Chapter 19: Knots

This chapter is about the different types of knots and its components. The general definition for a knot is a closed curve that is embedded in a three dimensional space and it is usually modeled with rope. The study of knots is a mix of different mathematics such as algebra, geometry, group theory, matrix theory, number theory, and more. One of the simplest knots is the trefoil knot which is when the ends of the rope are joined. The reason that it would be the simplest knot is because it can be diagrammed with a minimum of three crossings. The next simplest knot after the trefoil knot is the figure eight with four crossings. There are many ways to divide knots into different classes. One of the ways is alternating and non-alternating. Alternating knots can be diagrammed so that it is possible to go over and under the crossings. Another way of division is prime and composite knots. Prime knots cannot be manipulated to make two or more separated knots. The last basic binary division is called the invertible and the non-invertible knots. Invertible is being able to manipulate the rope so that the structure stays the same but the arrows of direction point the other way. A pretzel knot, on the other hand, has no crossings which proved that all pretzel knots are noninvertible if the crossing numbers are distinct odd integers with absolute values that are greater than one.

In L. Kauffman’s “Knots and Physics” book, she was using a lot of examples to show the different types of knots that there were. She wasn’t using a small variety and I would have wished that Gardner would have included more examples such as a clove hitch and a bowline. In Kauffman’s book she also explained the physics behind the knots not just the mathematics, which is what I wished that Gardner would have included along with all of the different types of mathematics. Friction and tension seem to be some of the main components in making the knot form. While it would have been cool to see all of the different types of knots in the book, I am glad that Gardner didn’t put in a step-by-step for every single structure. Some of the structures are basic and other make you want to think but as I was reading Kauffman’s book I wanted to try to solve the knots myself but it was a bit hard to do so since she included the instructions in both the writing and the images.

In the book “Introduction to Knot Theory” by R.H. Crowell and R.H. Fox, there is a lot of explanation for the mathematics behind the knots. Gardner briefly talks about the mathematics but he does not go into greater detail about how mathematics can be used to form knots or to work with knots in general. As a reader, I would wish for more information regarding how exactly mathematics is involved. I am glad that Gardner didn’t include anything about wild knots because as I was reading about wild knots in Crowell and Fox’s book, it made me very confused about the wild knots and what their purpose is to the talk of knots.

In K. Murasugi’s “Knot Theory and Its Applications,” the author goes in depth on what an actual knot is and the trefoil knot. I wished that Gardner would have explained more about a knot itself rather than give a brief explanation and jump onto different divisions. I would have also wanted for Gardner to talk more about the trefoil knot since it is the simplest knot. The short summary that Gardner provided made me a little confused about the trefoil knot; however, Murasugi’s explanation was a bit longer and more in depth and led me to understand more about the trefoil knot. One more thing that I wished that Gardner had included was a better explanation of mirrored “images” with knots that involve the different mathematics. I’m glad that Gardner omitted parts about the elementary knot moves because that part of Murasugi’s book confused me. There is a lot of mathematics being thrown into that one concept of the elementary knot moves that was hard to follow.

This chapter, overall, was very interesting to read and after reading more into the three author’s works besides Gardner’s, it had made me want to further look into the different types of knots that exist and how some of them are formed. I wanted to try forming them on my own as well. It also had made me want to read more into the more in-depth mathematics and the different formulas used to make these knots.

Chapter 48: Mathematical Zoo

This chapter deals with an idea of creating a so called zoo that displays animals of interest to people. This so called zoo would be divided into two sections: one side with all the living creatures and the other with creatures that are imaginary. One room in the living section would contain microscopic animals such as the radiolaria, which are one-celled organisms in the sea. this creature can be described as spherical, with six claw-like extensions. one particular kind of radiolaria is the Aulonia Hexagonia. This sphere is covered with regular hexagons packed closely to each other. But there is a problem with this as it raised the question of whether it was possible to cover a sphere with hexagons. The correct answer is no and is explained with Euler's formula of F+C-E=2 where the letters stand for faces, corners, and edges. Someone did claim to have seen this kind of radiolaria even though it's not possible and it was clarified when if you zoom close enough, you see the shapes with less than six sides. Other examples include the many viruses that are shaped like icosahedra, a solid with three dimensions of triangular faces, and the twisted horns of certain sheep and goats.

Another section would contain animals that violate bilateral symmetry. Such examples are the Crossbill, a small red bird with its upper and lower beaks crossed. The aquarium of this zoo would have a tank full of male fiddler crab, which has either an enormous left or right claw. The flatfish is even weirder as it is bilaterally symmetric when they are young, but as they grow older, one eye moves over the top of its head to the other side. In a different tank, there would be a species of hagfish. This fish looks like an eel, has four hearts, and has teeth on its tongue. It's unique in that it can tie into a knot which is helpful to escape predators or to tear food from a large fish. The insect room would contain bees and wasps. Bees have a unique ability to create honeycomb nests that would use the least amount of wax and hold the most amount of honey inside. This is something that many people are shocked to know about bees and their amazing skill at this.

The Imaginary section of this zoo contains creatures that can only be thought of as fantasy. Some include the palindromic beasts, which have identical ends. An example of this is a palindrome dog, with heads at each end and legs. Imaginary creatures on wheels for transport would be another room in this section. Examples would be the Wheelers, a race of four-legged humanoids with wheels instead of legs and an Ork, a species of bird with propellers in the back that enable it to fly. In a different room, it would contain the Woozy. It is a block-headed, thick-skinned, dark blue creature with its body parts shaped like blocks. This beast behaves friendly unless a certain phrase is said which is "Krizzle-Kroo" in which it turns angry that causes its eyes to shoot fire. Another topic mentioned is the Buckyball. This carbon molecule is shaped spherically and resembles a soccer ball. The buckyball is the world's smallest atom shaped like a soccer ball that has 20 hexagon faces and 12 pentagons. This molecule was created my chemists in 1989 and belongs to a class of symmetrical molecules called fullerenes.

