The Four Color Problem

This week’s installment of mathematical problems is going to be about the four color problem. This problem has to with the involvement of filling in maps with different colors. The problem started with physical maps of countries and boundaries, and then shifted into a more abstract mathematical field. The problem will be covered first, followed up with a five color solution proof, and finally some history.

The four color problem is, if there is a map can all the countries or states can be filled in with four colors, in that no two same colors come into contact with each other. This was purposed in the mid 1800's and since then people have been filling in actual geographical maps with four colors. No matter the complexity of the geographical map, it will not require more than four colors to fill in all the boundaries. The map of the world can be done, the map of the Untied States can be done, and so can the map of the United Kingdom counties. All geographical maps fit these criteria. This is where the realm of mathematics steps in. Instead of using maps that already exist, can a map be created that can disprove this conjecture? This is because for the average person, it is a lot simpler to come up with a counter argument rather than proving it. So people started creating maps, and filling them in. There arose a problem of repeating maps being checked. Though two maps look different if the share the same type of relation between borders, then they are in essence the same map. So instead of filling areas on maps countries were represented with colored dots. If two countries touched reached other, they were connected with a line. This allowed for redundancy to be removed, and allowed for deeper analysis. This deeper analysis comes from the fact that the problem has been shifted from a coloring book problem, to a networking problem. In the 1970's this problem took off with the newfound interest in computer science, that arose in the time period. There is some rather useful insight about networking that comes in handy in solving the problem. If there is a country, call it country x, country x can be connected up to five other countries. Meaning in a given map every country can only connect with five other countries and still fit the parameters of a map. That is a very useful bit of information to have, because it allows the problem to simplified, and narrowed down. This will be apparent in the proof of at least five colors can be used.

In the last part it was mentioned that a country must have a network with anywhere from one to five connections in it. That part has its own proof, which was not covered. Going off of that information to prove five colors, start with seven colors, and work down to five. If there is a map with seven colors, assuming it is the smallest map with seven colors, there has to be one of those country x's that follows that networking rule. If that country is removed, or pulled from the map, then there is a smaller map. The smaller map would only require six colors then, and if the country x is put back into place, now there is an extra color that country x can now take, reducing the number of colors needed. This process can be done down to five, though five requires a tad more information and proofing, but can be done using the same logic.

In the mid 1800's a map creator was coloring in the counties of Great Britain. He tried to optimize his map by filling it in with as few colors as possible, without the same color touching. He worked at it, and did it with four colors. He assumed that four colors might be enough to fill any map then. He purposed the idea to his brother, who in turn passed the idea onto a mathematician by the name of Augustus De Morgan. Augustus is a famous mathematician, who is responsible for putting the problem together, and getting it some attention. The problem remained unsolved for one hundred and twenty five years after its proposal. Kenneth Appel and Wolfgang Aachen did not solve it until the 1970’s. There proof was controversial because it was the first mathematical proof to be done by the use of computer power. Which at the time was a not considered a guaranteed source.

Sources https://www.youtube.com/watch?v=NgbK43jB4rQ

                  http://www-groups.dcs.st-and.ac.uk/~history/Biographies/De_Morgan.html

                  https://www.youtube.com/watch?v=laMkuPrad3s&feature=youtu.be

The Importance of Pure Mathematics

            In today’s world of mathematics, there are two main fields of mathematics. Those fields are pure, and applied mathematics.  Pure mathematics in its simplest definition is math purely for the sake of math. Pure mathematics would include developments is number theory, Algebra, Control theory, and Calculus analysis. Applied mathematics in its simplest definition is math that has a purpose outside math itself. Applied mathematics would include math done in, physics, chemistry, biology, and any other math intensive science. For the most part pure mathematics is not seen as an important part of society. Sometimes pure mathematics is given the label of useless, or pointless. Although it might seem like a waste of time to some people, pure mathematics has an immensely important part, in the everyday workings of society as a whole.

            Pure mathematics is important because it develops the tools for applied mathematics. The groundwork of all Calculus, Algebra, Topology, and mathematics as a whole, rely on pure mathematics. Farida Kachapova wrote in the Journal of Mathematics and Statistics that, “ . . . without fundamental mathematics there would be nothing to apply.”(Kachapova) This means that all of the applied mathematics need pure mathematics to facilitate them. For example, a huge chunk of Newtonian physics relies on the understanding of basic algebra.   G. H. Hardy was once quoted saying, “Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” (Hardy). What Hardy is saying is that the tools used for mathematics are more important then the products of mathematics. The tools of mathematics can be used to make all the results, while the results only help with that specific issue. That is why pure mathematics is important to applied mathematics.

