Talk:Sphere eversion

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could someone try to create a page for this? I would, but I'm not a math genius :p

Looks like someone’s done so!
—Nils von Barth (nbarth) (talk) 23:31, 5 May 2009 (UTC)

In practice?[edit]

Could someone describe a case where this paradox has practical relevance? Cunya (talk) 06:46, 18 November 2007 (UTC)

Hi Cunya,
I don’t know any direct practical applications of this, or in fact of immersion theory: like much of math, it’s motivated by internal considerations of math.
The main interest of this result within math is that it is a basic result in understanding immersions, and is both surprising, beautiful, and can be visualized in 3 dimensions, hence easily communicated – if you haven’t looked at the videos, I’d encourage you to do so: they were a milestone in mathematical visualization.
—Nils von Barth (nbarth) (talk) 23:31, 5 May 2009 (UTC)


I linked 'Embedding' to the main article, but I confess I don't know if there's a better (more specific) place to link it. Zero sharp (talk) 21:49, 15 May 2008 (UTC)


In what sense is this a "paradox"? It's surprising, sure, but not self-contradictory.

It’s a veridical paradox – surprising, as you say. I’ve noted this on the article page – thanks!
—Nils von Barth (nbarth) (talk) 23:31, 5 May 2009 (UTC)

This article disappointed me. I wanted an explanation for laymen. Why is it a "vertical paradox" and what does that even mean. —Preceding unsigned comment added by (talk) 08:18, 9 October 2010 (UTC)

Anyone using the term "veridical paradox" in an article about mathematics is unclear on the concept of explaining things clearly. (The obscure word used here is "veridical" — which means "truthful" — not the word "vertical".)

Every paradox is some situation that involves two apparently contradictory aspects. There are basically two types of paradoxes: The kind where there is an explanation for how the apparent contradiction is not really a contradiction, and the kind where the contradiction does not have any obvious explanation and may actually be impossible. A "veridical" paradox just means a paradox about a "true" situation: an apparently contradictory situation where the apparent contradiction is not really contradictory.Daqu (talk) 21:02, 13 July 2015 (UTC)


The lead says that "Smale's paradox" is an artifact of this encyclopedia. If so, the current title is not notable. But under the title "Sphere eversion" the article is notable. Therefore the article should be moved over the current redirect. Some pruning or tuning will also be necessary.Rgdboer (talk) 20:54, 21 April 2013 (UTC)

I think it was vandalism, since a whole series of edits was made by one editor on April Fool's day this year. I have reverted. Chris857 (talk) 21:29, 21 April 2013 (UTC)

Searching Mathematical Reviews there is one review by Etnyre, now in References, using "Smale paradox".Rgdboer (talk) 20:23, 25 April 2013 (UTC)

Where to take this[edit]

Well, I was reverted by User:Sutilcareh, so I think this issue should be discussed. The thing I dislike most about the article's current version is how self-referential it is. If "Smale's Paradox" is in fact not commonly used, I think the article should probably be moved to sphere eversion and the focus shifted. This would keep most of the same content but without unduly focusing on Wikipedia in a Wikipedia article not about Wikipedia. The lead could then potentially be something like the following:

In differential topology, sphere eversion (eversion means "to turn inside out") is the process of turning a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease. Such a turning was once thought impossible, but the opposite was shown by Stephen Smale in 1958, and as such this veridical paradox is sometimes referred to as Smale's paradox. More precisely, let

be the standard embedding; then there is a regular homotopy of immersions

such that ƒ0 = ƒ and ƒ1 = −ƒ.

What do others think? Chris857 (talk) 00:07, 24 May 2013 (UTC)

Agreed. Web searching shows that the usage "Smale's paradox" appears to be a neologism, largely created and supported by the naming of this article. Moving to sphere eversion, the name generally used for this by the mathematical community -- The Anome (talk) 11:16, 2 August 2015 (UTC)


I don't know if anyone is aware, but there is a popular demonstration of this concept on youtube, one which is quite well done. Can it be included in the page somewhere, perhaps as a reference? Apologies, I've never edited before. RoflCopter404 (talk) 11:45, 2 October 2013 (UTC)

Paradox is inexplicably left mysterious[edit]

The article's statement of the paradox and its resolution is as follows:

"Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong (Levy 1995). His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of S1 in R2. But the degree of the Gauss map for the embeddings f, −f in R3 are both equal to 1, and do not have opposite sign as one might incorrectly guess."

But just stating that the degrees of the Gauss maps of a) the standard embedding, and b) the antipodal embedding, of the 2-sphere in R3 are equal does not clarify why they are equal (or why they are not equal in the corresponding situation for the circle in R2).

Is the purpose of an encyclopedia to mystify the reader? I don't think so.Daqu (talk) 20:39, 13 July 2015 (UTC)

Others turning inside out[edit]

"S0, S1, S3 and S7 are the only spheres with trivial tangent bundles, that answers your question. i.e. you can only turn S0, S2, and S6 inside-out."by Ryan Budney --Takahiro4 (talk) 09:52, 31 July 2015 (UTC)

Move to "sphere eversion"[edit]

I can't find any WP:RS that describes the topic of this article as "Smale's paradox". I've moved the article to "sphere eversion", the name universally used in the mathematical literature, and removed the term "Smale's paradox" from the article completely. As far as I can tell, the term "Smale's paradox" is a rarely-used neologism that appears to have been largely created and supported by the naming of this article. -- The Anome (talk) 11:27, 2 August 2015 (UTC)