In Jarrod Diamond's article "Why Animals Run on Legs, Not On Wheels", he elaborates more on how animals haven't evolved to be able to have wheels to use to move around as opposed to human beings. It would have been good if Gardner included the three main reasons why animals are better off without wheels to show the other side of the argument. I'm glad he didn't include the history of how much we use these wheels for everyday life as it would distract from his main point on imaginary creatures with wheels.

In the article "Mathematics and the Buckyball" by Fan Chung and Shlomo Sternberg, it explains how this molecule drew interest from many different scientists in the many different areas of their expertise such as chemists, physicists, and chemical engineers. What made it so impressive was how it was symmetric that had various properties that can be explained with mathematics. It would have been good if Gardner included the different topics used to explain the buckyball and its properties. It was good that he didn't add in the complex mathematics involved with the buckyball as it would've confused the reader.

In the article "Great Balls of Fire" by Richard E. Smalley, he tells the background in creating the buckyball and the various methods used to make it work which eventually led to the creation of it. It would have been good if Gardner had added the moment when the chemists had succeeded in creating the buckyball. I'm glad he didn't add in the long process it took to create it has it would have made the reader less interested in the concept and would've been confusing to understand the details.

This chapter went into great detail on the various examples used in this mathematical zoo. It was interesting to read about the unique living creatures as well as some fantasy creatures that would be part of this so called zoo. Even though there were some topics that were hard to understand at first, rereading it made it easier for me to understand it better. 

Newcomb's Paradox

Chapter 44: Newcomb’s Paradox

Newcomb’s paradox is a thought experiment involving a game between two players, where one of these players is able to predict the future. The game is mainly operated by the “Predictor”, who is supposedly always accurate and incapable of error. The discussion by Nozick says only that the Predictor's predictions are "almost certainly" correct, and also specifies that "what you actually decide to do is not part of the explanation of why he made the prediction he made”. I wish Gardner would have included more examples and situations to make the paradox easier to understand, like I found in one of my sources. Franz Kiekeben uses the example of a  hypochondriac wanting to be a basketball player to show that there is no casual interaction in the case of Newcomb’s paradox. However, Kiekeben also gets very into detail with the definition of prediction, which I am glad Gardner did not include in the chapter because it doesn’t seem very relevant. I found in my source from Texas A&M University that free will also plays a big role in this paradox, which was not really mentioned in the book. I feel like human beings and their ability to make their own independent decisions is an important part of the game. The Texas A&M University source also added Newcomb-like problems which I found to be irrelevant as well. In my opinion, Newcomb’s paradox is problematic on several levels. The fact that a physical impossibility is included (the ability of predicting a human action) shows that relative prediction is possible indeed.

The Game

The player of the game is presented with two boxes, one transparent (box A) and the other opaque (box B). The player is allowed to take the contents of both boxes, or just the contents of box B. Box A hold $1000, which is visible. The contents of box B are not visible however, and they are determined before the start of the game. The Predictor makes a prediction as to whether the player of the game will take just box B, or both of them. If the Predictor predicts that both boxes will be taken, box B will not contain anything. But if the Predictor predicts that only box B will be taken, then it will contain $1,000,000. When the game begins, and the player is called to choose which boxes to take, the prediction has already been made, and therefore the contents of box B have already been determined.The problem is called a paradox because two strategies that both seem logical give conflicting answers.The first strategy is that regardless of what prediction was made, taking both boxes will give the player more money. However, even if the prediction is for the player to take only B, then taking both boxes yields $1,001,000, and taking only B yields only $1,000,000. Therefore, taking both boxes is still a better choice, regardless of the prediction made.

References:

Galef, Julia. "Newcomb’s Paradox: An Argument for Irrationality." Rationally Speaking Aug.-Sept. 2010: n. pag. Print.

Mackie, J. L. “Newcomb’s Paradox and the Direction of Causation”. Canadian Journal of Philosophy.

"Newcomb." Franz Kiekeben. N.p., n.d. Web. 27 Oct. 2014.

"Newcomb's Paradox." Texas A&M University. N.p., n.d. Web. 27 Oct. 2014.

Godel, Escher, Bach

Kurt Godel, M.C. Escher, and Johann Sebastian Bach, were the three men involved in the “trip-lets” And essentially what a “trip-let” is, it casts a shadow of three letters that have been carved into two blocks that are perfectly aligned above each other. Yet what is very interesting about the shadows that are casted is that top block casts a shadow of “GEB” (Godel, Escher, Bach) but, the bottom block casts a shadow of “EGB” (Eternal Golden Braid). This image of a “Trip-let” was the cover of “Godel, Escher, Bach: an Eternal Golden Braid” (660). The author is known as Douglas R. Hofstdter. Dr. Hofstadter had the idea to write a pamphlet about Godel’s theorem before the book but, he realized that he had to include Bach and Escher because of their work.

“the letters (preferably uppercase) must all be conventionally shaped, and they must fit snugly into the three rectangles that are the orthogonal projections of a rectangular block” (661).

