The Final Research Paper - Some Ideas

These topics are suggestions. I think each will "work" just fine. If you have another idea, suggest it to me and if I approve it, then you may do that topic.

Paper should contain both mathematical and "other" content. It should be at least 2000 words (8 double-spaced pages, 1.25" left/right margins, 1" top/bottom margins, size 12 Times New Roman). It will be due at the end of the regularly scheduled final exam and will take the place of a formal final exam. You'll do in class a presentation, the last day of class, of a bit of your paper. Since it in all likelihood not be done by then, it can't be about your final version (nor would we have time – each presentation would be 15-20 minutes). There is a good possibility that I may ask each person to present just to me some of the mathematical content of the paper. This could occur during the final exam period, or earlier if you are ready.

You are encouraged to show me a draft of your paper and/or to meet with me about it to discuss its mathematical content.

"Mathematical Content" means the paper must contain the very careful statements of some substantial theorems and then some arguments: full, heuristic, or partial proofs (i.e., a special case is proven or a piece of a theorem is proven or some examples given) of some theorems. I've indicated those I very strongly really think ought to be in a given topic by (!) or (!!) or (!!!) or (!!!!). [You are welcome to ask me if some other theorem/proof you have found in your research would be appropriate.] The number of !'s indicate the difficulty. You should certainly include those with (!) and, when possible, those marked (!!) – the better papers of course would include (!!)'s. It's icing on the cake to include those with more !'s than 2. Icing is good, of course.

"Other Content" means history (of people and/or cultures), "stories" about some theorems (like who proved them and when and why), commentary, reflections, connections with other areas of mathematics, ….

Each paper ought to contain a definition/vocabulary list. This is in addition to the 6 (or so) 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 goes 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 included/stated.

(1) All about the number π

• Formulas using π (like area or surface area or volume), with arguments, when possible.
• Formulas/expressions for that that give the value of π.
• Some People: Archimedes, Lambert, Hermit, Lambert (and discuss the Greek problem of "squaring the circle").
• Some words: Irrational, Algebraic, Transcendental, Series
• (!) Use the fact that π is not algebraic, show the circle can't be squared.
• (!!) Derive: Archimedes' estimation of π

(2) A Focus on Archimedes

• The story of his life and times.
• The story (and explanation) of the "crown."
• State/discuss his Theorem on the Cylinder and the Sphere (it's 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).
• (!!) Archimedes proof that the area of a circle = πr².

(3) Fibonacci Numbers (named for Leonardo Pisano Fibonacci)

• Connection with Art, Architecture, Biology, and Music
• Formulas for the n^th Fibonacci number
• Relations among Fibonacci numbers
• A discussion of recursion relationships
• Golden Rectangle and the number Ф (phi).
• (!) Derive a formula for Ф using the quadratic formula
• (!!) Prove ratios of consecutive Fibonacci numbers approach Ф.
• (!!!) Derive a formula for the n^th Fibonacci number.

(4) Logic: The Foundations of Good Thinking

• Truth Tables: "and," "or," "not," "implies"
• Quantification (uses of the phrases "for all" and "there exists")
• Aristotle (who created logic), Boole, Turing, Russell, Godel
• Direct proofs, indirect proofs, contrapositive, converse, tautology, syllogism.
• Describe Russell's Paradox
• Describe Godel's Incompleteness Theorem.
• (!) Use truth tables to prove that (A→B)↔(not B → not A) is a tautology
• (!!) Discuss proofs via Mathematical Induction and give an example.

(5) Pythagoras and the Pythagoreans – Version A

• The Pythagoreans as mystics: philosophy, music, and science
• Maybe start with a bit on Zeno's Paradoxes – connect with the Pythagoreans.
• (!) Prove the Pythagorean Theorem (find an "easy" proof).
• (!!) Prove: Pythagorean Theorem via Euclid's original proof (in many geometry texts)

(6) Pythagoras and the Pythagoreans – Version B

• The Pythagoreans as mystics: philosophy, music, and science.
• Maybe start with a bit on Zeno's Paradoxes – connect with the Pythagoreans.
• (!) Find a formula which generates (produces) Pythagorean Triplets, that is triplets of whole numbers (a, b, c) such that a²+b²=c². Here are two such examples: (3, 4, 5) and (5, 12, 13) and then show that this formula indeed always produces Pythagorean Triplets.
• (!!) Show that every Pythagorean Triplet is of the form described by your formula, above. That is, show that if one is given a Pythagorean Triplet, it is produced from your formula, above.

(7) Fermat's Last Theorem – Version A

• History of the problem, including the history of its final solution.
• A bit about Fermat (including, maybe, Fermat's Little Theorem) and some other work he did.
• (!!) Prove: The square root of 2 is irrational, using Fermat's Method of Descent
• (!!!) Prove: There is no solution to a⁴+b⁴=c⁴ [You can and should assume the Theorem on Pythagorean Triplets] using Fermat's Method of Descent
(7') Fermat's Last Theorem – Version B
• History of the problem, including the history of its final solution.
• (!!) Prove: The square root of 2 is irrational, using Fermat's Method of Descent
• (!) Find a formula which generates (produces) Pythagorean Triplets, that is triplets of whole numbers (a, b, c) such that a²+b²=c². Here are two such examples: (3, 4, 5) and (5, 12, 13) and then show that this formula indeed always produces Pythagorean Triplets.
• (!!) Show that every Pythagorean Triplet is of the form described by your formula, above. That is, show that if one is given a Pythagorean Triplet, it is produced from your formula, above.
• (!!) [optional ]Prove: The square root of 2 is irrational, using Fermat's Method of Descent
• (!!!) Prove: There is no solution to a⁴+b⁴=c⁴ [You can and should assume the Theorem on Pythagorean Triplets] using Fermat's Method of Descent

(8) Solving Equations with one unknown: The Theory of Algebraic Equations

• It's one of the great mathematical problems of Greek Antiquity.
• A bit about the situation for degree 1, 2, 3, 4 and then degree 5
• Contributions by different cultures: Greek, Western, Arab, Indian
• The History: Babylonians, Euclid, Arabs, Omar Khayyam, Al-Kashi, Niccolo Fontana (Tartaglia), Girolamo Cardan, Francois Viete, Lagrange, Gauss [Fundamental Theorem of Algebra], Niels Henrik Abel [a major player], Evariste Galois [the "winner"].
• (!) Derive the solution for degree 2.
• (!!) State the solution for degree 3.
• (!!!!) Derive the solution for degree 3

(9) The Mathematics of Motions and Cosmology: Kepler and Conics

• A brief history of Cosmology: Ptolemy, Copernicus, Brahe, Kepler
• The Conic Sections: Parabola, Ellipse (and the special case of circle), and Hyperbola.
• Tell how an ellipse is made with two pins, a piece of string, and a pencil.
• Discuss Kepler's Three Laws of Motion.
• Give the algebraic equations of the parabola, ellipse and hyperbola.
• Describe the reflective properties of the Parabola, and give an application (so that you need to discuss the role of the "focus" of a parabola).
• (!) Graph the ellipse x²/(3)² + y² /(4)² = 1. [Just to be clear, this is the same equation as x²/9 + y² /16 = 1. When you research this problem, you'll see why I wrote 9 as 3² and 16 and 4².
• (!!) Prove the reflective property of the ellipse (there is a nice, geometric, proof that does not use calculus.