**Course Description** -- Math 300 -- Spring 2017

16467 Monday, 3:00 - 3:50, Taft Hall 309

**Instructor:** Louis H. Kauffman

**Office: **533 SEO

**Phone: **(312) 996-3066

**E-mail: ** kauffman@uic.edu

**Web page: **http://www.math.uic.edu/~kauffman

**Office Hours**: 4PM on M, 3PM on WF.

This is a course devoted to writing mathematics, and writing about mathematics. You will write a number of essays, with plenty of time to complete them. We will later make the decisions about when the essays are finally due. You will be asked to hand in drafts of each essay and there will be ample feedback and discussion before the work is completed. In this way, each essay will give you an opportunity to improve your writing skills.

See UIC Library. This is a website provided by Librarian David Dror, especially for Math 300. It is devoted to showing you many resources for finding information in the UIC Library, and on the Web. Browse this site and try some of the different pathways to information that it provides. You can contact David Dror through this website, and we he will be happy to help you in your research.

While this is a one-class-hour per week course, there will be a lot of work to accomplish. This course is an important opportunity for you to improve your ability to write and to communicate. Skill in writing and communication is invaluable for all jobs and all professions.

FOR THE LAST CLASS: Please have records of all your completed assignments. This is for checking the instructor's records. Just hand in those assignments that are due at that time.

**Remark. **Please keep copies of the drafts of your writing assignments
that have been approved for no further rewrites.

**Remark. **All essays must be produced using a word processor and printed out when handed in. It is recommended that you learn the mathematical
word-processing system, LaTeX. However, MSWord and other processors will be sufficient.

**First Assignment:** (Due January 23, 2017.) Write an essay on geometry based on the lecture given in the first class. This is an
exercise in turning class notes into a more complete form. We request that you choose a title for your essay and write it in complete sentences, using
diagrams where it is appropriate. If you use material from a book for from the web, make reference to that material with a footnote
or with a list of references. The work you hand in for this assignment will be read and you will (most probably) be asked to re-write the essay
in the light of the comments. See Geometry.
This is a summary of some of the remarks that we made about geometry in class.
See On Writing Mathematics. This is an article with guidelines for writing mathematics well.
Please read this article before the second class meeting. Note that in writing mathematics you should use complete sentences and you are allowed
to use algebra and you can and should include diagrams and refer to them.
The art of writing good mathematics is to achieve a balance between formulas, diagrams and the written text.

**Second Assignment:** (Due January 30, 2017.)

Write an essay proving the Pythagorean Theorem.

If you can, find someone with whom you can discuss this topic or to whom you can teach the topic. Discuss or teach as the case may be. Then write a two page explanation of the topic, in your own words.

**Third Assignment:**

Write an essay on a topic of your choice. (Due February 6, 2017.)

**Fourth Assignment:**

Write an second essay on a topic of your choice. (Due February 13, 2017.)

**Fifth Assignment:**

Get caught up on your rewrites. Viete An excerpt from "A History of Pi" by Petr Beckmann, proving the remarkable formula of Viete for pi.

** Sixth Assignment:**
Write a third essay on a topic of your own choice. Due Feburary 27,2017.

** Seventth Assignment:**
Get caught up on your rewrites for March 6, 2017.

Here is an interesting article about "infinitesimals". Henle. Along with this, you may like to look at my notes on Formulas for Pi.. These notes discuss, as we did in class, a way to get limit formulas for Pi by using square roots of commplex numbers. The notes also discuss how to put our results in the form of infinite and infinitesimal numbers, using a system like that shown by Henle in his paper above.

** Eighth Assignment:**
Write an fourth essay on a topic of your choice. Be sure to write a self-contained essay that has
mathematical content (proofs, reasoning, examples) and correct references. Your essay should give evidence that you
have explored the subject your are talking about and not that you just looked up something and are reporting on it.
Go beyond just reporting to actually doing something of interest to you. Due March 13, 2017.

** Ninth Assignment:**
Write an fifth essay on a topic of your choice.Be sure to write a self-contained essay that has
mathematical content (proofs, reasoning, examples) and correct references. Your essay should give evidence that you
have explored the subject your are talking about and not that you just looked up something and are reporting on it.
Go beyond just reporting to actually doing something of interest to you. Due March 27, 2017.

** Tenth Assignment:**
Read the paper Fermat. and print a copy of it.
On your copy notate all mistakes, stylistic and mathematical that you can find in the paper.
Discuss the following argument in relation to what you read in the paper you are correcting: Suppose that we could prove that for any postive rational number P/Q then ((P/Q)^n +1)^(1/n) is
irrational for all n>2. Then we could prove the Fermat Conjecture. For Suppose that there were positive integers P,Q and R so that
P^n + Q^n = R^n for some n>2. Then, dividing by Q^n, we have (P/Q)^n + 1 = (R/Q)^n. Thus ((P/Q)^n +1)^(1/n) = R/Q would be a rational
number. This contradicts our assumption that ((P/Q)^n +1)^(1/n) is irrational.