Hofstadter has written in the book in which has both “Achilles” and “Tortoise” start off with the dialogue. And the structure of each dialogue is to mimic a composition constructed by Bach. They are both placed in Zeno’s paradox, “Achilles must catch Tortoise” (663). And yet this is just the first dialogue in Hofstadter’s book but, for the second dialogue it incorporates both Achilles and Tortoise again. What Achilles attempts to prove is “Z”. Yet although he attempts to prove it Tortoise refuses to believe his statements, until he sets rule which tend to prove his proof. This proof is typically known as a theorem of Euclid’s. Another dialogue within the book is know as “Crab-Canon” which has both Achilles and Tortoise’s sentences interspersed. Eventually they both use the same sentences but in reverse order where the Crab then appears to knot the two essential ideas together. The most substantial between these three characters is that the first letter of their names are three of the letters of the four nucleotides of DNA. And two of the nucleotides that are paired together are adenine and thymine; which is essentially just like both Achilles and Tortoise in the book.

Further into the chapter the term “formal system” arises. Hofstadter gives an example in his book using symbols such as M, I, and U, which then construct theorems. Yet there were rules such as:

“1. If the last letter of a theorem is I, U can be added to the theorem. 2. To any theorem Mx, x can be added. (For example, MUM can be transformed into MUMUM, and MU can be transformed into MUU.) 3. If III is in a theorem, it can be replaced by U, but the converse operation is not acceptable. (For example, MIII can be transformed into MU, and UMIIIMU can be transformed into UMUMU.) 4. If UU is in a theorem, it can be dropped. (For example, UUU can be transformed into U, and MUUUIII can be transformed into MUIII.)” (666)

There is also one “axiom” in the system which states that “M” must always be placed in front of the entire theorem.

The book then closes with “Six-Part Ricercar” “which is simultaneously patterned after Bach’s six-part ricercar and the story of how Bach came to write his Musical offering” (668). In this dialogue there is a computer in which pioneers Turing. Babbage another character “improvises at the keyboard of a flexible computer called a “smart-stupid”” (668). And there is a fued between both Babbage and Turing which consisted of figuring out which was real and which one was the program. They both decided to play the “Turing game” which is by asking shrewd questions to figure out who is real. At this point Hofstadter walks into the scene to inform all of the characters that they are all part of his imagination but he also states that he is fake as well.

A reference that that had been included in this chapter is “Exploring the Labyrinth of the Human Mind”. This had written about how the brain naturally makes a decision. It essentially states that when a human is reconstructing a word when they have a jumble of letters. And so the difference between humans and computers is that humans can figure out the word when the letters are jumbled together and also humans can also predict the next number in a sequence. These two actions are considered to be done instinctively and essentially computers cannot do what humans can do naturally. Yet the information on what humans can do compared to computers was interesting I had also found the GEB shadows interesting as well. That’s why when reading this reference I had found the extra information on the GEB shadows interesting.

CHAPTER 38: THE NEW ELEUSIS

The new Eleusis is a card game that is played by at least four to eight players (beyond 8 the game becomes long and chaotic). The game is played with an ordinary deck of cards, and was named for the ancient Eleusinian mysteries, religious rites in which initiates learned a cult’s secret rules. It was invented by Robert Abbott and it is of special interest because it provides a model of induction, which is the process at the heart of the scientific method.

The Start

The dealer has to make up a secret rule which is confined to the sequence of legally played cards in standard play, and must be neither easy nor too hard for players to figure out. When the secret rule has been established, two standard decks are shuffled together and the dealer deals 14 cards to each player excluding himself. The dealer places the starter card on the table and the chosen player begins the game which continues clockwise in a circle.  The cards are placed along the main line if correct (If they conform to the rule)- to the right of the starter card, otherwise on the side line as indicated in the diagram below. The dealer calls the play “right” or “wrong” as the players show their cards, and issues penalty to wrong cards played.

 

The Prophet

In case a player thinks they have figured out the rule, they can declare themselves prophet, in which case they take over the duties of the dealer; dealing cards, calling plays “right” or “wrong”, and issuing penalties to wrong plays. The judgments of the prophet are approved by the dealer, and in case the prophet makes a wrong judgment, he is declared a false prophet and overthrown immediately and the dealer resumes his duties.

In general, players score points by either ridding their hands of cards by making correct plays or becoming successful prophets. Incorrect plays and false prophets receive penalties of additional cards, which reduces their chances of winning. After a certain number of plays, players are expelled and the game ends when either all players are expelled or one player has successfully played his hand. Now the game is scored and the secret rule is revealed.

From “The Methodology of Knowledge Layers for Inducing Descriptions of Sequentially Ordered Events” by T. Dieterich, Martin Gardner has incorporated a brief summary of Eleusis from this masters’ thesis into the chapter, which is good and clear to the reader. It would have been good if he had shown more examples of possible secret rules which are included in this reference to enlighten the reader more about these rules. However, it is good that he did not mention the type of rules and the type of rule which each individual rule represents or steps of making the rule and checking its complexity; this would cause too much confusion to the reader.

Rules of Eleusis are also depicted in R. W. Schmittberger’s “New Rules for Classic Games”, which is a book of rules for many card and board games such as chess, monopoly, scrabble etc. From this Martin Gardner just picked the concise rules of Eleusis, such changes in rules with the presence or absence of the prophet etc. It is good that he did not mention general rules which would include other games to avoid confusing the reader.   

