explanatory Essay

1881 words

The Mathematics of Map Coloring

The four-color conjecture has been one of several unsolved mathematical problems. From 1852 to this day, practically every mathematician has studied the problem long and hard, but to no avail. The conjecture looks as though it has been solved by Wolfgang Haken and Kenneth Appel, both of the University of Illinois. They have used computer technology to prove the conjecture. The calculation itself goes on for about 1200 hours. The staggering length of the computation of the proof is what creates some controversy in the mathematical world. The Appel-Haken Theorem is based on numerous assumptions, “that there is an overwhelmingly great probability that their method of proof must succeed.” [3] It assumes that the theory itself is correct, but the theory itself is also an assumption. You can see why this issue has been wreaking havoc for many years.

It all started back in 1852 when Francis Guthrie was coloring a map of England. He wanted to know the least amount of colors, or chromatic number, it would take to color the map so no two adjacent regions are of the same color. He found the chromatic number to be four. He then studied arbitrary maps and wondered if all maps could be colored with four colors. Francis’ curiosity would be in the minds of all mathematicians to come. He then passed this question on to his brother, Frederick. He then submitted this to his professor Augustus deMorgan as a mathematical conjecture.

deMorgan was fascinated by the Four-Color problem and wrote in a letter to his colleague Sir William Rowan Hamilton the next day after seeing the conjecture. Hamilton was less enlightened by it, and never worked on it. It was through deMorgan that the Four-Color problem was made known, thus deMorgan has incorrectly been dubbed the originator of the problem.

Eventually the hype surrounding the conjecture died down in the early 1860’s. This down time, during which interest in the problem was minimal, only lasted about twenty years. A lawyer by the name of Alfred Bray Kempe proposed a solution in The American Journal of

- Explains how to delete the vertex and all edges leading into it until 6 vertices remain.
- Explains how to color the vertex with a color that isn't used on adjacent vertices.
- Introduces rudolf and gerda pritsch's the four color theorem.
- Explains that the four-color conjecture has been one of several unsolved mathematical problems.
- Assumes the opposite of the lemma, that no vertex on the graph has order five or less.
- Explains that the proof of the five-color theorem is not necessary since it has very similar characteristics to the six-colored proof.
- Explains that coloring a map is important because it can be applied to areas outside of cartography.

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- The Four Color Theoremexplanatory essaySince hundred years ago, when people started to make maps to show distinct regions, such as states or countries, the four color theorem has been well known among many mapmakers. Because a mapmaker who can plan very well, will only need four colors to color the map that he makes. The basic rule of coloring a map is that if two regions are next to each other, the mapmaker has to use two different colors to color the adjacent regions. The reason is because when two regions share one boundary can never be the same color. Another basic rule of coloring a map is that if two regions share only one point, then they do not necessary have to be colored differently. Many evidence showed that coloring a map required at least four colors but no more than five. Then mathematicians started to asked questions, such as “ Is it true that using only four colors are enough? Is there any exception that one has to color a map that requires more than four colors? Or is it has to do with a special sequence of arrangement that involved with different regions in order to make the theorem true?” However, the first mathmatician who asked these questions is a man named Francis Guthrie. He was the first one who posed the four color problem in1852.
#### In this essay, the author

- Explains that the four color theorem has been well known among mapmakers since 100 years ago, when people started to make maps to show distinct regions.
- Narrates how francis guthrie was a mathematician at university college london when he noticed that there should be at least four colors to color regions in order to make two adjacent regions have two different colors.
- Explains that alfred bray kempe, a mathematician, was the first one to give proof on the four color theory and linkage in 1879.
- Explains how danilo blanusa proved the four color theory in 1946. heesch, kenneth appel, and wolfgang haken developed a program to prove the theory.
- Opines that the four color theorem was proved by the computer, but the independent verification convinced people that it was finally proved.

