Mathematics E302  Fall 2007

Teacher: Andy Engelward
Office: 51 Brattle St, Cambridge, room 707
Email:engelward@math.harvard.edu CA: Jeff Joiner

Email:jjoiner@post.harvard.edu

Someone asked me about higher dimensional objects so note that although there will be no questions asking you to visualize higher dimensional objects (like a four dimensional object passing through three dimensions), you should still be able to figure out properties of higher dimensional objects that we've studied (like the hypercubes) by creating tables of information as we've done in class and on homework.

Also, when you're memorizing your proofs for the Pythagorean Theorem make sure that you can replicate them on paper if someone asked you too - i.e. know how to draw out the essential parts as best you can.  So, it might not make sense to memorize an incredibly involved proof (such as DaVinci's proof!)

Here's the Exam Review sheet I passed out in the last class, along with a practice test from a previous year.  After you try out the practice test, here are the solutions to check against. Also, all the homework solutions are posted on the assignments at this point. Good luck with all your studying!  Please don't hesitate to call me at home 781 676-0676 if you'd like to go over anything before the final.

I've had a couple of requests to see the Midterm Review Guide again, so here it is!.

Finally - a quick update on Kepler - yes, for a long time he was very attached to the idea that the five Platonic solids somehow explained the orbits of the six planets and did publish articles about it, but Patricia was of course correct in pointing out that he did eventually give this idea up after working with Tycho Brahe's astronomical observation data (when he came up with his theories about elliptical orbits).  For more on Kepler you might want to check out this Lecture on Kepler.  As for mysticism being used by scientists to explain physical phenomena, Kepler was hardly alone in this regard - for instance it's thought that Newton decided that there are seven colors in the rainbow (the infamous Roy G. Biv - red orange yellow green blue indigo and violet), even though the color graduations are of course continuous as he felt very attached to the number seven as being a critical number to explain nature (seven continents, seven seas, seven openings in our head, seven days of the week, etc.).  In any case, Kepler's lifetime astronomical accomplishments far outweigh his earlier fascination for the erroneous nested Platonic solids explanation for planetary orbits!

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