the unexplained lines and curves in the pictures.

Much better would be to limit the picture to the most common model, the Poincaré disk model, and to limit the curves to only one or two that are fully explained in the caption.

— Preceding unsigned comment added by 2601:200:c082:2ea0:79bd:1116:def:265c (talk) 18:17, 22 August 2023 (UTC)


 * I disagree with your characterization and criticism. The current image showing the relation to multiple models was very helpful to me. But as always on Wikipedia, feel free to make a better one. –jacobolus (t) 22:56, 22 August 2023 (UTC)

One illustration has the caption "Deforming the pseudosphere to Dini's surface. In differential geometry, this is a Lie transformation. In the corresponding solutions to the Sine-Gordon equation, this deformation corresponds to a Lorentz Boost of the static 1-soliton solution"

But nobody knows what a "Lie transformation" means, and there is no link or explanation of what it means (and no article by that name in Wikipedia).

Furthermore, the illustration, which depicts the evolution of the pseudosphere to Dini's surface, only goes as far as a small portion of Dini's surface. So the caption is simply wrong as stated.

(Also, the caption's second sentence is missing a period.)

— Preceding unsigned comment added by 2601:200:c082:2ea0:79bd:1116:def:265c (talk) 18:17, 22 August 2023 (UTC)


 * Hi, I created and uploaded the animation and wrote the caption. Thank you for pointing out these errors, I hope the newer version is satisfactory for you.
 * By the way, you can also make corrections like these yourself. This comment is rather harshly worded and I just wanted to inform you that my intention is to help, not to confuse.
 * Zephyr the west wind (talk) 22:34, 22 August 2023 (UTC)