Tag Archives: Topology

Inside Sphere Outside Sphere

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.

Sources

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

Graph Theory

Graph theory is a branch of mathematics that looks at the relationship between objects. I know this might sound a little vague, but to get the basic picture of what graph theory is, I’d like to take a look at the following image:

The bridges of Königsberg. Image: Bogdan Giuşcă, via Wikimedia Commons.

The bridges of Königsberg. Image: Bogdan Giuşcă, via Wikimedia Commons.

The image above is a drawing of the Königsberg bridges, located in Königsberg, Germany in 1736. This problem was the beginning of graph theory and no one even knew it. It can be defined by the following:

The Königsberg bridge problem asks if the seven bridges of the city of Königsberg, formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began.

Proof by Euler

Leonard Euler, in the year 1735, introduced a counter proof to the problem described above. He realized that the only important issues in the problem were the bridge crossing sequences. Using this abstract he created two new terms, “vertex” or “node”, and “edge.” In the above picture each unique landmass would be classified as a vertex, while each bridge is classified as an edge. Euler’s resulting image would look something like the following:

The graph that represents the bridges of Königsberg. Image: Riojajar, via Wikimedia Commons.

The graph that represents the bridges of Königsberg. Image: Riojajar, via Wikimedia Commons.

The image above is described as a graph. Within a graph, it does not matter the exact location of each node. However, the connection(s) formed between nodes does matter. Edges between nodes can either be curved or straight, because it’s not important to the graph structure, they are simply showing how each object is connected.

Euler then describes the process of his counter proof by first describing what is now known as the “degree” of a node. A degree of a node is described as how many edges it has (one, two, etc.). His proof says that within a connected graph (there can’t be any nodes that are alone, i.e. not connected to any other node) there can be either zero, or two, nodes that have an odd degree. This proof was monumental in his time and was later validated by Carl Hierholzer.

Traveling Salesman Problem

One of most studied graph theory problems in the field of Computer Science is the Traveling Salesman Problem or TSP for short. The basic problem definition states that given a list of cities and the distances between them, find the shortest path between all cities, where each city is visited exactly once and the “salesman” must return to the original city. Currently there are no algorithms that can solve this problem in polynomial time (polynomial time is described as O(na), where n is the number of cities and a is a constant integer). Active research in solving this problem and many algorithms come from general heuristic solutions, which are currently the fastest ways to solve the TSP. The TSP is used in many different fields including logistics, transportation and even microchip production.

Unsolved Problem

There is one major unsolved problem in graph theory known as the “Longest path problem.” The problem is described as finding the longest path in a graph, without repeating vertices, by counting the number of edges the path contains. This problem is classified as NP-hard, implying that it cannot be solved in polynomial time or solved at all (this definition of NP is also one of the Millennium Prize Problems, asking if P = NP, check out http://en.wikipedia.org/wiki/P_versus_NP_problem for a quick definition).

Conclusion

Graph theory is a very interesting subject and it has a lot to offer. Many mathematical and computer science problems have been solved using graph theory. I believe that with continued research, it will be known if the “Longest path problem” can be solved. There are many companies such as FedEx and UPS that are very interested in the traveling salesman problem, because it could greatly reduce the amount of fuel used delivering packages, thus reducing global emissions and saving natural resources. I find graph theory to be a very interesting subject and I hope in due time humanity will have all of the answers to graph theory’s most problematic questions.

References

http://plus.maths.org/content/maths-minute-bridges-konigsberg

http://www.math.uwaterloo.ca/tsp/apps/

http://en.wikipedia.org/wiki/Graph_theory

http://mathworld.wolfram.com/LongestPathProblem.html

http://world.mathigon.org/Graph_Theory