This chapter is titled “Paper Folding” and opens up
with the problem of determining the number of ways in which one can fold a pre-creased,
rectangular map. Gardner elaborates, explaining the possible permutations that
each cell (or, individual rectangle created when the map is folded) can
achieve. He later explains “recursive” and “nonrecursive” procedures which
essentially refer to being repetitive or not in regards to the “1 x n rectangle”, a problem involving the
folding of a strip of stamps along their perforated edges. This problem
addresses the thousands of probable permutations that can occur by folding the
stamps every which way, creating a variety of different combinations by use of
factorials. For example, when each stamp is numbered “1, 2, 3, 4”, some
combinations include: 1234, 2341, 3412, 4123, etc. Since there are four stamps,
by use of the factorial formula, there are 16 total ways in which the stamps
can be folded.
Also
discussed in this chapter is the “amusing pastime”, as described by Gardner, of
ordering six-letter words on a 2x3 map reading from left to right and from the
top down, the object being to discover the scrambled word. Similarly, paperfold
puzzles include 5 images on a 3x3 square that can be folded into a specific
picture. The example given in the chapter includes a picture of Hitler, Mussolini,
Tojo, and two jail cells. When folded in the correct ways, one will find one of
the dictators behind bars.
An
important facet of this chapter is the explanation of the unpublished, yet
quite famous and somewhat unsolved Beezlebub puzzle derived by Robert Edward
Neale, who was a Protestant minister, professor, origami extraordinaire, and
magician. Neale also discovered the way to master the folding of the
tetraflexagon. Much of this chapter is a step-by-step narrative of directions
on how to fold the various paper figures.
Upon
searching for the sources listed in Gardner’s bibliography for Chapter 32, I
had no luck finding a single one of the books in the library. Subsequently, I
researched a few articles online with relative concepts mentioned in the
chapter. Because Gardner’s Paper Folding chapter includes mainly just
instructions on how to construct various paper structures, I feel he could have
described and defined the tie to mathematics for more interesting structures - the
paper crane in origami for example. I discovered and article that mentioned a
concept called “Huzita’s Axioms” which I found quite interesting and is
something I feel would have been intriguing for Gardner to include in the
chapter.
Robert J. Lang elaborates
on Huzita’s Axiom, created by Humiaki
Huzita and Benedetto Scimem, reporting that this axiom “identified six distinctly different ways
one could create a single crease by aligning one or more combinations of points
and lines (i.e., existing creases) on a sheet of paper. Those six operations
became known as the Huzita axioms. The
Huzita axioms provided the first formal description of what types of geometric
constructions were possible with origami”. I feel this axiom provides a good
base for the understanding of more complicated origami.
Conversely, I am happy Gardner didn’t continue to include
specific and lengthy instructions on how to fold additional shapes, let alone
include the complicated, and sometimes even lengthier, mathematical formulas
which they may entail. I feel even the included instructions are a bit
monotonous and tiring after a while, neglecting to really focus on explaining
in depth how to actually understand the roots that tie them to mathematics.
From my extended research, I read of the endless shapes and the paper folding
instructions that follow. I’m also not particularly skilled at equations
involving complicated function formulas, which I came across as well, so I’m
glad Gardner didn’t go to those great lengths in this chapter.
Bibliography
1. Ahler,
Franz G. and Nilsso, Johan. "Substitution Rules for Higher-Dimensional
Paperfolding Structures." 21 Aug. 2014. Web. 26 Oct. 2015. http://arxiv.org/pdf/1408.4997.pdf
2. Hull,
Tom. "Origami & Math." Origami & Math. N.p., n.d. Web.
27 Oct. 2015.
3. Lang,
Robert J. "Huzita-Justin Axioms.". 2014-2015. Web. 27 Oct. 2015. http://www.langorigami.com/science/math/hja/hja.php
4. Weisstein, Eric W. "Folding." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Folding.html
I think that Monica’s chapter on paper folding was very interesting. I have never in my life been good at origami or anything like it, so I was happy to see that there were a lot of complicated formulas that went along with it. It made me feel slightly better about myself. I agree with Monica in that it would have been nice if Gardner had gone into greater detail of the mathematical significance of the paper folding. I understand that the formulas described the possibilities that came with the folding of the map, but it would have been nice to have a comprehensive explanation. I would also have liked to know more about how you can apply the problem of the number of possibilities to fold the edges of a stamp in to the map problem.
