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Ideas for Great Theorem Topics

Our biggest assignment is the “Great Theorem” report and presentation. We will interpret the term “great theorem“ liberally to allow papers on the development of mathematics in a particular culture or time period. The list below is only meant to be suggestive—you may propose a topic not on the list, subject to my approval. Individually you will write a report on a great theorem or topic and prepare a presentation for the class. Pick your top three choices for exploration and send the topics to me via email in order of preference before class on Friday, February 15. Topics will be assigned on a first-come, first-served basis, so the sooner you send the email, the more likely it is that you will get your first choice.

1. Incommensurables 2. Goldbach conjecture 3. Logarithms 4. Trisecting an angle 5. Doubling a cube 6. Golden Ratio 7. Problems from the Rhind Papyrus 8. Limits and continuity 9. Continued Fractions 10. Demoivre’s Theorem 11. Central Limit Theorem 12. Rolle’s and mean value theorems 13. Taylor series 14. Mathematical induction 15. Platonic and Archimedean solids 16. Linear programming 17. Karush-Kuhn-Tucker Theorem 18. Non-Euclidean Geometry 19. More of Archimedes’ Contributions 20. e is transcendental 21. Cauchy Residue Theorem 22. Desargues’ Theorem 23. Ptolemy’s Theorem and related results 24. Trigonometric functions and identities 25. Diophantine equations 26. Concept of function 27. Constructible polygons 28. Distribution of prime numbers 29. Kepler’s laws of planetary motion 30. Heron’s formula 31. Perfect numbers (with abundant and deficient numbers) 32. Mathematics of navigation 33. Fractals 34. Tessellations 35. Four color theorem 36. Kepler’s Conjecture 37. Fundamental Theorem of Algebra 38. Euler’s Formula 39. Godel’s Incompleteness Theorems 40. Continuum Hypothesis 41. Nonstandard Analysis

MTH 482–GUIDELINES FOR GREAT THEOREM REPORTS

1. A chronological survey is one possible way of organizing your work. An alternate approach might focus on the mathematicians who contributed to the subject or on the cultures in which the work was done.

2. Clearly identify your topic and place it in a broader historical and mathematical setting. Trace its development through time and across cultures.

3. Include some interpretation of the historical and mathematical facts. What is the mathematical and/or historical significance of your great theorem for the overall de- velopment of mathematics?

4. Include some mathematical details; i.e., do some mathematics. For example, show how a particular theorem was proved, solve a relevant mathematical problem, derive a formula, or explain how an algorithm works. Make it clear whether these these mathematical details are your own work or something you took from another source. If the mathematics is more than a page long, just include it in an appendix and refer to it in the body of the paper.

5. Document statements of historical fact and opinions that are not your own. Do not plagiarize, but give credit to authors whose ideas you use. Citations in parentheses, for example: (Burton, 2006) if you used your own words; or (Burton, 2006, p. 24) if it is a direct quote, should refer to items in your reference section. See the APA Citation Style Guide at the Miami University Libraries website.

6. You should consult a variety of references while doing your research. Do not rely too heavily on web sites or on general history of mathematics textbooks. Try to find specialized books and journal articles that address your topic. Try to include excerpts from an original source.

7. Expository writing should be clear, well organized, unified, and coherent. Choose words carefully and proofread your work to correct errors in spelling, grammar, or punctuation.

8. Your report should be double-spaced, in 12-point type, with one-inch margins. Con- cerning length: these reports are typically in the range of 6-10 pages. Remember that quality is more important than quantity.

9. You will submit your written report by Friday, March 22, at 5 PM, and 10-12 minute oral reports summarizing your topic will begin on April 29. (A sign-up sheet will determine the order of presentations.) Plan to speak for at most 12 minutes, followed by a few minutes for questions. Please prepare a brief handout for the class, summarizing the main points of your talk.

10. Don’t procrastinate!