Finally, the article “Simulating Scientific Inquiry with the Card Game Eleusis” by H. C. Romesburg, shows how the game Eleusis relates to the Scientific theory. In the chapter, Martin Gardner mentions briefly that the game Eleusis provides a model for the Scientific method and that it is an analogy to life and the search for truth but does not go into detailed explanation of how this is the case, which is good, because then the reader would lose the meaning of the game they are trying to grasp. However, it would have been good for him to mention some examples that show how Eleusis is an analogy to the search for truth even without going into detail, for example, Rule embedding, hypotheses in Science etc which are mentioned in the reference.

All in all I think the author gave a good brief and understandable explanation of the game Eleusis. The fact that it is brief and concise makes it interesting and ensures less confusion to the reader. I enjoyed reading about Eleusis and would surely like to play it sometime  

Chapter 6

This chapter mainly revolves around Piet Hein. Piet Hein had a number of different professions. One of his big projects was in Stockholm Sweden. The city decided to rebuild a series of old houses in the "heart of the city." They were struggling with this project so they had to ask Hein for some help. He came up with something called a superellipse, which wasn't too rounded or too orthogonal. In the chapter it was described as "a happy blend of elliptical and rectangular beauty." This design worked perfectly for the city and inspired other people to use it for different reasons. After working for the city of Stockholm, Hein was asked to use his theory to make superelliptical desks, chairs, tables, and beds. These were so popular because they had no corners.
The chapter then went on to talk a little bit about Columbus how he was able to make an egg stand by smashing a piece off of it, something that no one else at that banquet was able to do. This was continued when Gardner talked about his proof about how an oval is able to stand up on it's own. He proposed that if the center of gravity of the egg, and the center of curvature of the egg are vertically aligned then the egg will be able to stand up on its own. However Gardner thought that the center of curvature had to be a certain length about the center of gravity. It was proved later by U.S. Navy commander, C. E. Gremer, that the center of curvature can be infinitely high and the superegg will still remain standing.
Gardner was able to get a good amount of this information about Piet Hein from an old article from Life Magazine. This is where he got the information about his work in Stockholm. What he didn't really talk about was how popular he was in his home country, Denmark. I never really got the sense of his popularity from reading the chapter. He was known as a poet, and one out of every Dane owned a volume of his work. (Hicks) He had written over 7,000 poems and was known for many famous quotes. "In Scandinavian countries a clever after-dinner speaker is defined as one who can talk for 30 minutes without quoting Piet Hein."(Hicks) From reading the chapter I had no idea about the impact he had on this part of the country with his poetry. Since it's a math book I suppose that its is okay that this was left out, but still it is an amazing fact about his past. I am pleased that Garnder left about parts of his personal and schooling history however. In the article it talked a little bit about his schooling history, which wouldn't have been necessary for this chapter. It is already clear from reading this chapter that he must have had an amazing education. Its also talked about how he had been married three times and how he wasn't easy to be around. These facts also would have been unnecessary for the purpose that Gardner was trying to convey.
The source he used to learn about the superellipses was not so straight forward. The article from Mathematics Magazine mainly consisted of confusing calculus equations. I was glad that he did not add any of these complex equations in the chapter. There were a few equations in the chapter that were rather easy to understand, but most of the ones in the article were way above my head. The only one I was able to sort of understand was the integral. It used the integral to solve the area under the curve. I was taught this in calculus so I am able to understand the concept. If there were any equations from the article that I would have picked for him to put in it would have been the integral.

J. Allard, "Note on Squares and Cubes," Mathematical Magazine, Vol. 37, September 1964, p. 210-          14
J. Hicks, "Piet Hein, Bestrides Art and Science," Life, October 14, 1966 pp. 55-56.

Non-Euclidean Geometry

Chapter 14 talks about Non-Euclidean geometry, which comes from the Greek mathematician Euclid.  It starts off stating that Euclid’s Elements led to disagreements because of his assumptions, one of them being the non-euclidean geometry. Even to Bertrand Russell, Euclid’s Elements to him was known to be dull and a “tissue of nonsense.” Euclid wanted people to accept his not so famous fifth postulate, the parallel postulate that no line can intersect a point, without providing proof because it was so simple to understand.

 Non-Euclidean geometry consists of two geometries created on axioms. It is a system of definitions, proofs, and rules that define the points, lines, and its planes.  Two of the non-Euclidean geometries are spherical and hyperbolic geometry.  The difference is in their parallel lines.  In the Euclidean geometry, there is only one line through the point that is in the same line and never intersects.  In spherical geometry, there are no lines, and in hyperbolic geometry there are at least two different lines that pass through the point.

This topic of non-Euclidean geometry has been a controversy for many mathematicians for almost 2,000 years, which is to be able to remove the postulate by making it a theorem. For example, in the book “The Colossal Book of Mathematics”, there were many obsessions over this problem. Farkas Bolyai’s son, Janos became obsessed with this problem and did not give up until he could solve it. 

Resulting later on, he convinced himself that not only the postulate was independent from the axioms, but there was also consistent geometry that through a point, an infinite number of lines are parallel to that. It was Gauss, the Prince of Mathematicians and a really close friend of Janos, who actually created the term “non-Euclidean geometry”, which is now known as hyperbolic geometry. 

In the book “Euclidean and Non-Euclidean Geometries”, Marvin Jay Greenberg states that “according to Euclid, two lines in a plane either meet or are parallel. There is no other possible relation.” Gauss pointed out the error that was wrong in Janos’ discovery.  Even though Janos had discovered most of the non-Euclidean research, Gauss was seen as the mathematician to receive credit.    

According to D.M.Y. Sommerville’s book of “The Elements of Non-Euclidean Geometry,” Non-Euclidean geometry consists of infinite areas, from Bertrand’s proof.  The misconception is when applying the principle of superposition to an infinite of areas.