See also FermatLost. This is a time travel story about searching for the proof of Fermat's Last Theorem.

** Eleventh Assignment:**
Find a Science Fiction story that involves mathematics. Write a report on the story, how the mathematics
is used in the story and give a concise explanation of the mathematics itself. Thus your report should consist in two sections, one
devoted to the story and a second section devoted to the mathematics itself. THIS ASSIGNMENT IS DUE ON MONDAY, APRIL 17, 2017.

** Twelveth Assignment:**
Write an essay on a topic of your choice.Be sure to write a self-contained essay that has
mathematical content (proofs, reasoning, examples) and correct references. Your essay should give evidence that you
have explored the subject your are talking about and not that you just looked up something and are reporting on it.
Go beyond just reporting to actually doing something of interest to you. (You can delve further into the relationships of
mathematics and science fiction if you wish.THIS ASSIGNMENT IS DUE ON MONDAY, APRIL 24, 2017.

**Recommended Reading: ** Logicomix, Bloomsbury Press, by Apostolos Doxiadis and
Christos H. Papadimitriou. This book is available via
Amazon. Logicomix is a graphic novel about the search for foundations of mathematics at the hands of
Bertrand Russell, Gotlob Frege, David Hilbert, Kurt Goedel, Ludwig Wittgenstein and others, set in
the context of a political lecture by Bertrand Russell at the beginning of World War II.

Recommended Reading: The No Sided Professor. This is a topological short story by Martin Gardner.

Recommended Reading: The Feeling of Power. This is a short story by Isaac Asimov.

Recommended Reading: InfiniteHotel. A cartoon story about an infinite hotel that can seemingly accomodate any new influx of guests.

Recommended Reading: Math SciFi. Article by Alex Kasman.

Recommended Reading: Math SciFi. Greg Egan's Homepage

Recommended Reading: Tesseract. "And He Built a Crooked House", by Robert Heinlein.

Recommended Reading: Math SciFi. Dartmouth Course on Math and SciFi.

Recommended Reading: Mathematical Fiction General Access to Mathematical Fiction.

Recommended Reading: Math SciFi. Science Fiction Arxiv.

BEYOND THIS POINT ARE MANY SUGGESTIONS FOR READING AND PROJECTS.

See Isoceles. This is a discussion of a fallacious proof that all triangles are isoceles. You may wish to investigate this as an essay topic. Beware some of the discussions on the web that claim that this fallacy is a hole in Euclid's geometry. The proof is based on an incorrect assumption about the location of an intersection point. Once one does the work to prove the correct location of that point, there is no longer a problem.

See Mind. Here is a paper, published in the journal "Mind", that makes an apparently convincing argument that it should be published in "Mind". You may wish to analyze the argument in this paper. (It was published in "Mind".).

See Meander. This is a paper about "meanders", a type of design related to mazes and weaves and many aspects of art and mathematics. You could read this paper and report on it or explore some of its ideas and write about that.

You can let the mathematics be the game of Sprouts as explained in Sprouts. . In this case you should have some experience in playing the game, and your paper should define the game and give a proof that sprouts always ends in a finite number of moves.

Other Examples: how to solve a quadratic equation and its relationship with complex numbers, what is a set and what is an infinite set, Euclid's proof that there are infinitely many prime numbers, the concept of probability, what is calculus, what is projective geometry ...

See Euler's Formula. This is a 2-page description of Euler's formula V - E + F = 2 for connected plane graphs. Euler's formula is interesting in its own right and it is a powerful tool for solving many combinatorial problems. If you are puzzling over Brussel Sprouts, here is a hint. The final graph in the game has number of regions equal to the initial number of liberties. Use Euler's Formula and prove that a game of Brussel Sprouts that starts with N spots must end in exactly 5N - 2 moves.

See Missing Square Puzzle. This is a very nice application of geometry to a paradoxical puzzle.

See Phyllotaxis. This is an article by Douady and Chouder that explains how the Golden Ratio ends up controlling the growth and patterns of pine cones.

See Pythagoras. In his book "The Ascent of Man", Jacob Bronowski suggests that perhaps the proof indicated in this picture is the way Pythagoras proved his Theorem. You can discuss this proof, filling in needed detail. You may also want to look at the larger context given in the book by Bronowski.

