What Are the Limits of the Universe According to Cosmologists?

[fresh_42]

Wikipedia:

In the abstract interpretation, a pseudosphere of radius $R$ is a surface with constant Gaussian curvature $-\tfrac{1}{R^2}$ (more precisely, a complete, simply connected surface of this curvature), by analogy with a sphere of radius $R$, which is a surface with Gaussian curvature $\tfrac{1}{R^2}$.

The term was introduced in 1868 by Eugenio Beltrami in his work Models of Hyperbolic Geometry.

 

[Jaime Rudas]

What about this?
https://en.wikipedia.org/wiki/Pseudosphere

Yes, I think I was wrong. Although I don't know if it's possible to define curvature at the "edge" of the pseudosphere (i.e., at the points furthest from the axis of revolution).

 

[fresh_42]

Yes, I think I was wrong. Although I don't know if it's possible to define curvature at the "edge" of the pseudosphere (i.e., at the points furthest from the axis of revolution).

The German Wikipedia distinguishes under pseudosphere between hyperboloids, the tractrix (which you have in mind), and a theoretical surface with constant negative Gaussian curvature.

 

[Jaime Rudas]

The German Wikipedia distinguishes under pseudosphere between hyperboloids, the tractrix (which you have in mind), and a theoretical surface with constant negative Gaussian curvature.

The German Wikipedia says:

In der Differentialgeometrie wird der Begriff Pseudosphäre für verschiedene Flächen benutzt, die eine konstante negative Gaußkrümmung haben:

• ein Hyperboloid,
• ein Traktrikoid (die Drehfläche einer Traktrix) oder
• eine theoretische Oberfläche konstanter negativer Krümmung.

Which, translated, would be:

In differential geometry, the term pseudosphere is used for various surfaces that have a constant negative Gaussian curvature:

• a hyperboloid,
• a tractricoid (the surface of revolution of a tractrix), or
• a theoretical surface of constant negative curvature.

I understood that the hyperboloid didn't have constant Gaussian curvature. Am I also wrong about that?

 

[fresh_42]

The translation is correct.

I haven't calculated the curvature, but they claim it is constant and negative. I think this makes sense, as the Gaussian curvature only depends on the two radii of the principal curvatures, which should be constant: one from the rotation and the other from the hyperbola.

 

[pinball1970]

But completely understandable. Your English is excellent.

Better than mine.

 

[fresh_42]

Better than mine.

Too kind, but I doubt it. Except for those stupid mistakes and the - to me - untransparent comma rules, which are - to me - more logical in German ('and' replaces the comma in a list, commas in a sentence are used if they enclose a complete sentence with SPO), it is mainly the vocabulary I struggle with. Mine is significantly larger in German, and looking up the words doesn't help a lot since the dictionary doesn't tell me which one to use in what meaning.

 

[ShadowKraz]

Too kind, but I doubt it. Except for those stupid mistakes and the - to me - untransparent comma rules, which are - to me - more logical in German ('and' replaces the comma in a list, commas in a sentence are used if they enclose a complete sentence with SPO), it is mainly the vocabulary I struggle with. Mine is significantly larger in German, and looking up the words doesn't help a lot since the dictionary doesn't tell me which one to use in what meaning.

You are still doing better than most who speak English as their first language.

 

[PeterDonis]

I didn't use ChatGPT to give me the right answer, but rather to help me with certain steps to find it

ChatGPT is an unacceptable reference here no matter how you use it. If you use it to give you clues, you still need to check your work some other way--and the some other way is what you should reference here.

 

[PeterDonis]

Moderator's note: An off topic subthread spawned by a post about ChatGPT has been deleted. Please keep the discussion here on topic using acceptable references.

 

[Jaime Rudas]

I haven't calculated the curvature, but they claim it is constant and negative. I think this makes sense, as the Gaussian curvature only depends on the two radii of the principal curvatures, which should be constant: one from the rotation and the other from the hyperbola.

The Gaussian curvature $K$ of the hyperboloid $x^2+y^2-z^2=1$ is $\frac {-1}{(1+2z^2)^2}$. Thus, when $z=0$, the curvature $K=-1$ and when $z \neq 0$ the curvature $K$ remains negative, but greater than $-1$. It follows that the curvature of the hyperboloid is always negative, but not constant.