This chapter really had me thinking, compared to the other blogs I have worked on and read.  I had never heard of Non-Euclidean geometry until working on this section.  I found it interesting on how many mathematicians did not put much attention to this kind of geometry because for 2,000 years, no one had enough research and results. Still at this point, I stay in the middle of fully understanding this chapter, but that is why I continue to read on the topic. 

Chapter 22: Nontransitive Dice and Other Paradoxes

Chapter 22 deals with a paradox that relates transitivity and a group of paradoxes that originate from the principle of indifference. Transitivity is a relation such as if a=b and b=c, then a must be equal to c. Bradley Efron, a statistician from Stanford University, developed nontransitive dice. These dice violate transitivity and with any of these dice a betting game can be formed that is contrary to intuition.      

                                                  (Efron’s first set of nontransitive dice)

The four dice are numbered so that the winner has the maximum advantage. Lets say you and an opponent both select a single die. If your opponent picks one first then you have the remaining three to pick from. Both dies are tossed and the person who roles the highest number wins. You come to the conclusion that because your opponent picked first he has an unfair advantage. Regardless of the die your opponent picks you can always pick a die that has a 2/3 probability of winning. This paradox violates common sense and results from the assumption that the relation “more likely to win” must be transitive between the dice.  The red die beats the green die, the green die beats the blue die, the blue die beats the purple die, and the purple die beats the red die.  The probability of winning with the indicated pairs is then 2/3 because there are 36 possible throws on each pair and 24 possible outcomes on witch the first die (from right to left) beats the second die. Efron writes that 2/3 is the greatest possible advantage that can be achieved with his four dice.

The principle of indifference is the classical probability theory that drives from the 18^th century.  “The principle states that there must be no known reason for preferring one of a set of alternatives to any other.” (Keynes, 53) Suppose you have a six-sided die with six different numbers on it. The assumed probability of rolling on of those six numbers would be 1/6.  This analysis rests on practical view.  Now suppose you are given a shuffled deck of four cards, two black and two red. What would be the probability of picking two of the same colored cards?  Most people would think the probability to be 1/2. This is wrong. In fact the probability will be 1/3 because there are 24 possibilities and 8 cases in which there are matching cards. There are more confusing paradoxes than this one out there. The more confusing paradoxes lead to logical contradiction.

One of the more confusing contradictions is given by Sir Harold Jefferys in his book “The Theory of Probability”. The probability paradox goes as follows; suppose there are two boxes. The first box containing one black ball and one white and the second containing two black balls and one white ball. What is the probability of picking a white ball?  “There are five balls, two if which are white. Therefore according to the definition, the probability must be 2/5. But most statistical writers, including, I think, most of those that professedly accept the definition, would give 1/2 X 1/2 + 1/2 X 1/3= 5/12” (Jeffrys, 370) The actual probability contradicts most peoples hypothesis of what the probability was going to be.

Pascal misused the principle of indifference in his book “Pensées”. Thought 233 is worth quoting:

““God is, or he is not.” To which side shall we incline? Reason can determine nothing about it. There is an infinite gulf fixes between us. A game is playing at the extremity of this infinite distance in which heads or tails may turn up. What ill you wager? There is no reason for backing either one or the other, you cannot reasonably argue in favor of either…

Yes, but you must wager… which will you choose? ... Let us weigh the gain and the lose in choosing “heads” that God is… If you gain, you gain all. If you lose, you lose nothing. Wager, then, unhesitatingly that he is.” (Pascal, 66)

Pascal appeals the principle of indifference to a situation that has nothing to do with mathematics. Pascal is not the only one to make this ludicrous relation and he won’t be the last.

            This chapter was interesting because the probabilities given were not expected. The problems given seemed easy at first but ended up being harder than I first thought. I enjoyed the chapter a lot and could relate to it. It taught me to think more deeply into a problem because there is probably more to the problem than you thought.

Keynes, John Maynard. (1921) A treatise on probability, Macmillan and Co.,

Jeffreys, Harold. Theory Of Probability. Oxford: Clarendon Press, 1961. Print.

Pascal, Blaise,Trotter, W. F. ([1958) PenséesNew York : E.P. Dutton,

            

Chapter 47: Melody-Making Machines

Melody-Making Machines are something I had never heard of before or at least I didn’t think I did. In the first weeks of my ‘Computer Science I’ class we learned that all pictures, sound, and video could be expressed by a series of numbers. We were also informed that almost 80% of pictures, videos and sound would takes years and years to code in terms of single digit numbers.

            

 Chapter 47 in “The Colossal Book of Mathematics” brings reason to this subject and explores the different and nearly impossible algorithms to code music and paintings of today. In Science and Music (Dover, 1968) Sir James Jean talks about a curve that can define all symphonies. The sound that will come from this curve will be a very bland music but every now and then there will be a pleasing note. Since all curves can be coded very precisely, all sound can be coded with a very complex algorithm that may not ever be solved.

            

 The use of dice to produce any symphony or a symphony that no one has ever heard has been a problem that many composers and computer wizards have been playing with for hundreds of years. In the book Dicing with Mozart (Jones 26-29), Jones explains how other composers went about creating these symphonies that don’t exist yet. To explain how this “Mozart” system without over complicating things, you roll the 16 times. The first eight numbers, with the help of a chart, represent the first eight bars of the waltz. The second eight numbers, with the help of another chart, create the rest of the 16-bar piece. This system, with the help of some math that is way over my head, creates 11^14 waltzes. Jones, along with many other composers, believe this number is so large that it will play a waltzes never heard before.

