The question does not mean sphere eversion is intuitive to me! In fact, it is just the opposite and that is the purpose of this question. 


Recently, I was reading about [Smale's paradox](http://en.wikipedia.org/wiki/Smale%27s_paradox), the problem of sphere eversion (turning a sphere inside out). The wiki article is quite clear and gave me a good overview of the topic. I happened to see an animation of the eversion process as well. 


The problem of sphere eversion is to construct a homotopy between the inside and outside of a sphere in a three dimensional space. During the continuous deformation self-intersections of the sphere are allowed and creating creases is not allowed. 


Given that we can self-intersect the sphere while the process of eversion what could be a possible obstruction to the eversion? What exactly do we mean by self-intersection? Moreover, I find it difficult to imagine why a similar process cannot be employed in the circle case? Why can't we self intersect a circle with itself to turn it inside out? Is there an easy explanation for this phenomenon? 


This topic is new to me. I hope the question is not too naive. Thank you in advance. 



What is the 'non-intuitive' part in sphere eversion (turning inside out)?
Watch [Outside In](http://www.youtube.com/watch?v=wO61D9x6lNY&feature=gv) (something we should all do anyway, to commemorate Bill Thurston's passing).


To understand the mathematics behind sphere eversions, you should first get a good intuition for the concepts of [immersion](http://en.wikipedia.org/wiki/Immersion_(mathematics)) and [regular homotopy](http://en.wikipedia.org/wiki/Regular_homotopy). I recommend Guillemin and Pollack's "Differential Topology" book for starters.


To see why it is not possible to turn a circle inside out, you should read up on the Whitney-Graustein Theorem. It basically boils down to the calculation $\pi\_1(S^1)=\mathbb{Z}$, once you notice that the normalized differential of an immersion $S^1\looparrowright \mathbb{R}^2$ is a map $S^1\to S^1$.


You'll find a few more resources related to sphere eversions on my [web page.](http://homepages.abdn.ac.uk/mark.grant/pages/)


