Chapter 4: Curves of Constant Width

When given a job to move an enormously heavy load, one would think to move it with circular wheels. The axels on the wheels would buckle under pressure. The object should instead be placed on a platform with cylindrical rollers. The rollers would have to be picked up and placed in front of the object to keep it rolling. These rollers have what is called a constant width to keep the object from bouncing up and down.

Most people think that the only shape that has a constant width is a circle. This is a common mistake. There is actually infinity of curves that have a constant width. The simplest of these curves is the Reuleaux triangle. To construct a Reuleaux triangle, first draw and equilateral triangle with sides A, B, and C. Then with a compass draw and arc from A to B, B to C, and C to A. If two parallel lines forming a right angle limit a curve of constant width, such as the Reuleaux triangle, the bounding lines will form a square. If a Reuleaux triangle is placed within a square and rotated, this will be observed. In 1914 Harry James Watts took this into consideration and created a drill based on the Reuleaux triangle. This drill drilled square holes! The drill is simply a Reuleaux triangle made concave in three spots to allow for cutting. The Reuleaux triangle has the smallest area for the given width out of all the curves. The corners have angles of 120 degrees, which are the sharpest possible angles that a curve of constant width can obtain. Extending the sides of the equilateral triangle can round off these corners. This provides you with points D, E, F, G, H, and I.  Use a compass to first connect D to I, E to F, and G to H. Then use the compass to connect the letters that have already not been connected. This makes it a symmetrical curve of constant width. This process of can also be done with a regular pentagon.

It is possible to have unsymmetrical curves of constant width. One example is to start with a star of seven points. Draw line segments connecting each point to the point almost directly across from it. All of the line segments need to be mutually intersecting. Each of these line segments must be of equal length. Use a compass to connect the two opposite corners to form an arc. This will result in a curve of constant width.

A curve of constant with does not need to be made up of circular arcs. You can draw a convex curve from the top to the bottom of a square and touching its left side. This curve will determine a curve of constant width. This being said, the curve cannot include straight lines. The perimeters of all curves of constant width n have the same length. Since a circle is a curve of a constant width, the perimeter of any curve with a constant width of n must be pi n. This is the same as the circumference of a circle with a diameter of n.

Most people would assume that a sphere is the only solid on constant width. This is not true. There are many solids that when rotated inside a cube, the shape is touching all six sides of the cube. All solids with constant width are derived from the regular tetrahedron. It is a very common mistake to think that all solids of constant widths have the same surface area. This is definitely not the case. Contrary to this, all the shadows of solids with constant widths are curves of a constant width. In 1917 Sôichi Kakeya proposed the Kakeya needle problem. The problem is: What is the plane figure of least area in which a line segment of length 1 can be rotated 360 degrees? For a long time, many mathematicians believed that the answer to this problem was the deltoid. Ten years later Abram Samoilovitch Besicovitch, Russian mathematician, proved that this problem had no answer. He proved that there is no minimum area.

This chapter was very interesting to me. Prior to reading it I believed that circles were the only shapes with constant widths. To learn that there is infinity of shapes with constant width was really an eye opener. This chapter was sometimes hard to follow and I had to reread many sections more than once but in the end it was worth it.

Chapter 5 Rep-Tiles

Chapter 5 was about polygons and Replicating Figures. There are only three regular polygons. The equilateral triangle, the square, and the regular hexagon. These regular polygons can be used for making a floor that is very identical in shape and can be always repeated to cover the plane. But there are infinite numbers of irregular polygons that can provide this kind of tiling. Not many people knew about polygons about being made larger or smaller copies of themselves until 1962. In 1962 Solomon W. Golomb, who is a professor at the University of South California started working with replicating figures or as he called it "rep-tiles".

