Chapter 48: Mathematical Zoo

Chapter 48, Mathematical Zoo, is a particularly strange chapter. This chapter talks about the authors idea of "a zoo designed to display animals with features of special interest to recreational mathematicians". They go on to say that this type of zoo would be both entertaining and instructive. They say it would be divided in to two main wings one for living animals and the other for pictures, replicas, and animated cartoons of imaginary creatures. Many of these Imaginary creatures you would not think possible to exist however Gardner explains how they are not too farfetched. One of these beings is named A Wheeler, illustrated in figure 48.5. A Wheeler is an animal that has wheels instead of feet. Gardner quotes Robert G Rogers to help explain why the idea is not so insane. He quotes that if a wheel was "mounted on a bone bearing joint, with flexible veins and arteries, and a continuous series of circumferential pads(as on a dogs paw), the wheel could be wound back one turn by its internal muscles, then placed on the ground and rotated forward two full turns". If the wheel had a diameter of one foot this process would cause the Wheel to travel 6 and a quarter feet.

                Another organism Gardner would want in his zoo is a microscopic organism called radiolaria. These are one celled organisms that are found in the sea and have astonishing geometrical symmetries. Gardner cites an German biologist Ernst Haeckel who describes thousand of radiolaria in his Monograph of the Challenger Radiolaria. In this book there are 140 plates of drawings that display the geometric details of the different intricate forms of raiodlaria.

                Another portion of this chapter mentioned how the insect room at the zoo would display bees and there use of hexagonal honeycombs.  Scientists such as Darwin have marveled at bees use of honeycombs calling the ability to utilize them "the most wonderful of known instincts," and "absolutely perfect in economizing labor and wax.". While Gardner agrees that honeycombs are a great way of economizing wax he does state that there are better ways of doing it such as with a polyhedral cell.

                In  J. Diamonds "Why Animals Run On Legs, Not On Wheels," Diamond addresses the idea of why it is animals have legs instead of wheels. This article was actually very interesting. It talked about how bikes and other wheeled forms of transportation are actually more efficient than walking. So the question is why haven't animals evolved into having wheels instead of feet or legs. One thing that would have been nice to see in Gardner's Chapter 48 would have been the fact that wheels for feet would have made transportation for many animals near impossible. Such as with an ant, while their terrain looks relatively flat to us to an ant climbing the small hills would be near impossible. This bit of information would have been nice to see in the chapter because it shows the down side of having wheels instead of legs and how it is not optimal for all animals. One thing I was glad Gardner didn't include was the part where Diamond started talking about ancient civilizations. It isn't because this part was difficult to understand it was just the fact that it is irrelevant to the idea of a Zoo.

                In Jorge Luis Borges The Book of Imaginary beings Borges talks about many imaginary creatures, as you may have guessed by its title. This book in itself is interesting however much of it is irrelevant to mathematics. I'm glad Gardner did not include all of these beings due to many of them being irrelevant though I would have liked to see him include one named "The Leveler". This one resembles an elephant and has very wide flat feet. The leveler is supposedly 10 times larger than an elephant and would be used to level ground that was going to be built on. I think this would have been an interesting animal for Gardner to include in this chapter so he could have proved or disproved its ability to exist.

                L. F Toth "What Bees Know and What They Don't Know" is another one of the sources Gardner cites in his works. One thing I wish Gardner would have included was the formula Toth mentions in his work. In my opinion this would have helped to tie the honeycomb to mathematics. One thing I'm glad he included was why some scientists believe bees build the honeycombs in the shape they do. They believe it is less a result of evolution than an accidental product of how bees use their bodies.

Chapter 11: Spheres and Hyperspheres

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 4^th. 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.  

Chapter 41: Induction and Probability

In this chapter the author talks a lot about probability and gives examples of how there are many flawed hypothesis’s in relation to probability experiments. The author also talks about induction and confirming laws set in continuous patterns. This chapter is full of examples that are given to try and explain each concept. The first example tells us as the reader of a continuous- never ending carpet with a certain pattern on it. On this carpet are billions of tiny triangles and whenever a blue triangle is found, this blue triangle ha a red dot in one corner. The example goes on to say that after finding thousands of blue triangles, all with red dots in the corners, the experimentalists conclude all blue triangles have red dots. This conclusion is drawn without finding every blue triangle. Would you consider this to be true? How would you know for sure without completing the complete experiment? These are some of the questions that came to mind when reading this. Gardner goes on to explain how if there are no counter examples of the hypothesis, each example is a confirming instance of the law. Many more examples are given throughout the chapter about probability and drawing conclusions from these experiments. The base of all these examples given in the text explains to us as the reader that we can’t draw conclusive outcomes from certain tests when looking at the bigger picture. The best example given is the one at the end of the chapter. This example gives you the three tables, A, B and C. On all of these tables are two hats, a black hat and a grey hat. In each hat are a bunch of poker chips, either coloured or white. On the first table (table A) the black hat contains 5 coloured chips and 6 white chips. On the same table, the grey hat contains 3 coloured chips and 4 white chips. Your objective is to draw a coloured chip, which hat will you pick out of? The black hat is the better choice because the probability is higher (5/11 vs 3/7). The same is also true for choosing at table B. You would pick out of the black hat for a better chance to draw a coloured chip (6/9 vs 9/14). Now suppose you go to table C where all the chips from the black hat in table A and B are combined to get a total of 11 coloured chips and 9 white chips for the black hat on table C. The same is done for the grey hat and you now have a total of 12 coloured chips and 9 white chips. Now your choice will be different. You would pick for a coloured chip from the grey hat because its probability is greater (12/21 vs 11/20). This situation is called Simpsons Paradox by Colin R. Blyth. This is why you can’t draw conclusions until all experiments are done. This example contradicts the first example with the blue triangle and the red dot in a way because one would imagine you would pick the black hat on table C after picking it on both tables A and B, but after doing the full experiment the hypothesis is actually incorrect. So why should this be different in the case of the blue triangle with the red dot in the pattern on the carpet? This is a thought that crossed my mind after reading the chapter. 