 

[fresh_42]

The Gaussian curvature $K$ of the hyperboloid $x^2+y^2-z^2=1$ is $\frac {-1}{(1+2z^2)^2}$. Thus, when $z=0$, the curvature $K=-1$ and when $z \neq 0$ the curvature $K$ remains negative, but greater than $-1$. It follows that the curvature of the hyperboloid is always negative, but not constant.

For those of you who like fancy pictures:
http://sodwana.uni-ak.ac.at/geom/mitarbeiter/odehnal/talk/innsbruck_2014.pdf

 

[renormalize]

For those of you who like fancy pictures:
http://sodwana.uni-ak.ac.at/geom/mitarbeiter/odehnal/talk/innsbruck_2014.pdf

Is this material available anywhere else? Regarding your link, my security software states:
"Your connection to this web page is not safe due to an unmatching security certificate."

 

[fresh_42]

Is this material available anywhere else? Regarding your link, my security software states:
"Your connection to this web page is not safe due to an unmatching security certificate."

This is indeed a bit strange. It is an Austrian address, and the pdf says about the author University of Applied Arts Vienna, more specifically: 16th International Conference on Geometry and Graphics Innsbruck, Austria, August 4 – 8, 2014.

I can see the pdf, but I also get a security alert on the main address. It is a pdf (1.89 Mb) so it should be safe. Here is the upload and an example image:

 

[martinbn]

There is a theorem, i think Hilbert, that says that a constant negative curvature surface cannot be embedded isometrically in three dimensional euclidean space.

 

[A.T.]

There is a theorem, i think Hilbert, that says that a constant negative curvature surface cannot be embedded isometrically in three dimensional euclidean space.

See this answer:
https://physics.stackexchange.com/a/43061

Theorem: There does not exist a smooth immersion of the hyperbolic plane into Euclidean 3 space.

... despite the above, it is possible to embed "patches" of hyperbolic plane into Euclidean 3 space.

.... one of the more well-known is the tractricoid.

 

[cianfa72]

There is a theorem, i think Hilbert, that says that a constant negative curvature surface cannot be embedded isometrically in three dimensional euclidean space.

Even though the hyperbolic plane can't be embedded isometrically in 3D Euclidean space, can it be isometrically embedded in three dimensional Minkowski space ?

 

[jbriggs444]

Is this material available anywhere else? Regarding your link, my security software states:
"Your connection to this web page is not safe due to an unmatching security certificate."

My training as a network engineer (retired) has been activated...

On the face of it, the URL (http://sodwana.uni-ak.ac.at/geom/mitarbeiter/odehnal/talk/innsbruck_2014.pdf) is for plaintext HTTP. And indeed, a plaintext HTTP request does result in the PDF being retrieved.

GET /geom/mitarbeiter/odehnal/talk/innsbruck_2014.pdf HTTP/1.1
Host: sodwana.uni-ak.ac.at
Connection: keep-alive
Upgrade-Insecure-Requests: 1
User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/140.0.0.0 Safari/537.36
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7
Accept-Encoding: gzip, deflate
Accept-Language: en-US,en;q=0.9

HTTP/1.1 200 OK
Server: nginx
Date: Sun, 21 Sep 2025 21:13:13 GMT
Content-Type: application/pdf
Content-Length: 1987963
Connection: keep-alive
Keep-Alive: timeout=20
Last-Modified: Tue, 28 May 2019 09:08:26 GMT
ETag: "1e557b-589ef01ec3680"
Accept-Ranges: bytes

%PDF-1.4
[...]

However, one can see that the browser is invited to upgrade the insecure HTTP connection on TCP port 80 to a secure connection on TCP port 443. And my browser (Chrome) is happy to attempt that.

The TLS connection begins with Chrome sending a TLS 1.2 Client Hello with a Server Name Indication of sodwana.uni-ak.ac.at.

Edit: Digging further, the subjectAltName extension is present with a dNSName of tethys.uni-ak.ac.at.

The URL: https://tethys.uni-ak.ac.at/geom/mitarbeiter/odehnal/talk/innsbruck_2014.pdf seems perfectly functional and checks out as secure.

My browser responds with a TLS 1.2 Alert: "Certificate Unknown" and closes the TLS connection.

However, since a copy of the PDF was successfully retrieved using HTTP, my browser was happy to present that.