\thispagestyle{empty} \begin{center} {\Large \bf Alice in Mathland: A Mathematical Fantasy} \bigskip \bigskip \bigskip {\bf Senior Thesis Proposal} \\ \medskip Presented by \\ \medskip Sara Smollett \end{center} \vfill \begin{flushleft} {\bf Approved:} \\ \vspace{4mm} \begin{tabbing} % \= sets the tab; \> uses the predefined tab \rule{2.5in}{.1mm} \hspace{1in} \= \rule{.5in}{.1mm} \\ \vspace{-1mm}Allen Altman, Thesis Advisor \> Date \end{tabbing} {\bf Thesis Committee Members:} \\ Jamie Hutchinson \\ J. David Reed \end{flushleft} \pagebreak I'm proposing to write a thesis which combines mathematics and creative writing, a thesis which makes math accessible to a general reader and presents math as fun. Many people consider mathematics and creative writing to be extremely unrelated topics, but there is no reason that this must be the case. One of my main inspirations for my thesis is Lewis Carroll. Lewis Carroll is a well-known author. In addition to writing children's stories, he was also a mathematician and logician. He is an example of someone who was a successful writer and amateur mathematician. I will attempt to adopt his tone and style for portions of my thesis. My thesis will be an example of recreational mathematics. I want to create something which contains mathematical ideas, reads well (as a literary work, not a math text), and is entertaining and accessible to readers who possess little mathematical background. Ideally, someone might read my text and come away from it not only having been exposed to some new mathematics, but wanting to learn even more. I'd like to take a small step toward popularizing mathematics. What do I mean by my tentative title? Who is Alice and what is Mathland? Alice is a heroine created by Lewis Carroll. My Alice is a small, fantastical, somewhat mathematically-confused girl. Mathland is an alternative wonderland, a strange place that Alice finds after falling down a hole. It is a world of beautiful mathematics. The second part of my title is perhaps more confusing. What is a mathematical fantasy? What is mathematical fiction? These are questions that I have been struggling with, not because I don't know what mathematical fiction is, but because there are so few examples of such fiction. Perhaps there is so little mathematical fiction because math is not seen as a recreational topic. Math is generally considered to be something difficult, often something to be feared and avoided. That math is fun should not be such a radical idea. There is little mathematical literature because the audience for this genre is so small. At the same time, the audience cannot increase until there are more examples of texts. My thesis will contribute one such example. Although there aren't many examples of mathematical fiction, I am certainly not the first to write in this genre. One of the first to do so was Edwin Abbott who wrote {\it Flatland: A Romance of Many Dimensions}. {\it Flatland} is the story of a two-dimensional world inhabited by lines and polygons. It is an instructive fantasy. Other mathematical fictions include Dionys Burger's {\it Sphereland} (the sequel to {\it Flatland}), A. K. Dewdeney's {\it A Mathematical Mystery Tour}, and Hans Magnus Enzensberger's {\it Number Devil}. In his introduction to {\it Flatland}, Isaac Asimov writes: ``Fear not, however. It contains no difficult mathematics and it won't sprain your understanding. [It is] a pleasant fantasy. You will have no sensation of 'learning' whatsoever, but you will learn just the same.'' This is one of my objectives in writing this thesis. My ideas about both content and style have been shaped by the above authors. One other major influence is Douglas Hofstadter, whose {\it G\"{o}del, Escher, Bach} contains dialogues of Lewis Carroll's characters. Each chapter of his book begins with a short dialogue and is followed by a lengthy discussion of topics covered in the dialogue. This is the basis for the form of my thesis, though I would like to focus more on the fictional pieces than the explanation. Ideally, the