Re: [math-fun] Andrew Gleason
i was saddened to hear about andy gleason's passing. i took several classes from him last century, and each ranks very high on the list of classes in which i learned the most. i feel very fortunate to have had him as a teacher. i specifically remember, in his classical geometry course, he showed how desargues' theorem/axiom follows directly from the incidence axioms in 3 dimensional projective space, but not so in 2 dimensional space. he then remarked that if you can embed a 2 dimensional space into a 3 dimensional space, it sometimes gives you added information! indeed, there are projective planes that are "non-desarguesian" and therefore cannot be embedded in 3 dimensional projective space. this idea stuck with me and i've used it several times. one application is the following. the "radical axis" of two circles is the locus of all points in the euclidean plane that have equal "power" with respect to each circle. (the "power" of a point with respect to a circle is the square of the length of a tangent to the circle from that point; for points inside the circle, this must be modified to, say, the negative of the product of the two lengths of the segments into which the point divides a chord of the circle.) for two circles that intersect, it is easy to show that their radical axis is the line through the two points of intersection (or common tangent, if they intersect tangentially). for two circles (non-concentric) that do not intersect, their radical axis is still a line. i used gleason's idea to give a nice proof of this (it works better in the case that neither circle contains the other; otherwise one needs to handle the concentric case separately). i also recall, somewhat vaguely, that gleason's title was professor of mathematicks [sic] and natural philosophy. the obituary suggests that they have modernized(?) the spelling in the meantime. do anyone else remember this? or am i mis-remembering? i also note that ethan bolker, who is quoted in the boston globe article, is probably the same e. bolker listed in the references R recently sent for UPINT problem C5 "sums determining members of a set". mike
Whit Diffie passed along the news that Andrew Gleason has died. --Rich
<url is split> http://www.boston.com/bostonglobe/obituaries/articles/2008/10/20/ andrew_gleason_helped_solve_vexing_geometry_problem/
I was saddened to hear of Andy's passing. I knew him well both from when I was a graduate student at Harvard (he was department chair when I entered), and from my current employment at IDA. He was periodically on our oversight committee. I last saw him about a year ago when he visited our place. His mind was still as sharp as ever. If you can find it, it's well worth watching a film that he made for the MAA probably almost 50 years ago about NIM and games. It was a really beautiful lecture, and was indicative of the care that he put into his teaching. Victor On Wed, Oct 22, 2008 at 5:55 PM, Michael Reid <reid@gauss.math.ucf.edu> wrote:
i was saddened to hear about andy gleason's passing. i took several classes from him last century, and each ranks very high on the list of classes in which i learned the most. i feel very fortunate to have had him as a teacher.
i specifically remember, in his classical geometry course, he showed how desargues' theorem/axiom follows directly from the incidence axioms in 3 dimensional projective space, but not so in 2 dimensional space. he then remarked that if you can embed a 2 dimensional space into a 3 dimensional space, it sometimes gives you added information! indeed, there are projective planes that are "non-desarguesian" and therefore cannot be embedded in 3 dimensional projective space.
this idea stuck with me and i've used it several times. one application is the following. the "radical axis" of two circles is the locus of all points in the euclidean plane that have equal "power" with respect to each circle. (the "power" of a point with respect to a circle is the square of the length of a tangent to the circle from that point; for points inside the circle, this must be modified to, say, the negative of the product of the two lengths of the segments into which the point divides a chord of the circle.)
for two circles that intersect, it is easy to show that their radical axis is the line through the two points of intersection (or common tangent, if they intersect tangentially). for two circles (non-concentric) that do not intersect, their radical axis is still a line. i used gleason's idea to give a nice proof of this (it works better in the case that neither circle contains the other; otherwise one needs to handle the concentric case separately).
i also recall, somewhat vaguely, that gleason's title was professor of mathematicks [sic] and natural philosophy. the obituary suggests that they have modernized(?) the spelling in the meantime. do anyone else remember this? or am i mis-remembering?
i also note that ethan bolker, who is quoted in the boston globe article, is probably the same e. bolker listed in the references R recently sent for UPINT problem C5 "sums determining members of a set".
mike
Whit Diffie passed along the news that Andrew Gleason has died. --Rich
<url is split> http://www.boston.com/bostonglobe/obituaries/articles/2008/10/20/ andrew_gleason_helped_solve_vexing_geometry_problem/
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