After reading the chapter and the reference on Simpson’s paradox, I would of like to see some graphic representation illustrated in the test written by Gardner. I think it would of been a good way for visual learners to see the difference in how the paradox works. It would’ve been an easier concept to grasp if I was able to see how the averages of the individual experiments vs the combined experiments were shown on a table or a graph. On the other hand, I am happy that Gardner omitted Blyth’s boundaries for Simpson’s Reversals. I believe these boundaries would have made the example very confusing and very mathematical instead of explaining it so we as non-mathematicians could understand it. 

Blyth notes that from a mathematical standpoint, subject to the conditions

P(A/B&C) ≥ δ . P(A/~B&C)

P(A/B&~C) ≥ δ . P(A/~B&~C)

with δ ≥ 1, it is possible to have

P(A/B) ≈ 0 and P(A/~B) ≈ 1/δ.

These are Blyth’s boundaries for Simpson’s Reversal. 

The authors of “Sex Bias in Graduate Admissions Data from Berkley” show their data in a table which makes it easier to read and view thoroughly. Much of the statistics shown in the book by Gardner are embedded into the actual text, thus making it a little more difficult to read and gather properly. Using a table example such as the one of admitted and denied male and female students like the authors of “Sex Bias in Graduate Admissions: Data from Berkeley”, it would make the chapter more organized and a better read. Having said that, I am happy that Gardner did not include the formula for putting together the graph that shows a four-cell contingency for the male and female students. I think it would have been unnecessary and confusing if it was added. 

Lastly, I am happy that Gardner omitted the triplet of confirmation concepts.

1. Classificatory- e confirms h.
2. Comparative- e confirms h more than e’ confirms h.
3. Quantitative- the degree of confirmation of h on e is u.  

I think this concept was not important to the style of the authors explanation of the theory and would have made the chapter much harder and more complex. 

Overall, the chapter gave many good examples of the theory it was explaining which made the whole concept much easier to think about and process. These examples also gave the chapter a more fun and imaginative feel to the other ones I have read, making it a more enjoyable read. 

Chapter 20

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.

Chapter 22 Nontransitive Dice and Other Paradoxes

TABS The first thing I noticed as I started reading is that both chapter I have read so far have some sort of religion theme. This chapter’s main concern however was the topic of transitivity. It is the way we can rank things and compare them. If A is lighter than B and B is lighter than C, than A must be lighter than C. This logic thought does not work when it comes to nontransitive dice games. Even if you let your opponent pick their die first, you can still have 2/3 advantage to winning. The way we think that something is “more likely to win” is the cause for this commotion. In figure 22.1 it wouldn’t matter if your opponent chose a die with larger numbers, as long as you get a die that has more average numbers. 3 8’s offer a greater probability than 2 12’s.  When the chapter got into the probability of card deck color it reminded me of the games we play in class. The one game which involved 4 cards, 2 red and 2 black, and we guessed what the next color was going to be. And although I was terrible at this game, it made me realize how probability impacts our daily lives. Gardner also writes about Pascal’s Wager in life to compliment this probability discussion. He says that probability, trusting what we put our belief in, is exactly like this Wager. Just like in Pascal’s Wager with God, we all make wagers of life on a day to day basis. We took the wager to come to this college, some of us take the wager to believe in God, and others put a wager in sports.

TABS The James Cargile article had a very nice outline of what Pascal’s Wager truly is. The Cargile article states that the reason for Pascal’s Wager is not to say there isn’t a God, but instead to open up that question for debate. It says that there may or may not be God, but we have to take the chance in what we think. The part of the article I enjoyed the most was that Cargile included that many people who follow some faith are not doing it with a goal in mind, instead they are doing good just because it is outlined in in their beliefs. The author did a great job and integrating this part of the chapter, pulling from multiple sources, and not making it sound too pretentious.