            Pure mathematics is, and was used to develop many of the technologies used in the daily life of individuals.  The most commonly mentioned invention, which came about because of pure mathematics, is the computer. This is because the computer is revolutionary to society as a whole. Phones, laptops, Gameboys, and any other electronic available all rely on pure mathematics. Ben Orlin wrote about the invention of the computer saying that,” . . . one of the purest mathematical enterprises ever undertaken . . . It gave us the computer, which in turn gave us… well… the world we know (Orlin).  The invention of the computer changed the world, and it came from pure mathematics. Not only does the processing of the computer use pure mathematics, but also the components powering it use pure mathematics. Electrical engineering requires the use of imaginary numbers. For the longest time imaginary numbers were thought to be useless, it was thought that imaginary numbers meant nothing. Jim Lesurf wrote regarding complex numbers in electronics, saying that,” Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. “(Lesurf). This shows that even a pure mathematical idea can be used in today’s world. In understanding complex numbers, it allows for a better understanding of electronics, which are an integral part of technology today.

            Pure mathematics is an important part of society today, because pure mathematics is still relevant, and will always be relevant. Farida Kachapova also wrote that,” . . . even the oldest known mathematical formulae . . . known 2400 years ago by Babylonians, Chinese and later the Greeks …are the bread and butter of present-day elementary mathematics.” (Kachapova). This means that pure mathematics is still used today. The math the Greeks were using is still being used in the classroom today. Now not only is pure mathematics relevant today, but it will be relevant tomorrow, and for the future to come. The universe will never be fully explained, and there will always be new mysteries, new problems, and a need for new mathematics to try to explain it. Going back to G. H Hardy, “For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” (Hardy). Pure mathematics will always be the tool of applied mathematics. This meaning that no matter whatever problems arise, pure mathematics will be used. The fundamentals of math will always be used to understand new forms of applied mathematics.

            Pure mathematics is often times seen as non-useful form of mathematics. This is simply because pure mathematics is math for the sake of math. The common misconception is that there are not any benefits from working in pure mathematics, and only applied mathematics gets real world results. Pure mathematics is the root of all applied mathematics. Any applied mathematics goes back to number theory, algebra, or calculus roots. Although pure mathematics might not be making the break through, or discoveries, it is facilitating them. Pure mathematics is incredibly important in society today. This is because pure mathematics is the fundamentals of all mathematics, pure mathematics helped developed many of the technological advances of today, and pure mathematics with always be relevant in mans quest of understanding the universe.

           

      Works Cited

Kachapova, Farida. "On the Importance of Pure Mathematics." Science Publications. N.p., 2014. Web. 15 Mar. 2017.

Lesurf, Jim. "Complex Numbers." Complex Numbers. University of St. Andrew, n.d. Web. 18 Mar. 2017.

Orlin, Ben. "Why Do We Pay Pure Mathematicians?" Math with Bad Drawings. Word Press, 25 Feb. 2015. Web. 18 Mar. 2017.

Unknown. "Quotations of G H Hardy." History.mcs.st-andrews. N.p., Dec. 2013. Web. 12 Mar. 2017.

Squaring The Circle

            This week’s post is going to be about a problem called squaring the circle. The problem dates back to the time of the Greeks. The problem itself is not the difficult part, the difficult part of the problem is that one of the restrictions of the problem is to solve it only using the technology available to the Greeks. First, this post will cover the problem itself. Secondly, the post will cover how pi relates to the problem. Finally, the post will cover how pi was proven to be transcendental.

            Squaring the circle is a problem about finding a circle with the same area as a square. At first this problem may not seem hard, but not impossible. After a few minutes of thinking, it might become apparent that a circle of radius one, and a square of side length root pi would have the same are. That would be correct, but the Greeks did not have the square root of pi. The difficult part of the problem is that, the problem can only be solved with the abilities of the Greeks. That means there is no algebra, there are straight edges, and compasses. Straight edges, and compasses limit the ability of available math. With this Greek Technology the four basic functions of mathematics, addition, subtraction, multiplication, and division, can be performed. Addition and subtraction are straight forward. Adding is putting two line segments together, and subtraction is removing a line segment. Multiplication, and division require some scaling with similar triangles, but are also possible. There is one more operation that can be performed with a straight edge and a compass, and that is square rooting. Imagine a line of length x, and then a distance of one is added to it, then a semi-circle is drawn to the end of the line. If a line is drawn vertically from where the line x and line of length one meet, up to the semi-circle, the length of the line is the square root of x. Looking back at one of the solution to the problem, one might ask then why could the Greeks not square root pi? The answer is that pi is not a constructible number, a constructible number refers to a number that can made with a compass, and a straight edge.