explanations will fit naturally into the fiction. More in depth explanations, proofs, and discussions will follow every few chapters. A less mathematical reading can be achieved by treating these sections as appendices. The story I am writing is one of Alice and the Yellow Pig. The Yellow Pig is her guide (like Carroll's White Rabbit and Dante's Virgil) who leads her through Mathland and teaches math to her and the reader. Alice follows the pig down a hole to find herself in a garden of mathematics. The pig gives her a mathematical tour of the garden and introduces her to some of his mathematician friends. The two proceed over some bridges to the pig's private art gallery which is filled with math. Alice finally leaves mathland after answering a series of Carrollian logic riddles. Along the way, mathematical topics are covered in the areas of geometry, number theory, algebra, topology, graph theory, combinatorics, probability, game theory, and logic. The following tentative outline lists months in which the bulk of the creative writing part of the (rather short) chapters will be completed. The topics contained in single parenthesis are the mathematical topics to be explained within the chapters. The topics contained in double parenthesis are the more in-depth mathematical ideas which will be discussed in the explanatory sections. These will be completed and the chapters revised in the second month listed. \pagebreak \begin{flushleft} {\bf Schedule:} \\ \bigskip SEPTEMBER/JANUARY \\ \medskip * Down the Hole \\ Alice follows the pig down a hole. \\ (differential equations, the idea of change) \\ ((fourth dimension)) \\ \medskip * Into the Garden \\ Alice chases the pig. \\ (irrational numbers) \\ \medskip * What the Pig Said \\ Alice and the pig talk. \\ (rectangular lattice, slope, Cantor and infinities) \\ ((Minkowski)) \\ \medskip * A Pig and A Greek \\ The pig talks about geometry. \\ (distance, square roots, Pythagorean theorem) \\ ((Fermat's Last theorem)) \\ \bigskip OCTOBER/FEBRUARY \\ \medskip * Most Irrational \\ The pig discusses some important irrational numbers. \\ (pi, e, phi, Fibonacci, Lucas) \\ ((measuring irrationality)) \\ \medskip * The Golden Garden \\ The pig leads Alice through a garden that is rich in mathematics. \\ (math in nature, phi, Fibonacci) \\ \medskip * The Pig's Friends \\ The pig introduces Alice to some of his mathematician friends. \\ (Gauss and sums, Pascal and probability, Fermat, Ramanujan and number theory) \\ \medskip * Primes \\ The pig talks about prime numbers. \\ (infinitely many, Goldbach, twin primes, unique factorization, modular arithmetic) \\ ((Wilson's theorem, Fermat's little theorem)) \\ \bigskip NOVEMBER/MARCH \\ \medskip * Leaves on Kittens \\ The pig talks about combinatorics and pigeonholes. \\ (pigeonhole principle, Erd\"{o}s, probability) \\ ((more Erd\"{o}s)) \\ \medskip * Over the Bridge \\ Alice and the pig attempt to cross some bridges, color some countries, and shake some hands. \\ (Euler and the K\"{o}nigsberg bridge problem, graph theory, coloring, four color theorem, handshakes, Ramsey) \\ ((more four color theorem and Ramsey)) \\ \medskip * Through Another Door \\ Alice and the pig enter a room of tesselations and art \\ (art, tesselations, symmetry, groups, algebra, Chinese remainder theorem) \\ ((Galois)) \\ \medskip * Stepping into a Picture \\ Alice steps into an Escher painting and discovers the M\"{o}ebius strip. \\ (Escher, topology, M\"{o}ebius, Klein bottle) \\ ((colorings on different surfaces, v-e+f)) \\ \bigskip DECEMBER/MARCH \\ \medskip * And Another Picture \\ Alice steps into Dali's tesserect \\ (fourth dimension, Flatland, slices, cycloids) \\ ((hyperbolic geometry, proof of v-e+f )) \\ \medskip * Logicland \\ Alice and the pig continue on to logic land. \\ (strategies and the prisoners' dilemma, paradoxes, Hofstadter, logic and Lewis Carroll) \\ ((more puzzles from Lewis Carroll)) \\ \bigskip For more and up-to-date information about my thesis, refer to www.simons-rock.edu/\~{}sara/thesis. \end{flushleft}