Chapter 20
Doughnuts: Linked and Knotted
This
chapter explains that a torus is a doughnut shaped surface that is generated by
rotating a circle around an axis that is on a plane of a circle but doesn’t
intersect the circle. A torus is a
generated surface that occurs when revolving a circle in three-dimensional
space. If the axis of revolution does not touch the circle, the surface has a
ring shape and is called a ring torus. If the axis is tangent to the circle,
the result is a surface called horn torus. It is a topologically
equivalent to objects such as a bagel, a ring, a rubber band, a life preserver,
etc. A big misunderstanding with
topology is that many people believe that a robber model of a surface can be
deformed into a three dimensional space to make the topology equivalent model.
That situation is not the vase very often. That same dichotomy also applies to
knots in closed curves. The surface of the rope will end up being equivalent to
a knotted torus, which I thought was interesting!
They also
talk about what a topologist is the
mathematical study of shapes and topological spaces. It is an area of
mathematics concerned with properties of space that are preserved under
continuous deformations including stretching and bending, but not tearing or
gluing. This includes things such as knots inside of cubes and such.
A lot of controversial problems that
involve knots and links in tourses have been solved and published. Shown in the
picture below. Rolfsen published that the surface of the figure on the left is
topologically equivalent to the surface on the right! At first glance is
doesn’t look like it would at all, but somehow topologically it does. I found
it really interesting that it ended up that way. There was a lot of other
little pictures and diagrams. There was a bunch of little addendums. They are
like little practice problems and examples. Out of the 10 problems 3 of them
were really hard to understand but the other ones were easy to understand.
I found this chapter really
intriguing because at first it doesn’t make sense how something that looks to
be much longer then the other one can topologically be the same! It was really
confusing and hard to follow some of the pictures that were in the reading as
there was a lot. I also looked up some diagrams of a torus and they had a lot
of cool gifs of them and it was interesting to see a visual that made sense and
allowed me to have a better understanding of everything that was in the
chapter. This chapter
was very interesting to me. Prior to reading it I had no idea what toruses
were. This chapter was sometimes hard to follow and I had to reread many
sections and look at the pictures multiple times but in the end it ended up
being really interesting and ended up learning a lot.
You dd a nice job summarizing chapter 20! I also had trouble comprehending the pictures at times but I was definitely interested by them!
ReplyDeleteYou did a good job summarizing the chapter. I agree some of the pictures were hard to follow.
ReplyDeleteGood summary of the chapter. I had to watch videos on topology of toroids to help me to get a clear visual of the concept. I was also fascinated by sculptures by Keizo Ushio on the concept that I viewed; very interesting indeed!
ReplyDeleteVery interesting picture and summary. Hard to believe that the figure on the left is equivalent to the one on the right, topologically speaking and how it can be continuously deformed to the same shape. Really cool chapter.
ReplyDeleteThe pictures were hard to understand in the beginning. But your summary helped out to understand it. I also thought the sculptures were pretty cool to look at. Overall good summary.
ReplyDeletei must say that on top of learning and gaining a better knowledge of this subject, it made me want donuts.
ReplyDeleteI enjoyed reading your summary since I felt that I had a basic understanding of the chapter, but your summary really clarified the gist of this chapter. It was also really helpful that you posted a picture for people to see what it is that you are talking about. Yet although there were pictures it was hard for me to understand how they were similar since most of them looked completely different.
ReplyDeleteI was surprised that the figure on the left was equivalent to the figure on the right topologically. It reminds me of calculus where you are finding the area of a curve revolving around an axis. Good explanation on the concept.
ReplyDeleteGreat job summarizing the chapter! The illustrations were difficult to understand but they were also interesting.
ReplyDelete