Math IA
Four Color Theorem
Matt Reed
Four Color Theorem
I. Introduction
Ever since the beginning of travel and exploration, maps have helped people record the specifics of new and unexplored regions of the earth. The earliest maps were crudely drawn by hand and were rough estimates of geographic area based on interpretation of the land. Once people began coloring maps, to designate partitions within regions, the problem arose regarding the necessary number of colors it would take to color a map.
II. History
The theorem was first proposed by August Ferdinand Moebius in 1840. However, the problem was ignored until 1856. In 1856, Francis Guthrie proposed the conjecture to his brother Frederick who then brought the problem to the attention of his advisor; Augustus De Morgan. From this point on, a series of mathematicians sought to prove that it takes a minimum of four colors to color every region on a map without adjacent regions being the same color.
III. The Theorem
The theorem developed over a series of stages between the 1840s and the 1980s. Multiple false proofs and conjectures were proclaimed in the 1850s. However, they were all eventually disproven. A precursor to the proof, written in 1890 by Heawood, details the use of five colors to shade in the regions of a map. However, the proof involving four colors remained ultimately unproved until the 1970s. The proof was one of the first ever to be proven by a computer, which brought about a revolution in the solutions of mathematical conjectures. Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it. Kempe's argument goes as follows. “First, if planar regions separated by the graph are not trian...
... middle of paper ...
..., numerous renditions of the proof were disproven and built upon. Additionally, the Four Color Theorem was the first proof to be solved with aid from a computational device, creating controversy as to its legitimacy as a proof of its time period. However, now that computers have been integrated into our daily lives, it has become a full-fledged proof that will continue to be in use as long as new places to map are discovered.
Works Cited
O'Connor, JJ, and EF Robertson. "The Four Colour Theorem." The Four Colour Theorem. N.p., Sept. 1996. Web. 04 Feb. 2014.
Rogers, Leo. "The Four Colour Theorem." : Nrich.maths.org. N.p., n.d. Web. 08 Jan. 2014.
Thomas, Robin. "An Update on the Four Color Theorem." AMS.org. N.p., Aug. 1998. Web. 05 Jan. 2014.
Wilson, Robin J. Four Colors Suffice: How the Map Problem Was Solved. Princeton, NJ: Princeton UP, 2002. Print.
Newton, Henry, and William Winsor. "Spotlight on Colour: Flake White." Winsor&Newton. N.p., 2011. Web. 30 Apr. 2014.
The distinguishing uses of colours, hues, and shading that are illustrated in Fun Home are one manner in which Bechdel somberly relates
My 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.
invalidated by an earlier proof from 1578 now given access through a digital resource bank. Last
Ms. Robinson spent more than 20 years on one problem with Martin Davis and Hilary Putnam. Sooner or later then ended up calling the problem a Robinson Hypothesis.She soon became really desperate to see the problem solved or worked out before she died.
He set up his experiment by darkening his room and allowed only a single beam of sunlight to shine on a prism. When the beam of white light hit the prism, Newton saw a multitude of color exiting the prism. This experiment proved that white light is responsible for all the color we see in the world. His experiment also changed people's understanding of color because at that time, people believed that color was actually a mixture of light and
Pythagoras led an interesting life to say the least, but one of the most influential and important aspect of his life was the discovery and study of Pythagorean Theorem. If you’ve gone through almost any level of math you’ve most likely had to use this formula;〖 a〗^2+b^2=c^2. The equation is defined as the “area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.” Although the theorem is believed to be created by Pythagoras had been used by before his time by Native Americans and the Babylonians, but it is believed that he and his students did create the first proof. Pythagoras is also known for creating many theorems and mathematical discoveries such as;
His lectures developed an essay called “Of Colours” which was later revised into Book One of his Opticks. In his work in Opticks, Newton discovered that individual light rays excite sensations of colors in the retina when they strike the eye. He also discovered that rays refract at distinct angles creating the prismatic spectrum, “a beam of heterogeneous rays, i.e., alike incident on one face of a prism, separated or analyzed by the refraction into its component parts—and that phenomena such as the rainbow are produced by refractive analysis.” (Sir Isaac)
Historical geographer JB Harley wrote an essay on Map Deconstruction in 1989, in which Harley argues that a map is more than just a geographical representation of an area, his theory is that we need to look at a map not just as a geographical image but in its entire context. Harley points out that by an examination of the social structures that have influenced map making, that we may gain more knowledge about the world. The maps social construction is made from debate about what it should show. Harley broke away from the traditional argument about maps and examined the biases that govern the map and the map makers, by looking at what the maps included or excluded. Harley’s “basic argument within this essay is that we should encourage an epistemological shift in the way we interpret the nature of cartography.” Therefore Harley’s aim within his essay on ‘Deconstructing the Map’ was to break down the assumed ideas of a map being a purely scientific creation.
Ronald Lewis Graham is an American mathematician who comes from California, and who was born in 1935. His work revolves around looking for patterns in chaotic systems, and so he pursued a pattern involving joining up four points with six lines, all red or all blue, in a tube of many dimensions. This pattern is repeated many times in an all-blue cube, but Graham changed the color of individual edges, trying to avoid the pattern in just one color. His aim was to see if avoidance of the pattern in blue would force it to pop up in red. This doesn't happen in three, four, or even five dimensions. The number of dimensions required to guarantee this pattern turns out to be Graham's number. This part of mathematics is called Combinatorics, and Graham started looking into a more specific field of it, called Ramsey’s Theory. This theory could be explained the following way; this is an example of where complete disorder is impossible. In any large system, you've got to have a smaller set that has a lot of structure to it....
It did not matter if the light was transmitted, scattered, or reflected, the color stayed the same. These experiments lead to the discovery that colors are made up of light, and that the objects do not emit colors on their own. This is known as Newton’s Theory of Color.
Wilhem von Bezold discoveries contributed to the creation of the color systems we have today. He is best known for Bezold Effect or optical interaction of color. He found that he could change the entire appearance of his designs by substituting a different color for the color which occupied the most area. When one is looking at a specific hue, the hue ca...
The claim being discussed here is that the only way a map or a way of representing things can be useful is if it simplifies the knowledge that the actual territory gives, that is, if it reduces the salient i...
...not be able to be created. Although maps have developed from simple maps with east at the top to maps with depictions of different elevations, all throughout history the main goal of maps was to help further peoples understanding of a topic.