Sphere Turning
When you reach into your closet and pull out a shirt or pain of pants, there is a possibility that they are inside out. This is easily corrected, however, by reaching through the holes and pulling the “back“ side to the inside, resulting in the correct side facing out. Its really easy to turn anything with a hole in it inside out. Another example is when sewing a stuffed animal, you always leave a small space open. That way, you can reach in a pull the correct side out, therefore hiding any strings. A stuffed animal doesn’t have a hole that runs completely through it, it rather has a hole that connects two faces enabling the user to easily turn the fabric inside out.
What about 3d shapes with no holes in them? Is it possible to turn anything inside out?
Stephan Small, a differential topologist, published a theorem in 1959 that one could turn a sphere inside out following a couple of conditions: it was a regular map of a sphere in 3d space, the sphere could intersect itself, no creases or sharpy turns could be created.
What’s fascinating is that Smalls conjecture is correct. Even though there were no computer models for him to fully visualize his conclusion, it is possible to turn a sphere inside out.
Okay, that sounds great and all, but just because a topologist says something is true doesn’t make it visually conceptual. Like just because different dimensions exist doesn’t mean that most people would actually fully believe that. What is cool about this problem is that it can first be simplified into 2 dimensions. Can we turn a circle inside out?
Imagine that you had a circular race-car track. Now imagine that you are driving around that track in an anti-clockwise direction. Make on complete revolution so that you are back to where you started. All the time that you were driving around that circular track, you always had your left side facing the inside of the circle. Your right side never faced in. This circar would be classified as having a turning (or winding) number of 1. You only made one loop and you never changed directions.
Now pretend that you have a figure-eight race car track. You start driving on that track in the same direction that you started on the circle. Now though, when you cross that figure-eight section, you change directions so that your right side is facing the inside of the circle. Think of this as “undoing“ the turning of the original track. Once you get back to where you started, you would have made one anti-clockwise rotation and one clockwise rotation. These two rotations perfectly undo each other, and therefore the shape has a winding number of 0.
Every anti-clockwise rotation can be thought of as 1 and every clock-wise rotation would be thought of as -1.
Taking this principle, now imagine a close loop turning inside out. The car would now be traveling the opposite direction giving the new shape a winding number of -1. Because the original loop has a winding number of 1 and there new one of -1, the two shapes are not able to be turned to be each other. What is interesting is that any loop sequence you create with a turning number of 1 will be able to turn into that original sphere. This, of course, allows self-intersections as it is all still theoretical. As long as the loop is closed, the turning number will always be whole. This enables you to turn any loop into another compatible loop.
This process is similar for a 3d shape. A 3d shape can be simplified into a collection of concave up, concave down, and saddle surfaces (both positive and negative curvature). If you simplified the surfaces of a 3d shape to a collection of points, the concave up and down sections would form rings and the saddle would form X’s. When calculating the turning number of a sphere, the domes, and S’s cancel out.
Taking this theory, the surfaces of a sphere can be created to find their turning number, with is the same as an “inside out“ sphere. Coupling winding numbers with the conditions that the sphere can self-intersect, Smale’s theorem can be more easily visualized.
Today there are multiple animations of turning a sphere inside out, but during Smale’s time, he was able to prove the abstract concept with almost no visual aids. So next time you see a closed-loop, make sure to discover its winding number and manipulate it in a way to return to a simple closed-loop.
Sources:
http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm
https://mathworld.wolfram.com/ContourWindingNumber.html
https://www.youtube.com/watch?v=-tj190Lcw48