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21-101: The Four Pillars of Geometry |
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DescriptionFor generations of high school students, the word "geometry" conjures up images of straightedges and compasses. In fact the presentation in most high school geometry courses has changed very little -- at least in principle -- from that in Euclid's "Elements", published around 300 B.C. For over 2000 years Euclid and geometry were synonymous.Euclid's geometry, now called Euclidean geometry, was based on a set of axioms. The most famous of these is the so-called "Parallel Postulate", which states that given any line and a point not on that line, there is exactly one line through the point that does not intersect the line. For many centuries it was assumed that this axiom could be derived from the others. The 18th century, though, saw a description of a "hyperbolic geometry" which satisfied all of Euclid's axioms except the parallel postulate. (hyperbolic spaces have too many parallel lines.) Later developments included projective geometries, which have too few parallel lines. In this course, we will discuss three approaches (pillars) to understanding geometry. The first is constructive. These straightedge and compass constructions will likely be the most intuitive to most students. The second is algebraic. Introducing coordinates to a Euclidean plane can reduce complicated geometric arguments to simple calculation. The third approach involves "invariants" of "transformations". For instance, length and angle are invariants of the "rigid motions" of the plane. The fourth pillar is projective geometry, which describes why things look the way they do, and points toward some deeper connections between algebra and geometry. NewsTuesday 17 October: I'll hold a review session for the final exam during our usual class time on Wednesday (though this is not officially a class meeting). We'll meet in our usual class room. I'll have all your graded homework for you then as well.Tuesday 3 October: I've added Wednesday's homwork to the page for Week #6. Sunday 24 September: I've posted the reading assignments for Week #6. Follow the link from the Schedule page. Sunday 24 September: I've posted the reading assignments for Week #5. Follow the link from the Schedule page. Thursday 21 September: I've updated the Week #4 assignment page with the homework due for Friday. Friday 15 September: I've posted the reading assignments for Week #4 and the homework assignment for Monday. Thursday 14 September: I've updated the Week #3 assignment page with the homework due for Friday. Tuesday 12 September: Academic Development, the folks who provide the SI sessions and peer tutoring, are sponsoring several group workshops focusing on study skills. "For Freshemen Only" seems to cover a wide range of topics. They have a couple other sessions that address time management. Tuesday 12 September: The reading assignments for Week #3 have been posted. Also, for the sake of completeness, I've posted the homework assignment that was due yesterday. Wednesday 6 September: I've updated the assignment for Week #2 to include today's assignment (due Friday). Friday 1 September: I've posted the reading assignments for Week #2. Follow the link from the Schedule Page. Friday 1 September: After grading the first homework assignment, I've decided to postpone the due date for the second from today to next Wednesday (our next class meeting). You can take a look at your graded homework and the solutions. Use those as guidelines for writing up the next assignment. Wednesday 30 August: I've updated the Week#1 homework page with the problems I assigned in class today, and the reading for Friday's class. Follow the link from the Schedule page. Tuesday 29 August: I've updated the Week#1 homework page with the problems I assigned in class yesterday, and the reading for tomorrow's class. Follow the link from the Schedule page. Monday 28 August: David Joyce at Clark University has a very nice online translation of Euclid's Elements. The fugures are built using a Java applet that lets you to dynamically change the diagrams. It's worth a look. Monday 21 August: Welcome to 21-101 The Four Pillars of Geometry.
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