Volume VII, Issue II
May, 2016
XEVEX The Ramaz Mathematics Magazine
Volume VII, Issue II
Eratosthenes of Cyrene; First to Calculate Earth’s Circumference Matthew Hirschfeld ’17 Well over 2,000 years ago lived a man who relatively accurately estimated the circumference of the Earth. That man was Eratosthenes of Cyrene. Born in approximately 276 B.C. in Cyrene, Libya, Eratosthenes soon became one of the world’s most renowned mathematicians in his time. Eratosthenes has garnered the most acclaim for having made the first recorded measurement of the Earth’s circumference, which was also remarkable accurate. In great part, Eratosthenes was able to accomplish this feat due to his education in Athens, where he became known for his achievements in a diverse assortment of disciplines, including poetry, astronomy, and scientific writing. His reputation proceeded him so much so that Ptolemy III of Egypt decided to invite him to Alexandria to tutor his son. Later on, Eratosthenes would become the chief librarian of the Library of Alexandria.
Some Infinities Are Bigger Than Other Infinities Gabrielle Amar ’17 Imagine infinity–your reflections in opposing mirrors, the “infinite” moments that we imagine we experience. When we think of infinity, we think of immeasurable quantities that interminably increase or shrink. When we think of infinity, we wonder if it even exists and to what infinite size it can reach. German mathematician Georg Cantor (pronounced GayOrg Can-tor) opened a whole new door to this concept when he proved that one set of infinity can be larger than another set of infinity. In order to better understand this, it is important to first understand set theory. A set is a collection of distinct objects. Like a box with things in it. For example, take set s = {1,2,3}. This set contains the objects, or elements, 1, 2, and 3 and has a total, or cardinality, of 3 elements. But what if there were a box that contained all the counting numbers. Well in that
June, 2016 Such an opportunity surely would have exhilarated the insatiable intellect. At the time, the Library of Alexandria was a focal point for learning, drawing scholars from all across the known world. In this stately position, the mathematician was able to cross paths with the likes of Archimedes, while continuing his own learning. It was most likely in the Library of Alexandria where Eratosthenes read about an intriguing event that took place in Syene (now Aswan, Egypt) at the summer solstice. Syene was located to the south of Alexandria. At high noon, the sun would shine directly overhead, thereby causing no shadows to be cast from the columns. Eratosthenes noted, however, that at that same moment in Alexandria, the columns clearly did have shadows. Being a most astute mathematician, Eratosthenes decided he could utilize this knowledge to do a few calculations to figure out the circumference of the Earth. To accomplish this objective, Eratosthenes measured the shadow of an obelisk on June 21 at Continued on page 5 case, since the counting numbers go on to infinity, the cardinality of the set of all counting numbers is not a counting number itself, but rather it is an infinite number often referred to as aleph null (א0). This size of infinity is usually called countable infinity since the set of counting numbers could be listed infinitely. A subset of this set is all the even numbers ( e = {2, 4, 6, 8, 10…}) One may think that this even subset has a smaller infinity that the original set, however this is not the case because we realize that there is a one-to-one correspondence between the two sets, where the two sets have the same number of elements. The same goes with the set of all integers, both negative and positive (i ={...-2, -1, 0, 1, 2…}). You may think that this set is larger than the two sets previously discussed because it goes to infinity in both directions, however this is not the case! They still have the same cardinality, that is, Continued on page 4 2
Volume VII, Issue II
The Curious Art of Origami Gabrielle Amar ‘17 Since 1797 in Japan, Origami has been a lasting popular art form. However, only recently has the creation of simpler forms, such as the classic crane, transformed into the creation of more intricate forms with many more folds. What changed this art form to become highly detailed and naturalist is math. That is, people applied mathematical principles to origami to discover the underlying laws behind the art form. In origami, crease patterns act as the underlying blueprint for an origami figure. To draw these crease patterns, however, it is important to follow four simple laws. The first law is twocolorability: coloring any crease patterns with two colors without ever having the same color meeting. The