 

I believe there a two main concepts to this chapter that the author tried to communicate with us. The rep-2 figures and the rep-4 figures. The professor Golomb had some terminology.It was, replicating polygons of order k is one that can be divided into k replicas congruent to one another and similar to the original. Polygons of rep-k exist for any k, but they seem to be scarcest when k is a prime and to be the most abundant when k is a square number. There are only two types of rep-2 polygons. The isosceles right triangle and the parallelogram with sides in the ratio of 1 to the square root of 2.A parallelogram with sides of 1 and sq. root k is always rep-k proves that a rep-k polygon exists for any k. There is only one example, Golomb asserts, of a family of figures that exhibit all the replicating orders. When k is 7, a parallelogram of this family is the only known example. Rep-3 and rep-5 triangles also exist as well. Rep-4 figures, there are many of them. Every triangle is rep-4 and can be divided. Any type of parallelogram is a rep-4. There is only one type of rep-4 pentagon, which is call the sphinx-shaped figure. Golomb was the first to discover its rep-4 property. There are also three types of varieties of rep-4 hexagons. If the rectangle divided into four quadrants and one quadrant is thrown away, the re-maining figure is a rep-4 hexagon. There are no other known standard polygons with a rep-4 property.However, there are, "stellated" rep-4 polygons. A stellated polygon has two or more polygons joined at single points.One fact about the rep-4 polygon is that every known rep-4 polygon that is a standard type is also a rep-9 polygon. And it can also be the other way around every known standard rep-9 polygon is also a rep-4 polygon. Those are two main concepts I saw in the chapter. The rep-2 and rep-4 figures. 

 

This chapter wasn't that easy for me to understand the first time. I didn't really know what replicating figures were until I read this chapter. I had to read this chapter twice to fully understand it. I also thought the examples in the chapter also helped me as well. This chapter didn't really surprise me because I was already expecting something new to learn and difficult to understand. I thought this chapter was also pretty interesting to read as well because I learned a few new things about replicating figures.And last but not least this chapter did raise many questions for me the first time I read this so I definitely recommend reading this twice.​

Chapter 25: Aleph-null and Aleph-one

            Chapter 25 deals with the Set Theory; more specifically, it deals with the concepts of aleph-null and aleph-one. Aleph-null and aleph-one refer to subsets and types of infinity. Aleph-null, as implied by the term “null” or zero, constitutes a smaller infinity than aleph-one. Yes there are different sizes of infinity. The aleph-null sets are seen as finite sets of infinite numbers. Examples of such sets are the following: rational numbers, set of prime numbers, set of integers and even the number of atoms in the cosmos. These sets of numbers are infinitely long but we see them as finite infinite sets because, in theory, we can count all of these numbers. Aleph-sets are infinite sets of infinite numbers, or uncountable infinities. Examples of an uncountable infinity would be the square root of 2, pi. These numbers fall in the aleph-one set so does that mean that the square root of 2 is a higher infinity than all of the rational numbers? YES! The square root of two is an irrational number, indicating that it has nonrepeating and never ending numbers after the decimal, and seeing how we cannot reach the end of an irrational number it is viewed as uncountable. The example shown in the book to demonstrate the concept of aleph-one is similar to one that we did in class. Lets say we take the “n” amount of numbers in between zero and one. Now lets assume that these numbers are infinitely long and all different. Then let us stack the numbers on eachother:

0.01223…

0.23451…

0.95674…

0.01234…

0.32718…

This list in theory goes on forever and can contain an infinite amount of numbers. After you have the set of numbers written out, draw a diagonal line from the first number, “N”, after the decimal of the first number, then through the second number after the decimal of the second number and the third decimal after the third number etc. Upon doing this you will get a new number, according to the given set the number you get is 0.03638. Now surely this number HAS to be within one of the numbers in our infinite set. No! It can’t be within the numbers because the Nth number after the decimal will always be different than a number in your set. Professor Trevino did a better job of explaining this concept in class but I hope you catch my drift.

            I had to read this chapter a few times to finally wrap my head around the concepts of higher infinities. I really enjoyed reading this chapter because it pushed me to think at a more critical and higher level than I usually do. After learning about the different concepts of infinity I was inspired to research this topic some more and try and thinking of new categories and subsets of numbers that could qualify as aleph-null or aleph-one. If you are going to read this chapter I would strongly advise that you forget all that you think you know about infinity because you will quickly learn, as I did, that you don’t know what you think you know.

