Hello, Everybody!
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, 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.