** YouTube And Logic:** See Numberphile. You will find at this YouTube
site an enthusiatic explanation of the identity -1/12 = 1+2+3+4+.... But this very same sort of manipulation that they use can be used as
follows: Let S = 1+1+1+1+.... Then S = 1 + S and hence (subtracting S from both sides) we get 0 = 1. In other words, the sort of
manipulation used there is not consistent! Don't let the Internet bamboozle your brain! But do examine the Wiki on this same topic.
Sum

See Central Limit Theorem for notes on the binomial distribution and the Central Limit Theorem and an excerpt from the book "Mathematics for the Million" by Lancelot Hogben.

See Curved Spacetime. This is an excerpt from the book "Relativity Visualized" by Lewis Carroll Epstein.

Here is some informaton about the topology of closed bands. See Mobius. You may be interested to obtain scissors and tape and perform the experiments indicated on the sheet. You can write about the experiment and see if you can generalize the results to bands with n half-twists where n is an abitrary integer.

See Article. and Summary of Article. This is an example of an article and a summary of that article as it appeared in Mathematical Reviews.

See UMG. This is the website for the Undergraduate Mathematics Journal. It a good source of interesting articles. See Old Hats for a good example of a paper published in the Journal.

See Conway's Army for a short article explaining a solitaire game and a proof about its limitations.

See Existence for an existence proof and a story related to the question whether existence proofs should exist.

See Desargues for a short proof of the three-circles theorem using the Desargues configuration. This is an elegant sample of projective geometry.

Below this point are further suggestions for reading.

########################################################################See Geometry This is a summary of some remarks that we made about geometry in class.

See ContFrac. This is a calculator on the web that converts numbers to their continued fracations. Try it on sqrt(2), sqrt(3), e, pi and other favorite numbers.

See VanPoorten. This is an excellent article on continued fractions.

See Recreations. This is an access page to programs and mathematical recreations.

See Proof This is a remark about proofs in mathematics.

See PenroseTriangle This is an excerpt from an article by Penrose and some other writing, for class discussion.

See Proof of Pythagorean Theorem. This is a new proof recently published in the American Mathematical Monthly.

Below this point are a number of links you may find useful or amusing.

See The Library of Babel for a short story by Jorge Luis Borges.

See Euler's Mathematics This is an excerpt from a recent book introducing proofs and mathematical ideas.

**Recommended Book: ** What is Mathematics?, Oxford Univ. Press, by Richard Courant and
Herbert Robbins. This book is available via
Amazon.

See Boolean Notation. A short article about Boolean Algebra notation.

See Peirce. A excerpt from the writings of Charles Sanders Peirce, American philosopher and mathematician (1839-1914). In this article, Peirce gives his "sign of illation", a symbol for implication that is a combination of the Boolean negation sign and the Boolean plus sign. He also has a worthwhile discussion of the nature of logic.

See Boolean Algebra. A short article about Boolean Algebra and some ideas related to it, including switching circuits, Laws of Form, recursions and circuits that count.

See Even/Odd. An article by David Joyce about Even and Odd Numbers.

See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms. Compare these axioms with the discussion in the article by David Joyce on Even and Odd.

See Set Theory. This is an appendix from the book "Topology and Geometry" by Glen Bredon. The appendix is a concise but complete synopsis of basic set theory and goes beyond what is in Eccles. In particular, you will find discussion of well-ordering of sets, and in Theorem B18 the equivalence of a number of set theoretic principles such as well-ordering and the axiom of choice. You will also find a proof of the Cantor-Schroeder-Bernstein Theorem (B15 and B20). You do not need to read this whole appendix, but I DO ask you to look at Theorem B27. There you will find a beautiful proof that the real numbers have the same cardinality as P(N) where N is the natural numbers. The proof includes a proof that the set of finite subsets of N is countable (can you give an independent proof of this fact?). The proof of Theorem 27 uses the concept of continued fractions.

See Wang Algebra for a clever approach to graphs and spanning trees.

See ContFrac. This is a calculator on the web that converts numbers to their continued fractions. Try it on sqrt(2), sqrt(3), e, pi and other favorite numbers.

See PrimeFactor. This is a calculator on the web that factors numbers.

See Fractions and Decimals . This is a great page on fractions and decimals including a calculator for converting fractions to decimals and finding the perionds.

See Fraction to Decimal Converter . This is a sequel to the page above with a specific fraction to decimal converter.

See Roots of Two. This is a Wiki about the relationship of the twelfth root of two and the musical scale.

See Ulam Spiral 1. This is a Wiki about the structure of the primes and the Ulam Spiral.

See Ulam Spiral 2. This is a demo about the structure of the primes and the Ulam Spiral.

See Ulam Spiral 3. This is a YouTube Video about the structure of the primes, Ulam Spiral and the prime 41.