            

 David Cope, a professor of music at University of California- Santa Cruz, invented a music-imitating machine called EMI (Experiments in Musical Intelligence).  Requiem for the Soul (Holmes 23-27), talks about the music of David Cope and how the innovation of his work has changed the musical world forever. Cope used EMI to imitate “Mozart’s 42^nd symphony “, along with many other famous composers. Cope claims that even the greatest composer in the world couldn’t tell if the music was original or a computer imitation.

             

I think the author throughout the book did an excellent job explaining each chapter well. I really liked how the other uses examples that the average mathematical mind can relate to. I know the author used a lot of sources that explained concepts using everyday items such as dice playing cards. Some chapters were hard to follow but that was simply the difficulty of the subject but for the most part I enjoyed the readings assigned to me. Chapter 13, my first blog post, was the most interesting topic to me. Hypercubes dive into the unknown dimensions of the world. Since we talked about it in class and watched a movie about a similar topic, I feel like I can relate to that chapter more than others. Overall, I enjoyed the book and the array of topics covered.

R.Holmes, "Requiem for the Soul," New Scientist, August 9, 1997, pp. 23-27.On
    David Cope's music.

 K.Jones, "Dicing with Mozart," New Scientist, December 14, 1991, pp. 26-29.

Chapter 11: Spheres and Hyperspheres

          In Chapter 11 it mostly deals with spheres, circles, and hyperspheres. If it is a space of one dimension the 1-sphere consists of two points at a given distance on each side of a center point. The 2-sphere is the circle, and the 3-sphere is what is commonly called a sphere. Everything after three dimensions those are called hyperspheres.

A circles Cartesian formula is a^2 + b^2 + c^2 = r^2 (r stands for the radius). A spheres formula is a^2+b^2+c^2=r^2. A four sphere equation is a^2+b^2+c^2+d^2=r^2. And the ladder of Euclidean hyperspaces goes up so on. A circles surface is a line of one dimension.Also mentioned in the chapter was that circles would diminish. An example used was that if a flatlander started to paint the surface of a sphere which he lived on, if he extended the paint outward in ever widening circles, he would reach a halfway point at which the circles would begin to diminish, with himself on the inside, and he would eventually paint himself inside a spot.

A spheres surface is two dimensional and a 4-spheres surface is 3-dimensional. Einstein proposed that the surface of a 4-sphere is a model of the cosmos that is unbounded but finite at the same time. Another example Einstein suggested was that if a spaceship left the earth and traveled far enough in any direction , the spaceship would return to earth eventually. Also mentioned in the chapter was that circles would diminish.

Many hyperspheres are just what one expect by analogy with lower-order spheres. Spheres rotate around the center line, a circle rotates around a central point and a 4-sphere rotates around a center plane. A circle on a line is just a line segment.  Hyperspheres are impossible to see. Their properties can be studied by a simple extension of analytic geometry to more than three coordinates.

The main concept of this chapter is that it describes hyperspheres, circles, and spheres. And it explains the radius, dimensions and edges of the spheres, circles, and hyperspheres.

From my first reference "The Thirteen Spheres Problem" by A.J. Wasserman was very interesting to me. One thing that I wish the author would've included in the chapter was when he talked about Leech and his sketch. Leech in 1956  drew a sketch of an elegant proof that was presented. I feel like the sketch would've been nice to see in the chapter. One thing that the author omitted from the chapter that I'm happy about was that the explanation of the kissing number. I found it difficult to understand. The kissing number k(n) is the highest number of equal non overlapping spheres in R(n) that tough another sphere of the same size. In three dimensions the kissing number problem is asking how many white billiard balls can kiss(tough) a black ball.

Curves for a Tighter Fit by Ivers Peterson was easy to understand for me. One thing that the author didn't include that I'm happy about was when they talked about the eight-dimensional space. It said "they ponder, for instance, the most efficient way of packing the eight-dimensional equivalents of ordinary spheres into an eight-dimensional space." I didn't like it because I'm not much of a big fan of dimensions higher than four. One thing I wished the author would've included was the problem of filling a large shipping container with identical ball bearings. Spheres don't fit together as neatly as cubes. No matter how cleverly you arrange the balls, about one-quarter of the space in the carton or in any other container tightly packed with identical balls, will remain unoccupied. I found that problem very interesting because I haven't thought about that before and it was an interesting fact to me.

My final reference was Sphere Packings, Lattices, and Groups by J.H. Conway and N.J.A. Sloane was a reference I had to re-read and research a bit. Something that I wished the author would've included in the chapter was when the reference brought up Bounds for codes and sphere packing in chapter 9. I wish he would've included it because I wanted to know more about sphere packing. I'm glad he didn't include the spherical codes that are constructed from binary codes. I thought that would've been a little difficult to understand.

Chapter 15: Rotations and Reflections

    In Chapter 15 of “ The Colossal Book of Mathematics,” Martin Garden introduces the topic of rotations and reflections. The chapter brings up many examples of symmetry found in nature and how artists utilize different types of symmetry in order to create fascinating illusions. Thus, the readers are able to explore various cases of mathematical art in their daily lives.

    The chapter starts of by explaining that “[a] geometric figure is said to be symmetrical if it remains unchanged after a “symmetry operation” has been performed on it” (Gardner, 189). Also, the more operations you can perform of the figure, the “richer” it is. For example, the letter O is the richest out of all the other letters because it is unchanged by any type of rotation or reflection. Many artists from the late 19th to the mid 20th century have applied this technique to their artworks. Political cartoonist would draw famous public figures, and when the reader would invert the picture, they would see a pig or a donkey or something equally offensive. This device of upside-down drawing was taken to further heights by the cartoonist Gustave Verbeek. He worked for the New York Herald, and every Sunday he would draw a six panel comic strip. In order to read it, one first reads the newspaper right side up and then to continue the story, the paper must be flipped over, which held a new set of captions and took the same six panels in reverse order.

