The first thing I found interesting was that there are actually numbers that are triangular and square. This was intriguing to me because I thought of them as two different kinds of numbers. However, there are actually an infinite amount of triangular and square numbers, the lowest being 36. The chapter then discusses numbers found in 3D figures, like: pyramids, tetrahedral numbers found in 3 sided pyramids with a base and 3 sides all being equilateral triangles, and square pyramidal numbers found in 4 equilateral triangles with a square base. These numbers were very interesting since they gave you the equations for them: 1/6n(n+1)(n+2) for tetrahedral, and 1/6n(n+1)(2n+1) for square pyramidal. Where n= number of spheres along the base edge. I plugged in numbers and saw how the equations work and how there is a pattern. One example I found was with a tetrahedral number: if you plug in 4 you get twenty, if you then add figure 10.1 left (15 total spheres) to it you get the next number 35. That's obviously ingenious, but I found it fun since it helped me picture them 3 dimensionally.
The only question I have about this chapter is about the lowest density possible theorem. The design in figure 10.5 is very unnatural looking to me and I can not seem to find the reasoning in this. Why would you ever package something like that and waste all that space? Furthermore, what is the difference in figure 10.5 versus putting one sphere in the box and calling it the lowest density for packaging spheres? I am having a hard time visualizing and wrapping my head around the lowest density idea.
I enjoyed reading this chapter because of the multiple connections to mathematical theories and laws that we already know. The examples with the triangular and square models and their totals was interesting to me. Specifically, how the sum of the triangle would be a triangular number with the equation 1/2n(n+1) or 1/2n(n-1) would equal n^2 which is the sum of the square model. I agree that the example towards the end of the chapter (figure 10.5) seemed rather odd just because of the replacement with the smaller spheres. What I didn’t understand either was figure 10.3 just because the color and shapes do not match up with what the book was talking about. It was a good idea to plug in some numbers into the equation because it does help the reader have a better visualization of the 3D packing as well as just the square and triangular packing. you helped me understand more of what the square and triangular numbers mean in context as well as the tetrahedral numbers. Overall I thought the chapter was interesting and didn’t stray away from the topic. It was very specific on spherical models rather than switching to a polygon-based type of packing.
ReplyDeleteI had no idea that stacking spheres in a box could have so much mathematics behind it. Who knew there was a certain density one could reach? Chapter 10 was an interesting chapter for me, because there was so much I actually understood and so much for me to learn. I, like Frank, had never heard of triangular numbers. I also had never heard of tetrahedral numbers, so as I was reading through the chapter it grasped my attention because I was learning something completely new to me. The chapter expressed many new types of sets of numbers other than these that were brand new to me. Though I do not completely recognize the equations given ( 1/2n(n+1)), I am not sure exactly where I used them in my mathematics. I had no idea it related to different types of numbers. Reading this chapter really made me think of math as a long process where everything builds off of something earlier in order to create a new theory. I agree with Frank when he says figure 10.5 is a bit confusing. When looking I was also amazed with how much wasted space there was. Overall I really enjoyed reading this chapter because the different sets of numbers really intrigued me and I seemed to understand this chapter better than I had any of the others.
ReplyDeleteBeginning to read this chapter I found it really exciting that there are the things called “the ancients.” I probably should not have been as excited as I was. However, it amazes me that although we have built upon their knowledge, they laid the stepping stones to what we know now. This chapter brought me back to my childhood because I use to take all my dad’s golf balls out of the box and then get angry when they wouldn’t fit back in the same way. I never knew there were so many ways to pack spheres, and how complicated it can be. It is strange that only 74% of a box is used in packing of spheres, and that at no more than 12 balls touch central sphere at any given time. The part about the spheres changing into different shapes confused me. I didn’t understand how the spheres were getting sliced in half, or transformed into a completely knew shape. The compressing of the spheres would work if they were a soft object, but in the case of golf balls it seems pretty unrealistic that they could be packed into other shapes.
ReplyDeleteAfter reading this chapter, I found that the concepts explained throughout it are the most real-world applicable I have read to date from The Colossal book of Mathematics. The example of the ping-pong balls was a prime example of my previous statement. I'm also sure many other companies use triangular and square numbers when considering their packaging as it uses each shape respectively to perfectly calculate the number of items in the shape, depending on the number of items of course. Furthermore, at the end of the chapter the author talks about the density of packaging. This concept can also be applied to the real world. Knowing the absolute highest density for any package will allow companies to have cost efficient packaging and package at its lowest cost possible. This concept applies to the highest density theory, but doesn't explain the loosest density theory. This theory, as Frank mentioned, doesn't seem to have a meaning or real world application. For that reason, I share Frank's opinion of the loosest density theory and agree that it is a bit of a wasted study. All in all, for the real world applications, I found chapter 10 to be a thought-provoking and enjoyable read.
ReplyDeleteI never thought of different sized spheres stacking in this way. It is a totally different way of viewing such a simple concept. I found reading this chapter quite interesting and easier to follow because it is something we can visualize and understand easier compared to some other concepts in the book. You did a well job of explaining these triangular and square numbers clearly and concisely. Reading your blog and comparing it to the book helped immensely to grasp some of the terminologies and concepts presented in the chapter. Some visuals would have been nice along with the blog, but since we have the book we could just look there. It is quite exciting when there are connections between recreational mathematics and real world applications. This chapter is a perfect example of recreational mathematics possibly being applied to the real world. The highest density theory seems very useful for shipping companies or just every day people when packing objects. Along with the others, the lowest density possible theorem seemed a little confusing or irrelevant to real life applications. Overall, I think you did a very good job on summarizing the chapter and being able to portray difficult concepts presented to you in the chapter.
ReplyDeleteI never thought of different sized spheres stacking in this way. It is a totally different way of viewing such a simple concept. I found reading this chapter quite interesting and easier to follow because it is something we can visualize and understand easier compared to some other concepts in the book. You did a well job of explaining these triangular and square numbers clearly and concisely. Reading your blog and comparing it to the book helped immensely to grasp some of the terminologies and concepts presented in the chapter. Some visuals would have been nice along with the blog, but since we have the book we could just look there. It is quite exciting when there are connections between recreational mathematics and real world applications. This chapter is a perfect example of recreational mathematics possibly being applied to the real world. The highest density theory seems very useful for shipping companies or just every day people when packing objects. Along with the others, the lowest density possible theorem seemed a little confusing or irrelevant to real life applications. Overall, I think you did a very good job on summarizing the chapter and being able to portray difficult concepts presented to you in the chapter.
ReplyDelete