ReplyDeleteI also thought that the paper-folding puzzle games described in the chapter sounded fun. It made me wonder what the largest size of paper that one could use to create one of these games without it becoming excessive. I think that it would be a cool exercise to try and make them in class one day.
Gardner did seem to drone on at times as he went through explanations and instructions. However, I really like how he made it simple for his reader to understand how to create the things he described.
Origami was always something that had always interested me even though I was never really good at it. I always knew there was an art to it, but did not realize how much mathematics there was behind why this paper folding technique is so amazing. I found the different formulas and techniques of this very interesting. It honestly surprised me how many ways and formulas there can be in order to make different things from folding paper. I liked this chapter because it was the first one I have read so far that had a large relation to math, but could also be a fun past time. I can definitely see how such topic could be used recreationally. I thought Monica did a great job of explaining the chapter and making it less confusing for those reading this blog/chapter. I believe Gardner did a great job of including things Monica found in articles online. The chapter was kind of confusing at times, but nothing that was too hard to comprehend. Paper folding is definitely more complex than one would believe at times, but overall this chapter was very interesting and I would like to learn more and look into this topic a bit more.
ReplyDeleteI’ve never thought much about origami and all of the mathematics and tricks behind it. I am surprised by how much math it takes to fold a piece of paper. It makes sense that the formula would be 1* n rectangle and your explanation of it made it easier to understand. I am surprised from how many formulas and techniques go into folding a piece of paper not just into origami. Origami was invented in 105 AD and it shocks me that such a complicated technique with numbers could have been perfected without the technology we have now. This chapter was one of the easier ones to understand conceptually but there were still things I didn’t quite understand such as the “amusing pastime”. Monica’s explanation made it easier to distinguish the significance of the folding in order to place one of the dictators behind bars. I agree with Jo about the fact that Gardner seemed to drag on when this is an easier topic to explain. Some his explanations made the concepts harder to understand but overall the chapter was interesting. I can honestly say I did not know how much technique there was to paper folding nor the math behind it.
ReplyDeleteI’ve never thought much about origami and all of the mathematics and tricks behind it. I am surprised by how much math it takes to fold a piece of paper. It makes sense that the formula would be 1* n rectangle and your explanation of it made it easier to understand. I am surprised from how many formulas and techniques go into folding a piece of paper not just into origami. Origami was invented in 105 AD and it shocks me that such a complicated technique with numbers could have been perfected without the technology we have now. This chapter was one of the easier ones to understand conceptually but there were still things I didn’t quite understand such as the “amusing pastime”. Monica’s explanation made it easier to distinguish the significance of the folding in order to place one of the dictators behind bars. I agree with Jo about the fact that Gardner seemed to drag on when this is an easier topic to explain. Some his explanations made the concepts harder to understand but overall the chapter was interesting. I can honestly say I did not know how much technique there was to paper folding nor the math behind it.
ReplyDeleteThis chapter was interesting to say the least. I have always found paper folding/ origami very cool, but I never knew it had math behind it. It seems the math behind it has gotten stronger throughout its history by being able to figure out how many ways you can fold a map, which took me a while to realize how complex that problem actually is. When the chapter moved into the problems that involved letters and numbers in the squares, I became more intrigued. I think these would be fun problems or games, but only for the easier ones since some are very complicated. I like everyone else wish that Gardner could have provided more information on the history or why this is important. It seems it is only for paper folding? That could be the case, but it would definitely be interesting if it had some other use for it. Overall, this chapter was a little hard to pick up, but when I did the chapter was a lot of fun. I enjoyed the chapter just wish it had more useful information or real life situations. It was almost like it was just another kind of puzzle for fun.