second law is that the direction of folds at any vertex--the number of mountain folds or valley folds--always differs by two, two more or two less. The third law is that if you number the angles around the fold in a circle, all the evennumbered angles add up to a straight line and all the odd-numbered angles add up to a straight line. The fourth law is that the layers of form must be stacked so that a sheet can never penetrate a fold. To obey these laws in origami, we can take simple patterns like a repeating pattern of folds called textures. These textures are what bring the origami figure to life. However, in order to come up with the texture for the folds and ultimately the structure, we start out with the model itself and then a most abstract representation of it, like a stick figure. To go from the abstract representation to the texture and then to the simpler origami figure, the mathematical rules must come into play. The secret to creating arbitrarily complicated figures is to use circles to create
June, 2016 flaps. A simple flap is created by merely folding a square piece of paper in half (folding it diagonally) and then folding each halve in half, until the paper is long and narrow. At the end of that piece of paper (at the top corner in which the first crease was made), is a flap. If we open up the piece of paper and examine the blueprint, we can see that there needs to be a quarter-circle to create that flap (try it yourself!). There are other ways of making flaps too. It the flap is on the edge, it uses a half circle. So, no matter how one makes a flap, it needs some part of a circular region of paper. Therefore, to make a lot of flaps, we need lots of circles! In the 1990s, origami artists discovered these principles and realized we can make arbitrarily complicated figures just by packing circles. In addition, the art form of origami does not only produce beautiful and extremely convoluted structures for design and visual purposes, but origami is also used as a tool in medicine, in science, in space, in the body, consumer electronics and more. Origami is mainly used to solve the problem of transforming something that needs to be big and sheet-like at its destination, to a smaller form for the journey. For instance, airbag designers used the origami algorithms to get airbags to do their simulation-fitting flat sheets into a small space to flatten the airbag and inflating the airbag by having the origami creases form. The airbag-flattening algorithm evidently came from all the developments of circle packing and the mathematical theory that was really developed just to create animals using square pieces of paper. When mathematics is involved, problems that are solved for aesthetic value could be applied in the real world in a way that can even save a life.
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Volume VII, Issue II
Some Infinities Are Bigger Than Other Infinities Continued from page 1 countable infinity. To map this out, connect the first term in set e to 0 then connect the second term to 1 in set i then connect the third to -1, the fourth to 2, the fifth to -2 and so on. You will notice that there is a one-to-one correspondence between the elements of the two sets. Drawing on this conclusion, one may think that all infinities are the same size and that all sets have elements, that are listable, and that א0 is the cardinality of all sets of numbers. However, this is not the case for the set of real numbers. This is where Georg Cantor comes in. In 1891, Cantor proved that the real numbers have a greater infinity than the counting numbers. If we examine the real numbers–look really closely–we find that there are simply too many real numbers to fit even on an infinite list, meaning, an infinite set of real numbers does not necessarily list all of the real numbers. We can actually create any real number, out of an infinite sequence of digits, that could not possibly be on the list even if it is infinite. The proof to this is a proof by contradiction, shown in the chart. To the left of the chart, there is a list of countable numbers. To the right are the decimals of that
