Is it possible to invert a sphere

This was thought to be impossible to do, until the s. Then it was thought to be only impossible to see. It was not until , computer graphics, and the film Turning a Sphere Inside Out , that people could picture it.

Take a look at an everting sphere. Interestingly, it is impossible to turn a rubber-band-like circle inside out this way. You can turn a sphere inside-out according to these rules, but not a two-dimensional circle. Living in three dimensions has its positive aspects, in theory, and in computer animation.

Top Image: Soylent Green. We come from the future. By Esther Inglis-Arkell. This browser does not support the video element. It only takes a minute to sign up. 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? Watch Outside In 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 and 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. You'll find a few more resources related to sphere eversions on my web page. Sign up to join this community. The best answers are voted up and rise to the top. What is the 'non-intuitive' part in sphere eversion turning inside out?

Ask Question. Asked 9 years, 2 months ago. Active 5 years, 3 months ago. Viewed 3k times. This topic is new to me. I hope the question is not too naive.