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# Art and Consciousness Transfor NCLC 375

Mason

GPA 3.69

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This 3 page Class Notes was uploaded by Ulices Anderson on Monday September 28, 2015. The Class Notes belongs to NCLC 375 at George Mason University taught by Staff in Fall. Since its upload, it has received 26 views. For similar materials see /class/215076/nclc-375-george-mason-university in New Century Lrng Communities at George Mason University.

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Date Created: 09/28/15

NCLC 375 The Final Research Paper 7 The Topics These topics are strongly recommended suggestions I think each one would work just ne If you have another idea suggest it to me and if I approve it then you may do that topic This Final Research Paper takes the place of the Final Exam Requirements are three things 1 Group Class Presentation 7 not long maybe 12 hour containing some historycontext and the statements of some Theorems Maybe a tiny proof or indication of a proof of something This will be the last day of class In all likelihood your nal paper will not be done by then but this is ne for the hard part of your paper the part that will take you the most time is the preparation of the mathematics component and you are not really going to present this to the class 2 Group Paper Paper should contain both Mathematical and Other historical contextual content It should be 2000 to 2500 words this is approximately 8 to 10 pages double spaced 125quot leftright margins lquot topbottom margins size 12 Times New Roman excluding diagrams if you have any This is due by the day the Final Exam is scheduled 3 Individual Oral Presentation just to me during the Final Exam Period 7 or earlier if you are ready This is just the mathematics of your Paper and will in all likelihood will quotsimplyquot be the presentation to me of a particular proof NOTE To be clear both the Class Presentation and the Paper are group projects The Oral Presentation will be done singly You are encouraged to show me a draft of your paper and or to meet with me about it to discuss its mathematical content quotMathematical Contentquot means the paper must contain the very careful statements of some substantial theorems and then some arguments full proofs heuristic proofs or partial proofs ie a special case is proven or a piece of a theorem is proven or some examples given of one or more theorems I ve indicated those I very strongly really think ought to be include in a given topic by or or or You are welcome to ask me if some other theoremproof you have found in your research would be appropriate The number of s indicate the dif culty You should certainly include those with and when possible those marked 7 the better papers of course would include s It39s icing on the cake to include those with more s than 2 Icing is good of course OR 7 ifyou do a quothome runquot on a or then you don t have to do a quotOther Contentquot means history of people and or cultures quotstoriesquot about some theorems like who proved them and when and why commentary re ections connections with other areas of mathematics Each paper ought to contain a de nitionvocabulary list and a bibliography properly formatted These are in addition to the 8 to 10 required pages Some comments about these topics First they are all important both as part of the subject of mathematics but also in their effect on other disciplinary thinking in for example biology physics engineering and computer science Further their scope ranges from ideas that can be taught in high school to those from fantastically advanced mathematics so you ll have to insert yourself into the appropriate level Remember some you will be able to understand and include some proofs of the basic theorems but you will only be able to state without proof the more advanced ones But these latter theorems which are really at the edge of our knowledge need to be includedstated I m going to give some references below quotMath Univquot our text quotJourneyquot Journey through Genius The Great Theorems of Mathematics by William Dunham Penguin Books 1990 This is readily available in our library I m sure in various book stores and you may borrow mine to duplicate the few pages you might need 1 All about the number TE 0 Give a bit of the history of TE both old and new stuff 0 STATE Formulas using TE like area or surface area or volume with arguments when possible Any calculus book has these formulas often in a table inside the front of back jacket STATE Formulas expressions for that that give the value of TE There are lots of them These too can be found in any Calculus text and are readily available elsewhere as well They look like fancy sums or products for example Some involve trigonometry Discuss some people who have contributed to our understanding of TE Archimedes Lambert Hermit and discuss the Greek problem of quotsquaring the circlequot Some words you might include Irrational Algebraic Transcendental Series Use the fact that TC is not algebraic show the circle can t be squared Skim a bit of quotJourneyquot 7 pages 1113 and then look at middle of page 24 to top of page 26 Derive Archimedes39 estimation of TE See quotMath Univquot 7 start in the middle of page 28 and continue until the bottom of page 31 2 A Focus on Archimedes The story of his life and times The story and explanation of the quotcrown that was not solid goldquot Statediscuss his Theorem on the Cylinder and the Sphere it39s on his tombstone for both volume and surface area Derive the Theorem on the Cylinder and the Sphere from the formulas which you may assume for the volume and surface area of a cylinder and that of a sphere See quotMath Univquot 7 page 235 top Read from the top but discuss for sure from the paragraph towards the bottom starting quotSo this is the great theorem from quot Derive Archimedes39 estimation ofn See quotMath Univquot 7 start in the middle ofpage 28 and continue until the bottom of page 31 3 Fibonacci Numbers named for Leonardo Pisano Fibonacci There are just tons of references for this The web is a great resource Connection with Art Architecture Biology and Music Golden Rectangle and the number the golden ratio 1 phi Formulas for the nth Fibonacci number Formulas involving 1 Relations among Fibonacci numbers A discussion of recursion relationships Derive a formula for I using the quadratic formula Prove ratios of consecutive Fibonacci numbers approach I Derive a formula for the n3911 Fibonacci number The answer involves the square root of 5 You can just quote the relevant theorem on recursive relations You ll see the words quotcharacteristics equation or characteristics polynomialquot 4 Logic The Foundations of Good Thinking Any elementary book on Mathematical Logic is a good source Truth Tables quotandquot quotorquot quotnotquot quotimpliesquot Quantification uses of the phrases quotfor allquot and quotthere existsquot Axiomatic Systems People Aristotle who created logic Boole Turing Russell Godel Direct proofs indirect proofs contrapositive converse tautology syllogism Describe Russell s Paradox Describe Godel39s Incompleteness Theorem Describe the rule of inference called Monus Ponens Use truth tables to prove that A gtB lt gt not B gt not A is a tautology and do the same for R and P gtQ lt gt R gtP gtQ Discuss proofs via Mathematical Induction and give an example 5 Pythagoras and the Pythagoreans 7 Version A The Pythagoreans as mystics philosophy music and science Maybe start with a bit on Zeno s Paradoxes 7 connect with the Pythagoreans Prove the Pythagorean Theorem two specific quoteasy waysquot See quotMath Univquot 7 The Chinese Proof pages 9192 and the Similarity Proof pages 9395 Prove Pythagorean Theorem via Euclid s original proof in many geometry texts This is a wonderful proof If you choose this verify with me that you39ve found Euclid s proof 6 Pythagoras and the Pythagoreans 7 Version B The Pythagoreans as mystics philosophy music and science Maybe start with a bit on Zeno s Paradoxes 7 connect with the Pythagoreans Find a formula which generates produces Pythagorean Triplets that is triplets of whole numbers a b c such that a2b2c2 Here are two such examples 3 4 5 and 5 12 13 and then show that this formula indeed always produces Pythagorean Triplets Most elementary Number Theory books contain this

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