In chapter 20, Doughnuts: Linked and Knotted, Martin Gardner discusses a shape that we see almost everyday, which is commonly known as a doughnut shape. A torus is a doughnut shaped surface that is created by a rotating circle around an axis that is on a plane of a circle (figure 20.1). It is easiest to picture a ball rolling around in a circle continuously, if you cannot see it clearly in the picture. Additionally, a torus is topologically equivalent to the surface of rings, donuts, bagels, and more. A topologist is a mathematician who studies shapes and topological spaces. They are concerned with deforming inside a space to solve puzzles by bending, stretching, twisting, and multiple other ways. Therefore, they are saying a circle can be topologically equivalent to an ellipse by stretching it out and so on. I do not really understand the relevance of this kind of mathematics because I can not see where I would use it in the future.
A very common misunderstanding with topology is that many people think that a model of a surface can be deformed into a three dimensional space to make the topology an equivalent model. It took me a little while to understand what they meant by “deformed into a three dimensional space”, so I had to look up their example of the Möbius strip to make sense of it. Afterwards, I found out that I was going to have to research a lot more than this example.
Once the chapter started talking about the knots and torus’, I became very confused and stuck. I originally thought I understood what a torus is, but when they introduced it with knots I had no clue what I was reading. After re-reading the chapter, I had a better understanding for it. They were basically trying to solve the different kinds of knots/torus’ problems. I was so confused in the beginning because I could not wrap my head around how two shapes that were different, were identical topologically. The problems throughout the chapter and in the Addendum were a huge help to me understanding this chapter. If you did not understand it, then refer to the problem’s solutions in order to make sense out of each problem.
Overall, I did not like this chapter as much as others. Even though some of the things were intriguing and changed the way I thought about certain topics, I could not find the relevance in this kind of math. However, I did find this joke online about topology funny: What is a topologist? Someone who can not distinguish between a doughnut and a coffee mug.
The first source, Intuitive Concepts in Elementary Topology, is a book that explains topology in a simpler way to readers who are unfamiliar with the topic. It has several examples with in depth solutions. I think if everyone had this while reading this chapter, it could help them have a clearer understanding of topology because the book really simplifies the first steps in topology. I wish Garder would include more of the examples from this book in this chapter.
The second source, “Visualizing Toral Automorphisms”, I could not get my hands on in the library or online. But on the outside it seems to be a more in depth book about Toral Automorphisms. It seemed to set up the foundations for Torus shapes. Therefore, Gardner could have talked about where Torus’ came from.
The third source, “A Dozen Questions About a Donut”, is a harder book to understand than the “Intuitive concepts in elementary topology”. It included a little more writing about the history of Torus’. I wish I could have found the whole thing instead of a few pages online, however, some of the examples made sense when I first read them. I think Gardner also could have put some of the easier examples in the chapter from this book.
To be honest, I mostly chose to respond to Frank’s post because I saw the word doughnut in it. Much to my dismay, it was not a mathematical account of a delicious fried food, but rather a chapter on topology. To be fair though, it was a very interesting chapter. It is very perplexing to try to imagine the shape that would result by, “bending, stretching, twisting, and multiple other manipulations.” Another point of confusion for me in this chapter was understanding how different shapes could be topographically the same. I, like Frank, had to re-read the chapter a few times in order to fully grasp this main concept.
ReplyDeleteThe knot examples in this chapter were actually very helpful to me. They helped me to visualize what was happening in the problem much better. However, I wish that Gardner had started his descriptions of the examples a little bit more basic. I think that if he had given a better foundation for his examples, I wouldn’t have had to re read the chapter so many times to understand it.
I agree with Frank on the fact that the material covered in this chapter was hard to see an application for. It would have been nice had Gardner provided some background as to how topology is used in real life.
I, just like Jo, chose to reply to this post mainly because the third source was entitled, “A Dozen Questions About a Donut.” I found it interesting that so many real life shapes, like a straw, have topological equivalence to a torus. My favorite part of this chapter was finding out that many different shapes and rings can be inverted to create the same exact shape. I had no idea that there were so many ways the transverse knots and doughnuts to connect them into many different shapes. I was as equally as confused with the rest of the chapter as Frank was. Reading Frank’s post made it easier to understand for me personally because I never would have thought that they were just solving different knot/torus problems. Chapter 20 reminded me of a project we did during my sophomore year in high school. We were given a tube with string sticking out of four spots on the tube. We could pull the strings in any way we wanted. Our goal was to find out how the strings crossed inside the tube, and I never found out the true answer.
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ReplyDeleteChapter 20, Doughnuts: Linked and Knotted, seemed like it would be one of the simpler chapters turned into a chapter filled with complexity. Though interesting, chapter 20 is complex when it discusses doughnuts and knots together. It was a hard concept to grasp, but essentially what I got from it is that when you take an infinite amount of circles and create a cylinder you have nearly taken on the doughnut form. After stretching the cylinder and connecting it at both ends you have created a doughnut. After this concept is grasped the chapter goes on to discuss different ways that a cylinder can take form. For example, tying a knot before the circle is completed. This more basic part of the chapter was a little bit easier to understand. The chapter then goes on to discuss different ways that these cylinders can occupy space inside a cube. In RH Bing’s proof he displays six separate ways of how a cylinder can be manipulated inside of a cube. Towards the end there are a couple of riddles that were hard to solve. For example, if two doughnuts are chained together and one has a hole in it and is hollow is it possible unlink them? The answer is no, they cannot come apart but one doughnut could go inside of the other. All in all though the chapter was difficult to comprehend it was interesting.
ReplyDeleteI chose to read this chapter because it is connected to Chapter 19, the one I did my last blog post on. While chapter 19 talked about knots and their relation to mathematics, chapter 20 focuses primarily on doughnuts. To me a doughnut was always a delicious pastry, however chapter 20 proved to me that doughnuts are apparently much more. A doughnut is created by a rotating circle around an axis that is on a plane of a circle. While this is kind of hard to picture frank pointed out that it is easiest to picture this is by imagining a ball continuously rolling around in a circle. One part of the chapter I found really interesting was when they talked about getting an internally knotted torus to reverse through a hole. This part was kind of confusing to me at first but after rereading it a few times I understood how it worked. Another section I found interesting was the part where two torus's are linked and there is a hole in one of them, we are allowed to stretch and compress as "swallow" the other. I enjoyed this part of the chapter because t was relatively easy to understand and kind of interesting to try and think about and solve on my own. I liked this chapter because many of the concepts were interesting to read about even though I did not understand all of it.
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