This chapter was very confusing. To
start with it gave a definition of the different levels of spheres. A 1-sphere
is a line, a 2-sphere is a circle, a 3-sphere is your average sphere, and a 4,
5 or 6-sphere is called a hypersphere. The number is representative of the
number of dimensions that the sphere occupies. According to Gardner,
“hyperspheres are impossible to visualize”. However, he goes on to explain that
they can be studied through analytical geometry if you expand it to look at
more than three dimensions. The surface area of each of the types of spheres
also makes sense to me. The dimensionality of the surface of an n-sphere is
n-1.
After these points were established
I got really lost. I read the chapter 3 times and I still couldn’t wrap my head
around it. There was a situation proposed where there were 3 large hyperspheres
surrounding a small 4th. The problem was to find a formula for the
radii for the maximum number of mutually touching n-spheres. I really did not like the way that Gardner
presented the way to go about this problem. He did it in the form of the poem The Kiss Precise, written by British
chemist Frederick Soddy. The poem is a long, rhyming affair that uses weird
words to describe simple things.
The poem itself was one of the sources
I researched further. Through this research, I found that the poem was not
published alone, rather with proofs to go along with each stanza of the poem. I
would have liked if Gardner included some of these poems in his writing of the
chapter. I may not have understood what was happening in some of the proofs,
but the visuals that they provided would have been helpful for me in decoding
the answer to the spheres problem. I was happy Gardner did not include the
original roots of the solution based on DesCartes’s work. I feel like it would
have made the chapter more overwhelming than it already was.
The second source Gardner used that
I chose to research more in depth was the article “Leech Lattice, Sphere
Packing and Related topics” by J.H. Conway and N.J.A. Sloane. The language in
the book was kind of confusing, but it made sense overall. I am happy that
Gardner only included basic definitions from this book in the Chapter. Since it
was a book, there were a lot of problems that were tangent to the topics in the
chapter, and could have easily been woven in, but were not. I was disappointed
however, that Gardner did not include more information about sphere packing. He
covered it briefly, but Sloane explained it in a way that was easy to
understand.
The last of the sources that I
researched farther was “Kepler's
Spheres and Rubik's Cube”. I mostly chose to look at this article because I
liked the title, but I am glad that I did look at it because it was very
interesting. Gardner didn’t include this in the chapter, but Prepp talked about
the similarities between a hypersphere and a Rubik cube. I was sad that Gardner
did not include this comparison because it would have been a solid metaphor to
help understand what a hypersphere is. This being said, I was very glad that Gardner
did not include anything about icosahedrons. The math included with this shape
was very over my head.
Overall, this chapter was a tough one to get through.
The concept of hyperspheres is one that I will have to work hard on to
understand. Although the basic formulas are easy enough it is hard to apply
them to the questions being asked about hyperspheres. I don’t know how problems
concerning hyperspheres are applicable to real life, and that’s something I wish
Gardner could have included. It would have been easier to understand what was
happening if a connection could have been made.
This chapter was very complicated because it involved visualizing things that are impossible for us to see. Just like in the movie, Flatland, the line could not see the grandpa square unless the square moved into the same plane as the line. In the same way, it is impossible for a to visualize a hypersphere, for example, because it involves the sixth dimension when we can only see things in the third dimension. One thing in the chapter that confused me was the fourth small circle that was in between circles one, two, and three. I was not quite sure why the circles were placed where they were. I do think that the ironic math poem helped make the problem make more sense. I found it very interesting that the poem had a proof to go with every stanza. It was a very mathematical style to write a poem in. Towards the end of the chapter appear to be a series of squares inscribed inside of a circle. It follows an interesting pattern and seems to link with the “Solid Geometry” portion of this chapter. All in all, I found this chapter to be the most confusing yet. I had a difficult time grasping the subject matter.