            Pi is a rather important number in mathematics, but at the same time it is equally complex, and mysterious. Pi is a transcendental number. The name for transcendental comes from the idea that pi is beyond other numbers, that it has transcended to a high plane of understanding. Or in other words, it cannot be expressed as an algebraic number. This is why the Greeks could not solve the problem. Since pi cannot be an algebraic number, then automatically it cannot be a construable number. However, pi was not known to be transcendental until the late 1800’s. Even though pi was known, and could be calculated, it was still unclear whether or not it was an algebraic number. It was proven to be a transcendental number by a combined understanding from Euler, and a proof of e’s transcendentalism.

            Two important facts about the number e were proven, e is transcendental, and e to any algebraic term is transcendental. This was proven via contradictions. What that means is that if e was algebraic, certain properties would apply. Since those properties do not apply, e is transcendental. Now the same thing is done with pi, but there needs to be a relationship between e and pi. This is where Euler’s Identity comes to the rescue. Euler’s Identity is e raised to the (pi) x (i) is equal to negative one. Going back to the proof of e raised to an algebraic power, if pi is algebraic then e raised to it, should result in a transcendental number. Negative one is not transcendental, so by contradiction it must not be algebraic, so if pi is not algebraic, then it is transcendental.

            Going back to the problem itself, this means that there was not a way for the Greeks to find pi. Even if the Greeks tried to use a square with a side length not having pi in it, the radius circle would have to multiple by a ratio of pi to get a whole number answer. Pi is crucial to this problem, and so it was proven that under the given rules of the problem, Squaring the circle is an impossible problem to solve.

Sources https://www.youtube.com/watch?v=CMP9a2J4Bqw
              https://www.youtube.com/watch?v=seUU2bZtfgM
              http://mathworld.wolfram.com/CircleSquaring.html

Zeno's Paradox

           This post will focus on the workings of two of Zeno’s paradoxes, and how they relate to mathematics. First, the two paradoxes will be covered, and then the mathematical continuation into limits of infinite sums. Secondly, this post will cover how irrational numbers like Pi, e, and the square root of two, fit into the paradox.

            Zeno was a Greek philosopher that lived roughly around 500 B.C. He proposed a set of paradoxes, and the focus will be on two of those paradoxes, for their similarity. The first paradox starts out with a race between Achilles, and a tortoise. Since a tortoise is much slower than Achilles, the tortoise is given a hundred-meter head start of Achilles. After this head start, Achilles sprints a hundred meters to catch the tortoise. Now during the time it has taken Achilles to reach the hundred meters, the tortoise has move forward 10 meters. Again Achilles sprints ten meters to catch the tortoise, but again the tortoise has moved again another meter. This pattern will continue on forever. The paradox is that Achilles can run faster than the turtle, but cannot beat the turtle.

            The second paradox is similar, but it has to do with shooting an arrow. If an archer shoots an arrow at a target, surely it will travel the full distance to the target. Another way of thinking about it is the arrow must first travel halfway to the target. After traveling halfway, it must travel halfway of the new distance to the target. This process of halving the distance will go on forever, the arrow will just go another half the distance infinitely many times. The Paradox is that if there is an infinite number of halving, then how does the arrow ever hit the target? Surely it is known that the archer can hit the target, but then what happened to the infinite series?

            There is only one explanation that satisfies the two paradoxes. The sum of the infinite series must equal the finite distance traveled.  That may seem impossible, but it is completely possible. This process is used heavily in understanding calculus, the study of rates of change. To explain this answer we will start with partial sums, and then move into limits. To start the arrow travels half the distance, and then it travels half of the remaining distance, which would be a fourth of the total distance. So the sum of the fractional values would look like 1/2 + 1/4 +1/8 +1/16 . . . The sum of the first four terms .94 the total distance. So the more terms that are added to the series the closer the series approaches 1. This series, if summed all the way, with an infinite number of terms would equal one exactly. This idea is the introduction to infinite limits. A limit is a way of saying what happens as something approaches something else. In this case the limit as the number of terms approaches infinity would equal one. Many individuals struggle with the idea that there is a sum associated with an infinite series. If someone tried to put the series into a calculator, they would not ever be able to press enter, because then it would not be all of the infinite number of terms in the series. This idea is hard to understand, but at the same time it must be true because an archer can hit a target, people can clap their hands, people can drive from point A to point B. Every kind of movement requires an infinite sum of fractional parts of that distance, and yet everyone does it.

