Introduction:
In 1852, Francis Guthrie, a South African mathematician, experimented with coloring the counties of England while he was in school at University College in London. Guthrie proposed that four colors were sufficient to fill in all of the regions of a map without any two adjacent regions having the same color. Guthrie’s proposition became known as the Four Color Problem. This seemingly simple proposition perplexed the minds of mathematicians for over a century. Mathematicians made several failed attempts at both proofs and counterexamples until finally, in 1976, Kenneth Appel and Wolfgang Haken proved the statement using a computer. Their proof, however, was not accepted by all mathematicians since the bulk of it was proven with a computer and could not be verified by hand. This paper will present a brief summary of the historical failed attempts at a proof, as well as an overview of the proof by Appel and Haken and proofs following theirs.
Definitions:
In order to discuss the technical aspects of the proofs, we must first establish some formal definitions regarding the Four Color Theorem. These statements and definitions are taken from Georges Gonthier’s proof [2].
Four Color Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors.
This statement of the Theorem will be the formal definition to be proved. However, in order to facilitate a proof of this theorem, we must more technically define some of its terms.
Planar Map: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets.
Adjacency: Two regions of a map are adjacent if their res...
... middle of paper ...
...rely on computer verification for its hundreds of cases, more of this proof was able to be hand-verified which pacified some of the controversy amongst mathematicians who did not trust the computer assisted proof. The most recent attempt at a complete proof came in 2008 by Georges Gonthier of Microsoft Research, Cambridge. Gontheir’s proof was the first to use a “formal proof script” [1]. This computer script was the first to provide logical reasons for each step the computer made, just as a human would when constructing a proof.
Conclusion:
The Four Color Theorem holds the record as the first theorem to be proven true using a computer. Although not accepted by some, most mathematicians consider the theorem to be true based on the Appel-Haken proof in 1976. The proof of this specific theorem is just one example of the power a computer gives the mathematician.
They captured the natural beauty of Canada’s geography and in doing so gave Canadian’s a sense of artistic identity. The Group of Seven used dominating colours, like the greens and blues mentioned earlier, to not only had colour to their work but to also distinguish the Canadian style of art from European landscape paintings. In the 20’s, European paintings were very realistic landscapes and portraits, using brown as the dominant colour. The Group of Seven moved away from the European style of art with vibrant colours and loose brush strokes to create a new sense of Canadian identity through art.
The colors he uses to create this harmony are green, blue-green, and blue. The green color is mostly located on the bottom part of the painting and the left side, while the blues are present at the top in the sky and the middle as the river. Derain’s piece has complementary color harmony, which is when one or more pairs of colors that are located directly opposite of each other on the color wheel, dominating the painting. The colors used are yellow-orange, green, red-orange, and blue. The red-orange and yellow-orange draw your eye instantly in this painting and are located mostly in the middle, while the blue and greens are along the
colors to create one large color. For instance, in the sky he uses a mixture of
Alan Turning is known to be a pioneer of many facets of the computer age. The digital computer, artificial intelligence, memory subroutines, the Turning Machine, the Turing Test, and the application of algorithms to computers are all ideas somehow related to this man.
This fourth dimension effect was reached by stripping away the distractions. Malevich's art was made to be felt and he broke down complex characters into the simplest of geometric shapes. The colors that he chose for his subjects were not the ones that were true to life. He did this by choice to get past the human biased way of seeing an object. He was trying to transmit pure sensation. The actual subject is irrelevant; the feelings of it are the main focus.
In 1794 he was elected a member of the Manchester Literary and Philosophical Society. There he read his papers and identified the phenomenon of colour blindness, which he and his brother shared. When showed a colour spectrum besides blue and purple Dalton was only able to recognise one other colour, yellow. Or as he says ?that part of the image which others call red appears to me little more than a shade or deflect of light. After that the orange, yellow and green seem one colour which descends pretty uniformly from an intense to a rare yellow, making what I should call different shades of yellow?
Goldstine, Herman H. "Computers at the University of Pennsylvania's Moore School." The Jayne Lecture. Proceedings of the American Philosophical Society, Vol 136, No.1. January 24, 1991
When viewing Georges Seurat’s, A Sunday on La Grande Jatte (Fig. 31-37), perception is changed vastly depending on the viewer 's proximity to the piece. At close range, all that is visible is a mass array of countless circular dots and tiny lines in a vast range of colors. Greens, blues, reds, oranges, yellows, white, browns, black and purples are all visible in a multitude of intensities. The Divisionism technique utilized causes this piece to appear as an abstract collection of colors when viewed at close range. Yet when distance is between the piece and the viewer, these seemingly sporadic dots come together to create a complete and detailed scene. Primarily consisting of biomorphic shapes, Seurat’s incorporates in every inch of the canvas
Before delving into the philosophy of color and the questions posed by different pieces of art, the biological basis process of HOW color is seen should be explained. The retina is a neural sheet, containing the photoreceptors called rods and cones that is located at the back of the eye. Between the retina and the optic nerves leading to the brain are a series of cells that create a lateral inhibition network of the light/dark signals from the photoreceptors (1). This throws away a lot of the information generated by the photocells and gives the brain a "picture" of the edges of light and dark. The contrasts are created, leaving the brain to fill in the rest. Color vision is even more complex. Cones (the light adapted photoreceptor) contain three different photopigments, red, green, and blue, each corresponding to a particular wavelength of light (2). Color is a property of three things: the ratio of red/green activation, blue response, and value or lightness (3). The brain utilizes visual inputs to determine each of these, thus generating the characteristic color.
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.
This essay is written to introduce the Russell’s Theory on Definite Description. The main content of this essay including: the definition of definite description, the puzzles concerning definite description, Russell’s Theory on Definite Description, how this theory solves the puzzles, Strawson’s objection to this theory, my evaluation on the convincingness of Strawson’s objection and my evaluation on the convincingness of Russell’s Theory of Definite Description.
It is well known that in the past, Renaissance artists received their training in an atmosphere of artists and mathematicians studying and learning together (Emmer 2). People also suggest that the art of the future will depend on new technologies, computer graphics in particular (Emmer 1). There are many mathematical advantages to using computer graphics. They can help to visualize phenomena and to understand how to solve new problems (Emmer 2). “The use of ‘visual computers’ gives rise to new challenges for mathematicians. At the same time, computer graphics might in the future be the unifying language between art and science” (Emmer 3).
Between 1850 and 1900, the mathematics and physics fields began advancing. The advancements involved extremely arduous calculations and formulas that took a great deal of time when done manually.
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.