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- Explains that peter g.l. dirichlet was born in the 1800s, and his father was a postmaster in germany where he lived.
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- Cites allen, james, haslam, john, and neale, vicky. "theorem 11the pigeonhole principle."
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- Explains that gregor mendel wrote that genes are passed from parents to their children and can produce the same physical characteristics as the parents. he used pea plants pisum sativum to experiment with.
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- Explains that there have been many great mathematicians in the world, though many are not well-known. the bernoulli family contributed to mathematics, medicine, physics, and other areas.
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1722 wordsRead More - Pierre De Fermatanalytical essayPierre de Fermat Pierre de Fermat was born in the year 1601 in Beaumont-de-Lomages, France. Mr. Fermat's education began in 1631. He was home schooled. Mr. Fermat was a single man through his life. Pierre de Fermat, like many mathematicians of the early 17th century, found solutions to the four major problems that created a form of math called calculus. Before Sir Isaac Newton was even born, Fermat found a method for finding the tangent to a curve. He tried different ways in math to improve the system. This was his occupation. Mr. Fermat was a good scholar, and amused himself by restoring the work of Apollonius on plane loci. Mr. Fermat published only a few papers in his lifetime and gave no systematic exposition of his methods. He had a habit of scribbling notes in the margins of books or in letters rather than publishing them. He was modest because he thought if he published his theorems the people would not believe them. He did not seem to have the intention to publish his papers. It is probable that he revised his notes as the occasion required. His published works represent the final form of his research, and therefore cannot be dated earlier than 1660. Mr. Pierre de Fermat discovered many things in his lifetime. Some things that he did include: -If p is a prime and a is a prime to p then ap-1-1 is divisible by p, that is, ap-1-1=0 (mod p). The proof of this, first given by Euler, was known quite well. A more general theorem is that a0-(n)-1=0 (mod n), where a is prime...
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- Explains that pierre de fermat, like many mathematicians of the early 17th century, found solutions to the four major problems that created a form of math called calculus.
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858 wordsRead More - Alfred Binetexplanatory essayMy thoughts were that because of this discovery psychologist after him were able to expand on his research and make today what it is. Other people might say that it isn’t possible to know how intelligent a brain is from just one test but really the test is just estimation and shouldn’t be taken literally. Currently Alfred Binet works it still being used to base current intelligence tests of off. Alfred Binets' work has been used my many other psychologists to make other intelligence tests.
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- Explains that alfred binet created the first intelligence test and standardized testing. he became interested in child psychology by reading books written by child psychologists like charles darwin and alexander bains.
- Explains that alfred binet started child psychology when he was offered a job by jean charcot and was taught about psychology at the same time.
- Explains how alfred binet was asked to devise a test to test children's intelligence. once the test was completed, the children would either continue with school regularly or need more help by teachers trained to help them.
- Explains that the first intelligence test was created in 1905 by alfred binet and his assistant theodore simon to identify students who needed help coping with the school curriculum and to place them in the appropriate grade level.
- Explains that the stanford-binet test was developed by lewis terman, who used the binet test as a base but substantially revised it.
- Explains that the tests that were formed back then are being taken now by everyone who goes to school. they judge if students should go into harder classes, what book level they should be reading at, and how intelligent a person is overall.
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751 wordsRead More - Biography of Augustus DeMorganexplanatory essayAugustus DeMorgan was an English mathematician, logician, and bibliographer. He was born in June 1806 at Madura, Madras presidency, India and educated at Trinity College, Cambridge in 1823. Augustus DeMorgan had passed away on March 18, 1871, in London.
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- Explains that augustus demorgan was an english mathematician, logician, and bibliographer. he died on march 18, 1871, in london.
- Explains that augustus was recognized as far superior in mathematical ability, but his refusal to commit to studying resulted in him finishing only in fourth place in his class.
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- Explains that demorgan's law commemorates his achievements in laying the foundation for the theory of relations to prepare the way to modern symbolic, or mathematical logic.
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- Explains that demorgan's most important work was formal logic, which solved problems that were impossible under the classic aristotelian logic.
- Explains that demorgan published first-rate elementary texts on arithmetic, algebra, trigonometry, calculus, and important treaties on the theory of probability and formal logic.
- Explains that demorgan contributed many accomplishments to the field of mathematics. he devised a decimal coinage system, an almanac of all full moons from 2000 b.c and 2000 a.d.
- States that augustus wrote biographies of newton and halley and produced a dictionary of all the important mathematicians of the seventeenth century. demorgan felt that it was important for the students to know the history of mathematics.

700 wordsRead More - Spectroscopyexplanatory essayIn the early 19th century, scientists all over the world began to study light and colors. In the beginning, not much was known about colors and what caused certain shades to appear. It wasn’t until 1859 when two German scientists created the foundation for the study of spectroscopy and colors (Historical Introduction to Spectroscopy).
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- Explains that scientists began studying light and colors in the early 19th century. it wasn't until 1859 when two german scientists created the foundation for the study of spectroscopy.
- Explains that robert wilhelm bunsen grew a fascination with light when he first observed that certain colors were emitted when chemicals were burned under his flame lamp invention.
- Explains bunsen and kirchoff's discovery that each element had its own unique signature of colors revolutionized the study of spectroscopy and elements.
- Describes how bunsen and kirchoff used the spectroscope to look at mineral water's spectrum. they discovered cesium and rubidium using the same technique.
- Describes how spectroscopes are used to discover chemical composition in stars and planets million of miles away.
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- Cites the encyclopedia of spectroscopy by robert bunsen and gustav kirchhoff.
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649 wordsRead More - The Seven Bridges of Königsbergexplanatory essayThe bridges of the ancient city of Königsberg posed a famous and almost problematic challenge a few centuries ago. But this isn’t just about the math problem; it’s also a story about a famous Swiss mathematician named Leonhard Euler who founded the study of topology and graph theory by solving this problem. The effects of this problem have lasted centuries, and have helped develop several parts of our understanding of mathematics.
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- Explains that the bridges of königsberg posed a famous and almost problematic challenge. leonhard euler founded the study of topology and graph theory by solving this problem.
- Explains that euler is one of the most important and influential mathematicians ever, along with archimedes and newton.
- Explains that euler developed a new idea called geometriam situs, later called topology. topology is the study of non-rigid shapes.
- Explains that the city of königsberg, founded in the thirteenth century by the teutonic knights, was a major commercial and industrial city.
- Explains that euler dismissed the problem and said it was solvable by reason, but later began to consider it. he had the idea of a graph showing all the bridge locations by making every land mass represented as points, or vertices.
- Explains how euler created a way to find out if an euler walk is possible on any graph.
- Explains that a graph has an euler walk only when there are two vertices with an odd number with the walk beginning at one of these two.
- Explains that the civilians spent so long trying to find the answer to the königsberg bridge problem by looking at the graph of the bridges.
- Explains that an euler walk is possible with the five bridges left in königsberg, but only two of them are from his time.
- Explains that if a graph has zero or two vertices with any odd number of degree, it must have an euler walk.
- Explains that the brick wall puzzle is impossible to draw a line through every section of the figure. the graph shows all possible pathways by putting vertices in each brick.
- Explains how orly terquem created a graph connecting each of the six numbers as vertices to one another, allowing an euler circuit.
- Concludes that leonhard euler was a true genius, and in his career he discovered many principals, which revolutionized mathematics.
- Explains richeson's gem: the polyhedron formula and the birth of topology.

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