Q700 | 26805 | Hofstadter

2:30-3:45, TR, CRCC, 510 N. Fess Above section meets with Math M590 TOPIC: Mathematical Thinking in Geometry The goal of this seminar is to observe how creative and insightful mathematical thinking works, using geometry as the arena in which to look at the phenomena. Using the humble triangle as our springboard, we will first explore Euclidean geometry in some depth (inspired largely by the vast variety of so-called triangle centers and the unexpected wealth of relationships tying them together), and this exploration will naturally lead us into some contact with projective geometry and will even have us dip at least a little bit into non-Euclidean geometries of various sorts. All of this geometrizing will be done with an eye constantly kept on the nature of mathematical understanding and imagery, relentlessly focusing on the various ways in which deep intuitive, imagistic understanding of ideas, always pushing for greater simplicity, facilitates the invention of new ideas through non- obvious analogical leaps. The key role played, in such leaps, by some kind of esthetic drive, and the variety of qualities that go into this drive, will constitute a companion focus of our attention. Finally, the relationship between free-floating mathematical invention and the more grounded activity of proving of theorems will be scrutinized. This course can profitably be taken by both graduates and undergraduates, both in mathematics and not in mathematics. It does not have any advanced prerequisites (mostly just an enjoyment of math -- not even a prior calculus course!), but on the other hand it does require students to get their hands dirty in carrying out creative mathematical explorations. Those whose mathematical background is more advanced will naturally gravitate towards the more abstract, whereas those whose background in more elementary will tend to stay at the more concrete end of the spectrum. But thatıs fine -- it is equally possible for new insights into triangles, into mathematical thinking, and into the nature of creativity in general to pop up at any point along the concreteness/abstraction spectrum. The course will make use of the elegant computer program Geometerıs Sketchpad, which brings visual experience directly into the forefront. From the very start of the course, students will be encouraged to make personal explorations and discoveries, and in the second half of the semester, they will write a paper of five to ten pages describing both their geometrical discoveries and the cognitive pathways (both fruitful and fruitless) that they followed in making them. Textbooks: ========== David Wells, "The Penguin Dictionary of Curious and Interesting Geometry" (Penguin) Coxeter and Greitzer, "Geometry Revisited" (MAA) There will also be various handouts, some written by myself, others taken from various books and articles. For additional information please contact Helga Keller (htkeller@indiana.edu/855.6965)