ReplyDeleteI agree with Charlotte how it is very hard to visualize something impossible for us to see. If I wouldn't have seen the movie Flatland in class this chapter would have made no sense to me at all. Watching Flatland however I was able to visualize what they were talking about. I would love to see how they would make a fourth fifth and sixth dimension sphere however. Although as of tight now we can only see things in the third dimension I believe technology will soon advance enough to allow us to see things in the fourth dimension. Although the chapter was very confusing still I was able to understand parts of it. I also agree that I think it would have been a lot easier to understand if they compared it to a rubix cube. By doing this it was something you are familiar with and are able to picture in your head. I would like to know more about this subject because it is neat just to think about all the other dimensions we have not yet discovered and have yet to see. Although it doesn't state it I wonder if there would be any mathematical formulas that go with this. I am sure there is because if it has to deal with math its likely to have a formula with it.
ReplyDeleteI read this chapter after we watched the movie, Flatland. I agree that the chapter was not easy to understand just because it centered around things that are said to be impossible to visualize. If the fourth dimension is impossible to visualize then a hypersphere would obviously be just as impossible being that it is part of the sixth dimension. Of course the whole idea of a hypersphere and dimensions is theoretical so I think its interesting how Gardner goes on to describe it by using third dimensional thought. I agree with Jo about how Gardner presented the poem as a way to propose the spheres surrounding other spheres situation. I thought it could have easily been stated in a way that wasn’t so random even though it was fairly helpful to understand. I also wish he included how the problems and formulas could be applied in real-life terms. Jo did a good job of presenting the chapter although it was confusing and theoretical. Since I read the chapter after watching Flatland, I think it was easier to understand the whole concept than it would have been if I just read the chapter without the movie. With that being said, it was still hard to get a grasp on what was being described.
ReplyDeleteChapter 11: Spheres and Hyperspheres was an interesting chapter and really peaked my interest. A lot of what was discussed about the hypersheres existing in the fourth dimension and beyond had a lot to do with the chapter I just recently wrote about, Chapter 8: The Wonders of the Planiverse. This chapter talks about spheres that exist in a dimension that is above our and exists in a 4-dimensional plane, a plane that we cannot visualize so in turn we cannot visualize these spheres. When this chapter discusses 3 larger hypersphere surrounding a small 4th, I reminded a lot of the movie we watched in class on Friday, Flatlands, when little Hex stumbles upon the old ruins and sees the lines combined to make squares and the squares combined to make the cube. The same method is used in making these hyper shapes, combined a couple of cubes or in the case a couple of spheres and you can create the hypersphere that exists in the 4th dimension. Little Hex is unable to see the “cube” that is produced because it exists outside the 2-dimensional world that she lives in. Same as how we are unable to visualize these hypershperes, but we are able to understand them and how they work with the power of math.
ReplyDeleteI have to agree with Jo that this chapter was a bit confusing. The different levels of spheres is a different concept to think about. Some spheres, "hyperspheres", don't exist in any dimension that we know that can be imagined, drawn or proved. In turn, how could Gardner develop an idea of how these hypersheres would be shaped and combined with other spheres to be created? As Jo mentioned, I found it challenging to understand fully the points Gardner was trying to get across in his poem. This contributed to the chapter being more challenging than it started out to be and had to be. Although the poem was not the best attribute of the chapter, any time Gardner or any other scientist/ mathematician talks about the fourth-dimension and beyond, it is challenging to grasp the idea because of how unknown those concepts truly are and the inability to fully imagine what these shapes could possibly look like in these dimensions. All in all, chapter 11 continued to discuss concepts about dimensions beyond much of our understanding and thus will continue to be complex and completed to understand for myself and most of my classmates who are not experienced in this field.
ReplyDeleteI agree that this chapter was indeed quite confusing. Trying to think about problems involving a hypersphere is pretty difficult knowing that it is impossible to imagine one. Trying to imagine a 4th dimension sphere is hard enough in itself, then there’s a 5th and 6th. I too tried to re-read this chapter a few times to try understand some of the concepts of the hyperspheres. However, like many, I found it much simpler to apply the formulas to a 2-sphere or 3-sphere. I only picked up so many things from the chapter as it is hard to understand the terminology as well. This chapter didn’t feel as recreational as the others, it felt more math theory based. From what Jo mentioned about that sources Gardener used, I really think he should have included some of that information in this chapter to make it easier to understand. It nice to have more information in the chapter so the reader can learn things different ways like the poem. I am happy though that Gardener did not include the additional information about icosahedrons or DesCarte’s work. Overall, Jo did a good job on reporting the chapter and getting to the main concepts.
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