    Types of rotations cause also cause optical illusions. For instance, astronomers view photographs of the moon’s surface so that the light appears to illuminate the craters from above because if it was inverted or below, the craters would look like mesas jutting out of the surface. A very amusing illusion is that of a circular pie with a slice missing. If you were to turn the picture upside-down, one can find the missing slice. This shows how one is used to seeing things, like plates and pies, from above and not below.

    From the article “The Doodle Bug,” by Emily Bearn, she explains the type of rotational art called ambigrams. An ambigram is a word or art form whose elements retain the same meaning when viewed from a different direction or perspective. I wish the author would’ve included this small topic into the chapter because it is a very intriguing aspect of rotations and reflections in order to create art. The chapter is already short as it is, so it would’ve benefited by adding this into it. However, I’m glad Martin Gardner omitted going into detail about the lives of the artists he talked about. Bearn dives into the life of John Langdon, a professional in ambigrams. If he were to include the specifics of some people’s lives, it would’ve taken away from the main position of the chapter.

    In the article “Group Think,” the writer Steven Strogatz brings up an important topic, group theory. He explains how “group theory bridges the arts and sciences. It addresses something the two cultures share - an abiding fascination with symmetry.” Strogatz shows a practical usage of rotations and reflections in our daily lives, which is something Gardner didn’t cover, but should have. In his article, he explains how you can rotate a mattress three different ways in order to make it last longer. You can either flip it horizontally, vertically, or rotate 180 degrees. However, I’m glad that Gardner didn’t put in equations of rotations and reflections in his chapter. I believe it would’ve confused the readers and be redundant as the pictures already provide enough explanation.

    In my final reference, “The Turn About, Think About, Look About Book,” by Beau Gardner, it gives many pictures of the topics covered in the chapter. The book is made like a children’s book, so anyone can read it and enjoy it. What I think Martin Gardner should’ve included was more pictures of rotations and reflections because readers would’ve been entertained trying to solve what the illusion is. What I’m glad that Martin did omit was the childish aspect that’s from the “The Turn About, Think About, Look About Book.” His chapter gives more of an intellectual look than what Beau Gardner provides.

    This chapter was stimulating and very simple. It is one of the more easy to follow chapters in the book compared to other concepts. Also, it is a refresher from all the numbers and equations math is known to throw at you.

Chapter 7: Penrose Tiles

     There are two forms of tiling; periodic and non-periodic tiling. Periodic tiling is a form of tiling in which you can outline a section of the image and it will be the same when shifted but not rotated or reflected. Non-periodic tiling is a form of tiling that uses each individual tile that uses each individual tile to create a larger or smaller version of the tile (this is defined as tiling through inflation or deflation). Penrose tiling is a form of non-periodic tiling in which the patterns are created from the shapes and the shapes are cut in half and then "glued" back together to form the original shapes through inflation or deflation.
   
     Penrose tiles generally lack symmetry since they are non-periodic. However, they can be constructed with a high levels of symmetry. The problem is that we generally don't recognize it. The chapter uses the universe as an example; it contains a cryptic mixture of order and what seem to be deviations from the order when in the grand scheme of things, still comply to the order (or at least try to).

     In R. Penrose's "Pentaplextiy: A Class of Nonperiodic Tiling of the plane," Penrose goes on to explain all the different method's on how to create these grand penrose tiles. However, he also goes into detail as to how all of the math is involved into the proofs of how these work which weren't too easy to follow and understand.

     While going through C. Radin's "Miles of Tiles," he gives a wonderful explanation of exactly what penrose tiles are and gives a huge array of examples to help the reader follow along as he explains. For example, he uses the crystalline structures of elements to show the natural occurrence of periodic tiling. However, when he gets to a few other examples, he only uses coordinates and points with their restrictions which can allow the reader to stray away from the text out of utter confusion or complete boredom.

     With R. Berger's "The Undecidability of the Domino Problem," he uses an example of dominos creating periodic tiling that is straight-forward and easy to comprehend. This allows the reader to get a glimpse of tiling without getting flustered through all the problematic vocabulary and procedural aspects of periodic and nonperiodic tiling. However, like the other two sources, the problem comes when the math does. When they start explaining the math of it, the phrasing is very confusing and hard to follow.

     The biggest problem with the three sources was when it came to the math. With the math lacking examples to show exactly how the math itself works, all the explanations seemed to just be a cluster of confusion. However, this chapter really emphasized the beauty on penrose tiling rather than the math itself. It also showed how this beauty is found in nature. This chapter really accentuates that one should really appreciate the beauty behind these patterns that most people just find in coloring books.