TABS While reading the “Deciding for God,” I really like that Gardner didn’t include Turner’s hypothesis explanation. I found it to be extremely confusing, and didn’t explain Pascal’s Wager as clear as the Cargile article. Turner’s research required more background knowledge that I can only guess was given in the earlier pages of his paper. Turner’s paper was extremely math heavy, which made it hard to understand

 TABS The Richard Savage article entitled, “The Paradox of Nontransitive Dice,” is the prominently used in the book chapter. Using the 3 dice game example from this article perfectly explained the use of nontransitive dice. I also like how Gardner let the reader decide if they wanted the answers instead of just writing a bunch of math inside of the book. It made the chapter seem to flow more than if he included all the math that was present in Savages’ article.  

Chapter 32: Paper Folding

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

Chapter 25: Aleph-Null and Aleph-One

Chapter 25 focuses on set theory. Set theory is described as “the branch of mathematics that deals with the formal properties of sets as units and the expression of other branches of mathematics in terms of sets” (Google). It begins with Paul J. Cohen of Stanford University finding an answer to what was defined as “one of the greatest problems of modern set theory” (Gardner 327): Is there an order of infinity higher than the number of integers but lower than the number of points on a line? Georg Ferdinand Ludwig Philipp Cantor defined this infinity of the integers are aleph-null or aleph sub 0. An example that Gardner used for aleph-null was given a set counted 1,2,3 and so on and put in prime numbers in one-to-one correspondence with the integers. The prime set is an aleph-null because it is countable. Another example that was used was the method that Charles Sanders Peirce created. Starting with the fractions 0/1 and 1/0. If you add the numerators and denominators of the two you would get 1/1 and that would go in between the starting fractions. If you repeat this, you would continually put the new fractions in between either the 1/0 and 1/1 or the 1/1 and 0/1. If continued, each rational number will only appear once and in its simplified form. While doing this, you will see that each fraction that appears on either side of 1/1 will have its reciprocal on the opposite side. This set is also an example of aleph-null.

                  Gardner goes on to other examples such as the card and 3 objects method which he says could go over aleph-null but dials it down to make it countable in Figure 25.1. What he does is say every card that’s the same color on a diagonal is turned over. But this still does not create a subset that can be on the list because its nth card is not the same as the nth card of subset n. The set of all subsets of an aleph-null is a set with 2 raised to the power of aleph-null. This is proof that a set like this cannot be matched one-to-one with counting integers because it is an uncountable infinity/ a higher aleph.

                  Cantor also proved that the set of real numbers is uncountable as well. It is described as given that there is a line segment going from 0 to 1. Rational fractions are in between and obviously in between the rational points being an infinity of other rational points. Following the card demonstration and every facing up card is replaced by 1 and every face down card is a 0. When you put a binary point in front of each row, there will be an infinite list of binary fractions between 0 and 1. Cantor proved that the three sets- the subsets of aleph-null, the real numbers, and the points on the line segment have the same number of elements. He called this C or “the power of the continuum”. He also called it aleph-one, the first infinity greater than aleph-null.

                  The distinction between aleph-null and aleph-one is as follows: aleph-null is countable natural numbers while aleph-one is countable ordinal numbers. The distinction is important for geometry because if an infinite plane is encounters that is tessellated with a polygon, the number of vertices would be aleph-null. When using ESP symbols, all but one can be drawn aleph-one times on a piece of paper.

The plus symbol cannot be aleph-one replicated.

                  Physicist, Richard Schlegal, tried to relate both alephs to cosmology by contradicting the steady-state theory (the universe has always existed and has always been expanding with hydrogen being continuously created [Google]).

                  One of the articles in the bibliography was “Non-Cantorian Set Theory”. The article gives an example about the counting numbers and the prime numbers like Gardner included on page 328 but the authors also included the fractions. I’m glad Gardner did not include their fractions because it makes the whole concepts more complicated than it needs to be. The authors touched upon the idea that there are an infinite set between aleph-null and aleph-one. The authors went more in depth with this concept and Gardner did include this in chapter 25 and I’m glad he did so that the whole chapter wasn’t so one sided towards a theory that has not been fully proved yet.

                  Gardner also included the Farey Sequences which is the explanation for the fraction and their reciprocals set. He grabbed some of these ideas from “Farey Sequences” from the Enrichment Mathematics for High School.

                  Lastly, Gardner used theories described in “The Problem of Infinite Matter in Steady- State Cosmology” by Richard Schlegal. Schlegal describes how the number of atom spaces begins with aleph-null and exponentially increasing. Gardner includes this idea in chapter 25 and goes on to say that the as the universe increases then so does the number of infinite alephs. I’m glad he included this because it makes the idea of increasing infinite numbers easier to understand by connecting it to the always increasing universe which we presume to be infinite as well. I wish he included the equation for this idea such as Schlegal did on page 21. He represented it by N = No e¯bt. Gardner did not attempt to describe this equation even though it goes hand-in-hand with the whole concept of expansion where N equals the volume of atom spaces (aleph-null) and b is constant creation/expansion.

                  I thought that this chapter was hard to understand at first but after reading it a few times, the idea of a number being larger than infinity became easier to perceive with the connection to sets and statistics as well.