ReplyDeletePersonally I have always found folding paper, or more specifically origami to be beautiful and very creative. I never had the touch to be able to do it myself but I always liked watching people do it. Chapter 32 talks about a few very complex methods of folding paper, the example of the tetraflexagon is something I never thought possible to do with paper. Being able to fold a piece of paper so many times, then to be able to pull on two of the edges to get four 1-cells on the top and four 2-cells on the bottom is amazing to me. Another example that I found to be interesting was the Map fold problem with the faces of Hitler, Stalin, Mussolini, and two pairs of jail cell bars. When done properly you are able to get one of the dictators to be behind bars, being able to put three of the worst men behind bars by simply folding a piece of people is hilarious and I love the idea behind it. I agree with Monica, I am happy the author did not include more lengthy instructions on how to fold these complex patterns. I would have like to see more of the simpler but still challenging fun ones like the 2x4 map-fold puzzle.
ReplyDeleteI have always been amazed by some Origami that people can make. This chapter seems like a perfect example of recreational math. I knew Origami could be extremely complicated and the possibilities could be endless but I never thought about it in terms of mathematics. Its amazing how complex math concepts can be applied to such an ancient folding technique. It seems that the math portion of Origami can be just as confusing or difficult as the actual folding! It was nice to see step by steps when folding the complex shapes. What made me intrigued is when the chapter introduced the problems and puzzles with the folding and letters. It made me ponder on these problems for quite a while and trying to imagine it in different sizes and such. I agree with the others that Gardner could have tried to be clearer and included more information. The articles Monica found definitely seemed to have helpful information that could have been included within the chapter. Monica seemed to do a great job on writing about the content of her chapter. Reading what she said made reading the chapter a lot easier when breaking down the main concepts of the chapter.
ReplyDeleteAfter reading this chapter and Monica's summary and thoughts, I've found this chapter to be the most relatable to our course- Recreational Math. The idea of being able to fold a paper to create number combinations, words, and pictures is incredible and a very unique skill. This is the definition of recreational in the sense of it being a hobby and something people enjoy doing in their spare time. This is evident in Monica's response as well where she states it to be an "amusing pastime", quoting Gardner. Gardner also does a good job of keeping the steps simple enough to be able to follow and try and enjoy on your own. I agree with Monica's point on if Gardner had added more complex steps and equations to prove his point of how many different ways you can truly fold a paper. I believe that this would of created a very tedious and cluttered chapter instead of a fun and interesting one as it is now. This chapter was a fun and simple way to learn about equations to create something that amuses people and can amaze people at the same time. I think it was a great chapter to read and look at due to the fact it incorporated math in a fun and different way.
ReplyDeleteOrigami is a very interesting topic to think about if you try to do it mathematically. There are just so many ways that you are able to fold a paper and have it make something else. While reading this chapter I didn't have to go back and re-read anything, because it was all straightforward. As Monica mentioned I am also grateful that they didn't have very long and lengthy formulas to explain paper folding more in depth. On the flip side of that though I would like to know more about the subject and all the math that is related to it, because you can just do so much with this and go in many different directions. It would be interesting to see some of the math formulas used to do some of them. I feel like this is one of the things we would be capable of doing the class and everybody would know what they were doing. Some of the chapters are really complicated and don't make a lot of since, but this chapter is simpler and something we could do in the classroom. it is quite interesting when reading all these chapters not just this one is how much math is used for things that you didn't know math was used for.
ReplyDeleteI decided to read this chapter after reading the title, I remember in middle school seeing all the kids who knew how to do origami and paper folding and I always thought it was really cool but never really thought about the math side too it. I understood what a recursive function is however I was unsure of the 1xn formula. One part of this chapter I would have liked to have seen a visual for is the part with making an anagram with the six-letter words on a 2 x 3 map. I believe I understand what Gardner is describing in this as he gave a few examples, however I believe that a visual would have been nice to help me fully grasp the concept of it. I did appreciate the mention of in this chapter of the puzzle from World War II since that is one of my favorite history subjects. Another part of this chapter that amazed me was the section about Neale and the tetraflexagon. I am not going to pretend like I can imagine what that shape may look like but after looking at the visuals and reading this portion of the chapter I am fairly impressed that someone could fold a shape like that out of a piece of paper. Overall this has been one of my favorite chapters since it had a fun topic and was fairly easy to understand.
ReplyDelete