Topology is the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures. Topology is one of the most important fields in mathematics. Incredible mathematical concepts have been discovered, and other fields of mathematics often benefit from the various subsidiary discoveries when studying some bigger idea in topology. There are countless memorable theorems, conjectures, and concepts that are so famous that anyone who has studied mathematics for a moment in their life has heard about them. There are different sub classes of topology including general topology, algebraic, geometric and differential and they have applications in fields including but not limited to biology, computer science, physics and robotics. While it may not be as famous and may not have any major applications, Smale’s paradox is an example of a topological discovery that is mesmerizing and fun to think about.
Smale’s paradox states that is possible to turn a sphere inside out in three-dimensional space with self-intersections, but no creases (Wikipedia). Even though this concept is referred to as a paradox it isn’t really a paradox and is recognized primarily as spherical eversion in the math world. Stephen Smale discovered this paradox in 1958 and stunned the mathematical world with his proof of spherical eversion. Spherical eversion just means turning a sphere inside out. Without making any creases or poking any holes a sphere can be flipped inside out provided self-intersection is allowed. Spherical eversion is concerned with what kind of maps are homotopic to each other and so it is beneficial to know that two continuous functions from one topological space to another are homotopic, if one can be continuously deformed into the other. The deformation is called a homotopy. A regular homotopy is just a special kind of homtopy between immersions of one manifold in another.
There are a few recognized proofs for this. Smale’s proof was indirect. Smale identified classes of immersions of spheres with a homotopy group of the Stiefel manifold. Smale’s logic was that because the immersions in two dimensional space vanish in three dimensional space these homotopies become regular. This concept is sounds very complicated, but the first video below has a thorough explanation. The complex method for demonstrating this unusual veridical paradox was developed primarily by Bernard Morin (livescience). Other methods exist such as the Aitchison’s holiverse, Thurston’s corrugations and the Half-way models.
Ian Aitchison’s holiverse is the newest addition to the study of spherical eversion, and was recognized in 2010. Aitchison’s holiverse relies on topology and geometry. The video below covers some of the concepts at play in Aitchison’s holiverse.
The Thurston corrugations are topological and generic and rely on manipulating homotopies into regular ones. He uses an interesting technique; adding waves to the curve. Thurston’s method is the easiest to follow. Imagine adding waves to a sphere such that it looked like the waves of and ocean moving around the sphere and take a snap shot of the moment. Push the top and bottom of sphere through each other and before forming a crease turn the two sides counter clockwise with each other. Finally, push the center of sphere back through itself. This can be tough to visualize without aid. Luckily there are lot of magnificent visualizations online. In mathematics it is typically beneficial to visualize a problem in lower dimensions. In this case the problem doesn’t get any simpler if you try and break it down into two-dimensions. Circular eversion, the two-dimensional equivalent of spherical eversion, is not possible. This can be explained abstractly. If you have a circular track and start driving around the track, you can only turn in one direction, whereas in a sphere you can manipulate domes saddles and bowls by twisting, stretching and self-intersecting the sphere in such a way that the three dimensional equivalent of this concept would have no problems. The video below explains all of this, including the lower dimension pitfall, in great detail.
Developed in the 1980’s the Half-way models rely on special homotopies and understanding of fourth dimensional partial differential equations. The Half-way models method can be demonstrated using chicken wire constructions. Chicken wire can’t be pulled through itself though so it can only be demonstrated in phases or snap shots of the process. Computer generated videos exist that demonstrate the morphology of the process. It is much harder to describe the movement in each phase of this process than in the Thurston process.
In summary, turning a sphere inside out doesn’t have any direct applications, but does provide some insight on how we might think about things in higher dimensions and is fun to think about, literally, in and of itself. A few people spent sizeable chunks of their lives proving this and constructing visual aids for the process. It is interesting to see that the Thurston and Half-way demonstrations have noticeably different geometric manipulations. The processes differ completely.
Wolchover, By Natalie. “5 Seriously Mind-Boggling Math Facts.” LiveScience. TechMedia Network, 25 Jan. 2013. Web. 12 Mar. 2015.
“Smale’s Paradox.” Wikipedia. Wikimedia Foundation, n.d. Web. 12 Mar. 2015.
“Turning a Sphere Inside-out (1994).” YouTube. YouTube, n.d. Web. 12 Mar. 2015.
Spherical Eversion from green to red. http://www.youngwizards.com/ErrantryWiki/index.php/Eversion