Chapter 27: Fractal Music

Fractal music is basically any kind of music that exhibits self-similarity in respect to some of its characteristics. The word “fractal” is used when talking about images, landscapes, sounds, and any other pattern that is self-similar in nature. Self-similarity can be described as looking at one small part of something, but still getting a sense of the whole picture. This chapter revolves around the fact that it is extremely difficult to explain the meaning of harmony and rhythm, and also explains the relevance of imitation to music. Aristotle and Plato both agreed that music imitates nature. It could be said to imitate natural sounds like heartbeats and the sound of cicadas, and then the tone patterns come into play, which is where musical pleasure derives from. Tones could be made from wind blowing over an object or even a cat’s howl. On top of all the natural rhythms and tones, we have humans who sing and play instruments. Program music however, has a different kind of imitation than others. For example, “Slaughter on Tenth Avenue” imitates a police car siren and the shot of a pistol. These imitative noises are a “trivial aspect of music” and can imitate things like emotions and beauty. 

I was a little confused as to what fractal music had to do with math, until i got to the part where Gardner started talking about noises of different colors, dice, and spinners. White noise is a hiss that doesn’t change whether you make it go slower or faster. By using dice and spinners, white “tunes” could be easily composed. Brownian noise involves random movement of little particles suspended in a liquid and buffeted by molecules. Brown music can be made the same way as white music, but by varying the tone durations to make it more sophisticated. Noises can also be simulated by tossing dice. For example, the 1/f noise was simulated by tossing 10 dice. All of this might seem very confusing, but it is actually pretty simple. Its not very difficult to see how the algorithms produce sequences halfway between white and brown. The less significant digits, which are the ones to the right, change often while the more significant ones to the right are more stable.

I enjoyed reading this chapter, because music is one of my favorite subjects and I’ve never even heard of fractal music before. When I first started reading the chapter, I was slightly puzzled because it was such a new concept but as I got further into the reading, it got more interesting and started to make a lot more sense. I thought it was really interesting how Gardner made a reference to Hermes, which was a Greek legend who invented the lyre, just by finding a turtle shell who’s tendons would make noise when they were plucked. I thought it was amazing how much math had to do with the creation of noise and melodies, and also how brilliant these people must have been to discover it. It is easy for us to notice how painting and sculpture imitate nature, but music is way different.

Chapter 30 The Soma Cube

Chapter 30 is about the Soma cube. The Soma cube was invented by Piet Hein who was a Danish writer. He came up with the idea of the Soma cube when he was listening to a lecture given by a physicist, Werner Heisenberg. Piet Hein realized that it is possible to form irregular shapes by combining four cubes of the same size and shape and put together to make a bigger cube. The way that an irregular shape is described is that it has concavity and is usually three cubes joined together. He states that two cubes are able to be joined together, but only on a single coordinate. On the other hand, three cubes can have a second coordinate perpendicular to the first coordinate, and so on with four cubes. As Hein continued to discover new things about the cubes, he realized that twenty seven little cubes would be able to make three by three by three cubes. Ever since he tried making these seven components of cubes, he was able to confirm his research and it became known as the Soma.

            The Soma puzzle started getting very popular among people. It was easy to make but involved a lot of thinking. In order to make a Soma cube, you should start by trying to make a stepped structure. There are more than 230 different solutions for the Soma cube constructions. One of the strategies given was to first put your irregular shapes down first and then fill the structures in. As the cube was becoming a popular item among other people, they kept trying to solve all of the Soma problems. They solved so many that the shapes started becoming familiar to them and they were able to do it in their heads. A few tests were done by psychologists that showed that solving these puzzles was correlated with a person’s general intelligence. There is an example that as the ratios of the cubes change, it is more possible to build them.