June, 2016 number in the list. The first number of the subscript of d corresponds to the place of the number in the list and the second numerical subscript corresponds to the decimal position. All we need to do is to create a real number that isn’t the first number on the list, or the second number, or any number, no matter what the list is. To do this, we can take a diagonal mapping of these decimals (Cantor’s Diagonal Argument) to create a new number called n in which each decimal gets added one value. This number, n, conflicts with every single number on the list in at least one decimal place. In other words, our new number n does not match a single number on this infinite list, and yet it should be on the infinite list of real numbers because it’s just a real number made of decimals! Here we have a contradiction, which means that the real numbers cannot be mapped to the counting numbers and that they cannot have the same cardinality. This led Cantor to call the cardinality of the set of real numbers aleph one, א1, the next order of infinity. However, we still don’t know how much larger א1 is to א0. We don’t know if there are other infinities in between the two either. All we know is that there are some infinities that are indeed bigger than others. Through all this, we find that Cantor had truly opened math to the very thing it was supposed to save us from, irresolvable uncertainty. 4
Volume VII, Issue II
Eratosthenes of Cyrene; First to Calculate Earth’s Circumference Continued from page 2 noon. Using trigonometry, he discovered that the sun was 7°14’ from being directly overhead. He also noted that, since the Earth is curved, the greater the curve, the longer the shadows would be. Based on his observation, Eratosthenes hypothesized that Syene must lie 7°14’ along a curve from Alexandria. Furthermore, knowing that a circle contained 360°, Eratosthenes understood that his calculation, 7°14’, was roughly one fiftieth of a circle. Therefore, by the logic of simple proportions, Eratosthenes believed that if he multiplied the distance between Syene and Alexandria by 50, he would have the circumference of the Earth. The unknown factors thus far was merely how far away Syene was from Alexandria. Eratosthenes measured the distance using the unit for length of stadia. There is no precise modern day conversion for stadia, nor is it exactly clear which version of the stadia Eratosthenes was using. Regardless, based on what is known, Eratosthenes’s estimation was astonishingly accurate. There are two prevailing theories as to how the ancient mathematician was able to figure out the distance. According to one theory, Eratosthenes hired a man to walk to Syene and count his steps. The other idea is that Eratosthenes had heard a camel could travel 100 stadia a day, and it took a camel about 50 days to travel to Syene. Whatever was the case, it was Eratosthenes’s estimation that the distance between Syene and Alexandria was 5,000 stadia. If that were the case, then using his formula, the earth was 250,000 stadia in circumference. Due to the uncertainty surrounding the length that stadia represents and moreover, which stadia he was using, historians believe that Eratosthenes’s conclusion was between 0.5% and 17% of the mark. Even if the latter scenario were true, given the severely limited
June, 2016 technology he was dealing with at the time, his calculation was remarkably accurate. Being in Egypt at the time, however, many scholars find it most likely that Eratosthenes was using the Egyptian stadia (157.5 m). This would make his approximation only about 1% too small. While there had been previous efforts to discover the Earth’s circumference, none had been recorded until that of Eratosthenes, and, from what is known about them, they were all far less accurate. While calculating the approximate circumference of the Earth was unmistakably a monumental contribution to scholarship at the time by Eratosthenes, it was by no means his only one. Eratosthenes has also been accredited with developing a primitive method of mapping out the known the world, by drawing line northsouth and east-west – early latitude and longitude lines. While these lines were irregular and often drawn inaccurately, they provided a precursor for maps we use today. Eratosthenes is also famous for his Sieve of Eratosthenes, a simple algorithm that makes it easy to find all prime numbers up to a certain limit. Though none of Eratosthenes’s personal work on the sieve survives, Nicomedes in his Introduction to Arithmetic ascribes the creation of the algorithm to Eratosthenes. In addition, Eratosthenes approximated the distance to both the sun and the moon and measured the tilt of the Earth’s axis, all with stunning accuracy. Eratosthenes also wrote the poem Hermes, correctly sketched the route of the Nile, and even gave a more-or-less accurate account of why the Nile flooded, something that had baffled scholars for centuries. Despite his incredible accomplishments, Eratosthenes was often nicknamed “Beta,” the second letter in the Greek alphabet with signified Eratosthenes’s being second-best in everything he did. Eratosthenes died around 194 B.C. and is believed to have starved himself to death. Historians understand that Eratosthenes started going blind in his later years and, unable to continue his work, he simply stopped eating.