            This paradox can be seen again in irrational constants such as, pi, or the square root of the square root of two. Looking at pi, if a circle is draw with a compass, having radius of one, then the area of the circle is pi. This is concerning because pi goes on forever, so how can a finite area be an endless decimal? The same goes for the square root of two. Imagine a right triangle with two legs of length one. The hypotenuse is equal to the square root of two, a never ending decimal. Except at the same time the length of the hypotenuse does not go on forever. In both of these cases, it shows that some infinite series can be expressed as a finite number. In general, the numbers of the terms in these series must be getting smaller over time, or else they would just shoot off towards infinity.

Sources http://www.iep.utm.edu/zeno-par/

             https://www.youtube.com/watch?v=u7Z9UnWOJNY

Problems with Infinity

            This post is going to be varying slightly from the standard format of these posts. Instead of focusing on a specific problem, the focus will be shifted toward a number of problems and paradoxes concerning infinity. Infinity is not a number; it is a concept. So when infinity gets involved in problem, it sometimes causes the problem to break. This post will cover four of these problems with Infinity.

            The first problem concerns the Hilbert Hotel. This paradox was proposed by David Hilbert. First, imagine a hotel with an infinite number of rooms. In every room there is a mathematician. The problem starts out with a new mathematician walking in, and wanting a room. Well obviously the hotel is full, but the manager is smart enough to come up with a solution. He asks every mathematician to move to the room next to them, so room one moves to room two, and two moves to room three, and so on. So now room number is empty. Since there is an infinite number of rooms, there is never not enough room. In fact, even if a infinite number of mathematicians walked into the hotel they could all find a room. This goes against the initial understanding that the hotel is full, but at the same time there is always room.

            The second infinite paradox is called Gabriel’s Trumpet. The trumpet part comes from the general shape of the trumpet. A hollow cone like shape that tapers off, except this cone shape tapers off to infinitesimally small. A volume of this cone can be found, but yet the cone would have an infinite surface area. If there was a mathematical paint that had no thickness between molecules an infinite amount of that paint could be poured into the cone, but at the same time the volume of the cone can be calculated with calculus. This is the paradox, how can something take up a given space, but yet have infinite surface area?

            The third paradox concerns a dart board. Imagine a dart with a sharp point, that is in fact a single point thick. The problem arises when the dart is thrown at the dart board. The question is, if there is a hundred percent chance that the dart hits the board, what is the probability of hitting a certain singular point on the board? Since there are an infinite number of point in the surface of a dart board, the probability of hitting a certain point cannot be great than zero of the sum of probabilities would be infinite. Also the probability must be more than zero because if it was zero then the dart would not hit the board. The paradox is that a probability of hitting the board cannot be determined even if there is a hundred percent chance the dart will hit the board.

            The final paradox concerning infinity in this post, is about doubling money. Suppose there is a new casino game that has just arrived. The game is centered around a coin, and a pot of money. The pot starts at one dollar, and if a heads is flipped the money in the pot doubles. If a tails is flipped then the game ends, and the pot is collected. To the average individual it would make sense to put in a few pounds at a time, but mathematics has a different answer. Mathematics says to put as much money in as possible. This is because the sum of every outcome tends towards infinity. The paradox is that like any other casino game there is a chance to lose it all, but at the same there is a mathematical guaranteed chance of walking away with an infinite amount of money.

            As pointed out in these four problems, the concept of infinity creates some problems with mathematics. One of the biggest problems is getting around the idea of infinity. In the universe there is not anything that is infinite, except maybe the space of the universe itself, but that is another paradox in itself. In all of these problem there was something that could not exist in the real world. There are not any infinite hotels, infinite surface areas, and people hit dart boards. The answer is not that infinite needs to be removed, but rather that mathematics as a whole needs to work with infinity in another way. When some new idea is introduced to mathematics there is always a fight to hold on to what is known, and not to change. That will simply limit math as a whole. If no one ever stepped outside the box there would not be the complex number, or zeta functions, or countless other aspects. The point is infinity is not something in mathematics that cannot be dealt with. 

Sources https://www.youtube.com/watch?v=dDl7g_2x74Q

              https://nrich.maths.org/5788

              https://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/?_r=0