            Many of the readers about the Soma were so fascinated that they sent in their own sketches of their puzzles and complaints that they are now devoting their time to solving these Soma cube puzzles in their free time. Somas became part of the everyday life for teachers to give to students and psychological tests. Some cool examples included building stairs with cubes, a dog, a chair, a sofa, a scorpion, a bathtub, and many more. A Soma set consists of seven pieces normally; however, one man named Theodore Katsanis sent in a letter suggesting of a set of eight pieces formed with four cubes which form two by four by four rectangular shapes. He said that a person can make five cubes by putting together the twenty nine pieces and he called this a “pentacube”. The Soma pieces are a subset of polycubes which are polyhedrons that join unit cubes together by faces. It is also verified that there are 240 ways of making the Soma cube. A Rhoma is a slant version of the Soma made by a distortion of the large cube into a rhomboid shape. An example of a 27 cubed dissection would be that the cubes are either black or white and you have to make a cube that is checkered throughout the entire structure or just on its six faces. The cubes can also be made into different colors and you would have to form a cube with a specific pattern of the different colors on each face.

            I thought that this chapter was really interesting. There were many intriguing examples in the book with pictures of the different Soma cubes. It got me to think about different types of the cubes and shapes that the book gave. The chapter stated that out of the figures given in the book, one of them was not able to be made out of the Soma pieces. I thought that was very interesting because the picture showed that it was built so it was hard to tell, but at the end of the chapter when it told me that it was the skyscraper that couldn’t be made out of the Soma pieces, I looked over that figure and noticed that there are never two cubes exactly next to each other. It surprised me that I wasn’t able to figure out which one couldn’t be built just from looking at the picture, but the chapter did state that it takes many days for people to usually figure it out. I also enjoyed looking at the different shapes that were shown inside of the chapter. 

Chapter 9 The Helix

Chapter 9 was about the helix. A segment will always fit, no matter if the curve can be slid along the curve from one end to the other or if a straight segment fits snugly into a straight scabbard. The main question of this chapter was "Is it possible to design a sword and its scabbard that are not either straight or curved in a circular arc?" Most people answer no but are wrong according to Martin Gardner (the author).

A circular helix is a curve that coils around a circular cylinder in such a way that it crosses the "elements" of the cylinder at a constant angle. If you restrict the curve until the coils are close together you get a cramped wound helix resembling a Slinky toy then when you let go the helix topples into a circle. If you stretch the helix it is transformed into a straight line. Most manmade helical structures come in both right and left forms such as: candy canes, circular staircases, rope and cable made of twisted strands. Helical structures are also in living forms like parts of the human body. The genetic code carries information that tells each helix strictly "where to go". Linus Pauling's work is base upon the helical structure of protein molecules. Researchers found that there has been increasing evidence that every giant protein molecule found in nature has a "blackbone" that coils in a right-handed helix. A "blackbone" is a chain made up of units each one of which is an asymmetric structure of the same handedness. In the human ear, the cochlea is a conical helix that is left-handed in the left ear and right-handed in the right ear. On any occasion, a single helix is outstanding in the structure of any living animal or plant, the species usually environs itself to a helix of a specific handedness. The Devil's corkscrew is a puzzling typer of helical fossil, which is found in Wyoming and Nebraska. Helices in the plant world are common in the structure of stalks, stems, seeds, flowers, tendrils, cones, and leaves. The enmeshing of two circular helices of differing handedness is also elaborated in a remarkable optical-illusion toy that was sold in this country in the 1930s. The helix aspect of the neutrino's path outcomes from the fusion of its forward motion with it's "spin". Examples of this are: a point on a propellor of a moving ship or plane and a squirrel running up or down a tree.

This chapter to me was a very easy to read and easy to understand. I didn't know much about the helix before this reading and I can honestly say that I have captured a lot more after reading about the helix. I wouldn't say the chapter was surprising by any means, there wasn't anything that I read that was surprising to me.

Chapter 33 Ramsey's Theory

 
       Chapter 33 is about Ramsey's theory. It is named after Frank Plumpton Ramsey who died at the age of 26. His theory is based on graph theory. The main idea of this is studying sets of points joined by lines. In this theory it compares points to represent people at a dinner table such as six points representing six people. It compares the points drawn to lines with two different colors. These represent people who know each other at the dinner table and people who are strangers.