Chapter 12: The Church of the Fourth Dimension

 In this chapter the author took us through a story of an adventure into a church of four dimensions. In this church, an explanation of how a fourth dimension is possible is made. The sermon in the church covered the idea that four dimensions is possible and can be explained through the same logic that our three dimensional world is possible. After the dimensions are explained, the author takes us to a room with the pastor. The pastor demonstrates the idea of a fourth dimension by using magic tricks. Once of which is taking a piece of leather that has two slits in it but is connected at the end and braiding it. Another trick is taking a rubber band, placing it in the box, putting it under the table, and then revealing that the regular rubber band is now tied to another rubber band. Although both of these magic tricks can be done with slight-of-hand movements and logic, they do get the point across that a fourth dimension could be possible, and how it would work. The fourth dimension would make it possible to come into our third dimension at any point and exit it at any point.
  The Fourth Dimension Simply Explained by Henry P. Manning covered the topic of how a fourth dimension could be explained. One thing that I believe that Martin Gardner should have included within chapter twelve, is the explanation provided by Manning. This explanation was that one dimension could be described as a point extended outward from itself, and two dimension is that line extended outward from itself, three dimensions would be that plane extended outward from itself, and the cube that would be formed from that would be extended outward from itself to form a hypercube. This hypercube would represent the fourth dimension. I believe that this was a tremendous explanation that is easy to visualize. Something that I am happy that Gardner omitted from the chapter is the explanation of the triangles intersecting. The reason that I am happy that this was omitted from the chapter is because I feel that it was confusing, and would be difficult for many to understand.
  In Surfing Through Hyperspace by Clifford A. Pickover the three degrees of freedom (of our world) are explained. The three degrees of freedom describe the directions that something can move. In our three dimensional world, motion can happen in up and down, left and right, and forward and backward. What I liked was that Pickover gave the example of the fly. The reason that I wish that Gardner had included this example is that it makes a lot of sense and is a really good way to describe the limitations of three dimensions. It is explained that a fly could not move if placed in a box, so it would be in zero degrees of freedom. If the same fly was placed in a tube it would have one degree of freedom, forward and backward motion. This fly could have its wings removed and be allowed to walk along a flat (or curved) surface to demonstrate two degrees of freedom, forward and backward, left and right motion. A fly with its wings and not contained in a box that is the same size as it would have three degrees of freedom and would be able to use upward and downward, left and right, and forward and backward motion. I feel that this was very explanatory and easily visualized which would have added to the chapter in The Colossal Book of Mathematics. One thing that I feel was better left out of Gardner's book, was the whole story line. Although this chapter included a storyline, I feel like the story in Surfing Through Hyperspace included too many unimportant descriptions and got off track a lot.
  In Geometry of Four Dimensions by Henry P. Manning the definitions of a line, a plane, a hyperplane, and a space of four dimensions are given. The definitions are very clear, and assist in the understanding of the fourth dimension and how it is that a fourth dimension would exist. This fourth dimension would be a space made by five points that are not of the same hyperplane, but are collinear to two other points. The other descriptions are given before the description of a space of four dimensions and are very helpful. Something that I did not like was how Manning defined "a figure" among other words. I feel that it could prove confusing, and that he muddled his definitions. This, I feel, was better left out of Gardner's book.
  Overall, I feel that the chapter was good. The storyline helped keep the information easy to read, and created a good flow. This chapter seemed easy to read, but for a few of the concepts, it may have been necessary to reread a couple of times.

Chapter 40: Does time ever stop?

Chapter 40 tries to explain the inexplicable; Does time ever stop? Within this chapter not only the concept of time stopping but the concepts of time moving in all directions are explained. The chapter opens up multiple theories and viewpoints of time from different famed physicists and then it settles down and attempts to explain the concepts. Some concepts are explained through references to short stories and other science fiction novellas related to time travel.

            One of the ideas related to time stopping in the chapter talks about the universe we live in as being a part of normal time and that there is a realm outside of ours that experiences hyper time. The book compared our time to hyper time by explaining it in the following way: imagine that the realm we live in as being a 3d movie being played at some point in the realm of hyper time. Now let us say that the person watching the movie wanted to get up and get a snack. They would pause the movie and then play it when they got back. To them, there was a stoppage in time, but this stoppage of time was unnoticed to the people in the movie because to them time was stopped and started in the same instant. So if there is such a thing as hyper time then “for all we can know a billion years of hyper time may have passed between my typing of the first and second word of this sentence.”

            Theories involving this realm of hyper time commonly associate gods as the persons living in it. This idea of gods or God living outside our realm of time begged the question whether or not they could reverse and rewrite the past. The book talks about numerous people being on either side of this argument. One of the conclusions drawn was that a god cannot alter the past. This does not mean that gods are bound by time but that there are laws of time that gods or hypertime beings must adhere to. The book states it like this: “God can’t make a four sided triangle, not because god can’t make four-sided shapes, but because a triangle has 3 sides by definition.”

            From the article, “Time Without Change”, published the “Journal of Philosophy”, the author should have included the concepts of Mctaggartian change over time and Aristotelian change over time. The Mctaggartian change is a very literal, explained as the idea that everything is constantly changing because atoms are always moving; therefore, everything is always changing. The Aristotelian viewpoint on change is that nothing changes until someone notices that it has changed. Something that was left out for good reason is the idea of it being impossible to be aware of a changeless interval of time during that interval of time. it is possible to know of a changeless interval of time before or after the interval however.

            From the book, “Confirmation and Confirmability” by Schlesinger, something that should have been included were the different and opposing ideas of “time without change” and “change without time”. These viewpoints offer separate ideas of the relationship between time and change, and what it means for time or change to be stopped. I am glad that the author excluded what it means for something to be confirmable and the paradoxes behind confirmability.

            Lastly, from the Book “The Ambidextrous Universe”, by Martin Gardner himself, the author should have included his own ideas about how traveling backwards in time isn’t an impossibility but more of an improbability. I am glad that he did not include the Solipsist point of view, which dictates never being completely sure of anything.

            I thoroughly enjoyed this chapter. Concepts of time and space have always intrigued me. I liked reading about the different theories of time travel and time stopping.