Talk:Sphere eversion

message: This section should be replaced by a simpler gallery of how Bednorz's eversion is accomplished.  The current diagrams only confuse the reader and discourage them from pursuing the subject further:  start with a band of construction paper.  At least in that case the vast majority of readers would fully understand the central transormation and be incentivized to pursue the complete sphere.   Links to (simple) code such as Mathematica would also be helpful.Youriens (talk) 13:33, 6 September 2022 (UTC)
thread_title: Gallery and string models are not helpful and only confuse the reader 

message: 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)
thread_title:  Eversion? 

message: 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)
thread_title:  Embedding 

message: 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)
thread_title:  Paradox? 

message: 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 
:f\colon S^2\to \R^3
be the standard embedding; then there is a regular homotopy of immersions 
:f_t\colon S^2\to \R^3
such that ƒ0 = ƒ and ƒ1 = −ƒ.
What do others think? Chris857 (talk) 00:07, 24 May 2013 (UTC)
replies: text: 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)
thread_title: Where to take this

message: 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)
replies: text: I don't see why not. It's a relevant source and it explains the subject in very understandable language. Let me see what I can do.  TrippCeyssens (talk) 15:56, 25 March 2022 (UTC)
thread_title:  Youtube 

message: 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)
thread_title:  Paradox is inexplicably left mysterious 

message: "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)
thread_title: Others turning inside out 

message: 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)
thread_title:  Move to "sphere eversion"