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Volume VII, Issue II
The Monty Hall Problem Emanuel Dicker ’18 The Monty Hall Problem is one of the classic tricky probability problems that relates to lives today. It is based on the show “Let’s Make A Deal,” as Monty Hall was the host of the show for twenty years. The way the show worked was Monty would go onto the stage, and ask the contestant to choose one of three doors; behind two of them were zonks (the undesirable prizes), and behind one was a dream car. So seemingly, the contestant had a 1/3 chance of originally choosing the car. However, what Monty does is that after you choose a door, he decides to choose one of the other two remaining doors and reveal that behind it there is a goat, and he offers to you to either change your door to the only other remaining door, or to stay put. Now it seems like with this second chance, the odds are now even to either stay put or switch. Behind one door is a car, and behind the other one is a goat, so there is a 50% chance of getting the goat. However,
Explaining Probability and the Law of Averages Jasmine Levine ’17 Probability is the likelihood of something happening measured by the ratio of the favorable cases to the whole number of cases possible if we repeat the experiment many, many times. For example, when tossing a coin there are 2 possible outcomes heads (H) or tails(T). We say that the probability of a coin landing H is 1/2 or 50 % and the probability of a coin landing T is 1/2. The law of averages is the false belief, sometimes known as gambler’s fallacy. It makes the assumption that if any deviations in expected probability occur with a small number of consecutive experiments, that they will certainly ‘average’ out sooner rather than later. For example, William flipped a coin five times and got tails all five times. He told his friend John that he would bet him $100 that the next flip would be heads as he was ‘due for one.’ The truth is that each time William flips a coin
June, 2016 Monty helped you out even more than that, because mathematically, it is actually better to switch. Take this scenario: if on the first hand, you chose the door where the car was originally, which there is only a 1/3 chance of happening (of course you don’t know at the time that you chose the correct door) so when he reveals the other door, and there is goat behind it, if you were to switch to the third door, then you would choose the goat and lose. But if at the beginning, you chose a goat door, which was 2/3 chance of happening, now when Monty revealed the other goat, he was essentially telling you that the car was behind the door which he didn’t reveal. So if you were to switch in all of those occurrences – which are 2/3 of the total occurrences possible- you would get your dream car. If the contestant relies on the fact that he is mathematically more likely (2/3 likely) to originally choose the goat door, then when he switches, he is mathematically more likely to win. So in fact, it is in the contestants favor to always switch the door. (each independent trial) the probability is still 1/2. The probability of getting heads after 5 flips of tails is still 1/2. The coin does not remember that the last 5 flips were tails. But if William flips the same coin 1000 times, he will see that the experimental probability evens out to about 1/2 (expected probability) after all those trials. An example of ' gamblers fallacy’ is the following: Roulette is a game of chance and very popular in casinos. Jack decided to test his luck on a casino cruise. In Roulette there are a total of 37 colored numbers on the perimeter of the wheel. There are 18 red spots, 18 black spots and 1 green spot. Therefore there is a 47.37% chance that the white plastic ball will land on black and a 47.37% chance it will land on red. So the probability that the ball will land on black or red is the same. Sometimes the player will see that the ball has landed on black 3 times. Therefore, he will put all his money on red for the next spin. In reality, there is the same exact likelihood that the ball will land on black or red in the next spin. 6
Volume VII, Issue II
Gravitational Waves Derek Korff-Korn ’18 “Ladies and gentlemen, we have detected gravitational waves,” David Reitze, the director of the LIGO Laboratory, declared on February 11th, 2016, confirming Einstein’s 100-year-old prediction of ripples in spacetime due to gravity. In 1916, Albert Einstein hypothesized– based on his theory of general relativity–that gravitational waves transport energy, also known as gravitational radiation. This is radiant energy, a similar form to electromagneti c radiation. In Einstein’s theory of relativity, gravity is regarded as a phenomenon resulting from the ripples in spacetime. The presence of this curvature is caused by mass. The more mass that is contained within a specific volume of space, the greater the curve in spacetime will be in the outskirts of the volume. LIGO Laboratory used twin detectors to hear the gravitational ‘ringing’ formed by the collision of two black holes approximately 1.3