        This chapter talks about how Kn would represent the graph being that n represents the number of points the graph has and k is just a variable to represent the graph. It then talks about coloring the lines red and blue to represent two different groups and that colored with three would represent three different groups. It also talks about subsets and how each line connected to two points represents a subset. To further that a subgraph is a graph contained in the graph.

            Ramsey’s theory can be compared to a game. The most common is called Sim it is played on K6 ( a graph with 6 points). Which can be compared to the problem with the six people. It also can be put in R(3,3)=6 notation. Which the R represents Ramsey’s number, the three represents a triangle of one graph or a subgraph, and the other three represents a triangle of another color. The next thing talks about Knuth’s arrow notation which is 3 arrow 3 is the same thing as 3*3*3=27. Also 3 arrow arrow 3 is the same as 3 arrow 27 or 3^27 which is the same as 3^3^3. This number written as a number is 7,625,597,484,987.

            Ramsey’s theory poses a question. An example of a question it poses is while having a party and you want to invite at least four people who know each other and five who don’t, how many people should you invite? The answer is 25 people. The way it is answered is by drawing points where each point represents a person and each point is joined to all of the other points by a line. An example of this is if you color a line red it means two points represent two people who know each other. If the line is blue it means that the two people do not know each other. The way 25 is proved as the answer is because of asking the question what the smallest amount of points are needed so that there are four points joined by red lines and five joined by blue lines? This was not proved until recently. This can also be written in the R(4,5)=25 notation. Where the R represents Ramsey’s number, the four represents one subgroup, and the five represents another subgroup.

            In general there were some confusing ideas presented in this chapter and it was interesting that the graph could be compared to a real world situation. Such as inviting people to a party and how many to invite if you want a certain amount of people to know each other and a certain amount to not know each other. It was surprising in a way. Questions the chapter posed for me were things like knowing when a graph is six colored and weather you are certain to get a subgraph of one of the colors.

Chapter 43 The Unexpected Hanging

Chapter 43 mainly focused on situations where we invalidate our own reasoning. Michael Scriven, a professor of the logic of science, starts off the chapter by giving an example of how “The unexpected hanging” paradox works.  The puzzle starts off with a man who is condemned to be hung.  The man was sentenced on a Saturday and the judge stated that the hanging will take place at noon but on one of the seven days of next week, without the prisoner knowing.  It will become a surprise for the prisoner who is getting hung. The prisoner and his lawyer discuss about how the sentence cannot be carried out because the judge’s order seems self-refuting.  The lawyer explains that the judge cannot hang the prisoner on Saturday because it is already the last day of the week and that he will still be alive on Friday afternoon of next week. Since he will still be alive on Friday afternoon, Thursday cannot be the day of the hanging either because if he is alive on Wednesday afternoon, he will know already that Thursday is the day. 

The prisoner continues to convince himself and be confident that he will not be hung at all next week because he will know ahead of time.  But that is the problem, the paradox of “The unexpected hanging” is still a controversial topic because there are no correct deductions.  It is easy to guess what will happen next, but what assures you that it WILL actually happen?  You may guess correctly, but it does not always result being true. At the end, the hanging does occur. The prisoner assumed that his prediction will be fulfilled, but it was falsified.

“Future events can be known to be a true prediction by one person but not known to be true by another until after the event.” The judge knew already what day the hanging was going to occur, but the prisoner did not. It was ultimately up to the judge to decide what day of the week the hanging was going to be happen, without the prisoner knowing when his last day of life was going to be.  The judge was clear on keeping his word of the sentence happening the following week but the prisoners’ expectation was completely different. 

Even though Chapter 43 did not deal with a lot of Mathematics, it still had me thinking on various sections. It came to my surprise that there are many examples of how philosophers have not fully concluded on how to resolve these kinds of paradoxes.  There are no actual answers on how we expect things to happen. Guessing is one way of knowing what will happen next but it is not always assured right.  I learned that we cannot come to a conclusion of how unexpected things and situations will occur.  There are no correct deductions on guessing what day there will be a pop quiz or a fire drill. That will be a surprise that we are not expecting. I really enjoyed reading this chapter! Ps-Don't procrastinate!