The Mathematics of Map Coloring
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
In this essay, the author
- 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.
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- 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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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.
In this essay, the author
- 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.
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- Explains that augustus demorgan was an english mathematician, logician, and bibliographer. he died on march 18, 1871, in london.
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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).
In this essay, the author
- 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.
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In this essay, the author
- 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.
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