Squaring the Circle problem – the most famous problem of antiquity Jasmine Levine ’17 Squaring the circle is a problem proposed by ancient Greek geometers around 200 B.C. It is the challenge of constructing a square with the same area as a given circle with a compass and straightedge only. The ancient Greeks did not have algebra. Squaring a circle would require constructing the length of the square root of Pi. Pi is the ratio of a circle’s circumference to its diameter. A circle with radius 1 has area Pi. Hence a square with the same area must have a side of square root of
June, 2016 billion light years from Earth. Each of the black holes observed were both about 30 solar masses. After hundreds of years due to a loss of energy by emitting these gravitational waves, they merged into a single, more massive black hole that weighed 62 times the mass of the sun. The detection of the gravitational waves came from both campuses in the U.S. Each L-shaped interferometer spans 4 kilometers in length and uses laser light split into two beams that travel back and forth through, bouncing between configured mirrors. Without any changing conditions, the waves should travel at an equal rate for each beam and travel through the mirrors at the same time. But according to Einstein’s theory (which did happen) the Gravitational waves would affect the light wave’s pattern when they intersect. This discovery is going to be vital to the way scientists in the future understand the unknown aspects of the universe, including the beginning of time. Pi. In the 18th century, Johann Heinrich Lambert proved that Pi is irrational, that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. In 1882 Ferdinand von Lindemann proved Pi is not just irrational, but transcendental as well. That means it is not a root of a non-zero polynomial equation with integer coefficients. The fact that Pi is transcendental means that it is impossible to draw to perfection using a compass and straightedge a square with the same area as a circle (We cannot draw square root of Pi). 7
Volume VII, Issue II
Game Theory
Emanuel Dicker ’18 Game theory is the mathematical branch that studies how decisions are made when they depend on the decisions of another participant. Game theory usually determines the optimal state for the two participants as a whole, or the Nash Equilibrium, which is when each party has picked a choice given the choices of the other party. The perfect example is known as the prisoner’s dilemma. In this case, there are two criminals, George and Kevin. Both are taken in separately and are assumed to not be related in their crimes. They are told that they are both going to jail for two years for a drug charge, as they were caught with the drugs on them. However, one of the detectives starts to think that they actually worked together on a different crime, an armed robbery. So he decides to go to both of them and say the same thing: There are four options. One: if George were to confess to both of their involvements in the crime and Kevin weren’t, then George would go for 1 year, and Kevin would go for 10 years. Two: If Kevin were to confess and George weren’t, then Kevin would go for one year, and George would go for ten. Three: If they both confess, then they both go for three years. The last option is if both of them deny, and they both go away for two years for the drug charges. The best way to visualize this is a payoff matrix. Look at it from each of their points of view. In George’s point of view, if Kevin confesses, then George should also confess, because then he gets three years rather than 10. If Kevin denies, then George should confess, because then he gets one year instead of two. So, confessing is the way to go for George. The same rings true for Kevin. If George were to confess, then Kevin would feel inclined to confess so that he would get three years, and not 10. If George were to
June, 2016 confess, so that Kevin would get one year instead of two. So in this scenario, it makes more sense for both of them to confess, so we have reached a Nash Equilibrium because depending on the other person’s decisions, the other parties decisions are made in order to better their scenario, as in both of them getting the 3 years. So here, it is better for both of them to confess, and to take the 3 years, than to be risky and deny and maybe get the two years. The opportunity of them getting two years is an unstable state, as they could end up with either 1 or 10 years; thus, the most reliable route to go from both points of view, is for both of them to confess. Confess
Deny
Confess
George: 3 Kevin: 3
George: 1 Kevin: 10
Deny
George: 10 Kevin:1
George: 2 Kevin: 2
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Volume VII, Issue II
June, 2016
Editors: Ben Kaplan ’16 Ben Rabinowitz ’16 Gabrielle Amar ’17 Matthew Hirschfeld ’17 Jasmine Levine ’17 Faculty Advisor: Rabbi Stern
Writers: Derek Korf-‐Korn ’18 Matthew Hirschfeld ’17 Jasmine Levine ’17 Gabrielle Amar ’17 Emmanuel Dicker ’18
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