Re: [math-fun] Regular spherical polyhedra
I believe the condition that "No two adjacent polygons have the same image under F" takes care of Andy's concern, and in fact that is the reason this condition was included. --Dan Andy wrote: << I wrote: << . . . * For any edge e of any one polygon, there are exactly two polygons that have e as a full edge. Two such polygons sharing exactly one full edge are called "adjacent". No two adjacent polygons have the same image under F.
If I were making the definition, I would add the condition here that the image under X of the two polygons adjacent to e includes an open neighborhood of every point in the image of e except possibly the vertices e connects. Or roughly speaking, the two faces adjacent to e are on "opposite sides" of the edge. Did you intend to include this condition? Are there polygons that satisfy your definition that do not satisfy this condition?
________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
On Thu, May 20, 2010 at 1:49 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I believe the condition that "No two adjacent polygons have the same image under F" takes care of Andy's concern, and in fact that is the reason this condition was included.
Yes, you're right. This problem occurred to me when I read the edge condition, before I got to the regularity condition, which implies that each polygon is a regular polygon, so that the images of two polygons on the same side would be identical. Put another way, if you wanted to define "spherical polygon" rather than "regular spherical polygon", it wouldn't suffice to remove the regularity condition; you'd have to replace ""No two adjacent polygons have the same image under F" with something more like the condition I specified.
* Project the regular tetrahedron to the sphere and choose any two disjoint edges e, e' of it. Now take two identical copies T_1, T_2 of this spherical tetrahedron and "slit" each of them them along each edge corresponding to e and e', creating 8 loose edges. Now identify these in 4 pairs (as when creating a Riemann surface) so that one side of e on T_1 is identified to the other side of e on T_2, etc., so that no loose edges remain.
(In this case the topology of X is that of a torus.)
Puzzle (not that hard, but confused me for a few minutes): What's wrong with the following argument? That can't be, because the resulting map from the torus to the sphere is a double cover. But the sphere has trivial fundamental group, and therefore has no covering spaces. Answer below a f t e r s o m e s p o i l e r s p a c e The cover is ramified at the corners of the tetrahedron. Andy
(Not fun.) It is with great sadness that I report Martin Gardner died today. http://blogs.discovermagazine.com/badastronomy/2010/05/22/martin-gardner-191... When I visited him last October in Norman, OK, he was sharp, alert, and happy to offer a list of great ideas he thought would be good for the Museum of Mathematics. He demonstrated some card tricks, calculator tricks, and a rope illusion. He told a number of interesting stories about his life and said he was starting to work on his autobiography --- a book project which he predicted would take three years to complete. I thought that it took enormous vision to start a three-year project at the age of 95 and am sad that it will never be finished. George Hart http://momath.org http://georgehart.com
="George W. Hart" <george@georgehart.com> It is with great sadness that I report Martin Gardner died today.
Sad indeed--just days ago I was proposing writing a paper specifically because it seemed he'd enjoy it! He must have touched so many lives. I wrote some kind of a confused fan letter to him when I was 9 or 10, and he was kind enough to reply. It was hugely encouraging for me to keep trying to learn more about mathematics. Sure wish I still had it!
I'm sorry to hear the news, too. Like so many others, I was inspired and intrigued by his column every month in Scientific American. Who, if anyone, will be the Martin Gardner of the current generation of young people? Bob Baillie --- George W. Hart wrote:
(Not fun.)
It is with great sadness that I report Martin Gardner died today.
http://blogs.discovermagazine.com/badastronomy/2010/05/22/martin-gardner-191...
When I visited him last October in Norman, OK, he was sharp, alert, and happy to offer a list of great ideas he thought would be good for the Museum of Mathematics. He demonstrated some card tricks, calculator tricks, and a rope illusion. He told a number of interesting stories about his life and said he was starting to work on his autobiography --- a book project which he predicted would take three years to complete. I thought that it took enormous vision to start a three-year project at the age of 95 and am sad that it will never be finished.
George Hart http://momath.org http://georgehart.com
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George -- thanks for passing on the sad news. I wanted Martin to live forever. It's literally true that I never would have been lured into mathematics from physics if I hadn't been pleasantly distracted by Martin's Scientific American columns. (An unkind view of my switching would be that 'physics' gain was mathematics' loss'.) I met Martin only once. I spent a very satisfying half-day with him at his home in Hastings (or was it Dobbs Ferry?) in the late 70s. I'll perhaps tell you that story some other time (when we next meet). Alan On Sat, May 22, 2010 at 10:19 PM, George W. Hart <george@georgehart.com>wrote:
(Not fun.)
It is with great sadness that I report Martin Gardner died today.
http://blogs.discovermagazine.com/badastronomy/2010/05/22/martin-gardner-191...
When I visited him last October in Norman, OK, he was sharp, alert, and happy to offer a list of great ideas he thought would be good for the Museum of Mathematics. He demonstrated some card tricks, calculator tricks, and a rope illusion. He told a number of interesting stories about his life and said he was starting to work on his autobiography --- a book project which he predicted would take three years to complete. I thought that it took enormous vision to start a three-year project at the age of 95 and am sad that it will never be finished.
George Hart http://momath.org http://georgehart.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I find myself shedding tears for a man I never met, who lived on the other side of the world, with whom I corresponded briefly on only a couple of occasions [map-folding and Penrose tiling, since you didn't ask]. On one such his reply was accompanied by a substantial parcel of his books, an unlooked-for and unexpected gift which still occupies my shelves on those occasions when I can prise them back from people who regularly walk off with them. What is most astonishing to me about this unassuming giant is how he managed evidently to combine a verisimilitude of personal friendship to heaven knows how many thousand complete strangers, with what can only have been a prodigious work ethic. Quite apart from any more elevated considerations, simply to produce the amount of accurate, high quality material for which he was (as far as I know) single-handedly responsible must have demanded a submerged iceberg of tightly organised survey research which defeats the imagination. He would never admit to actually being a mathematician, but his address for many years 10 Euclid, Hastings-on-Hudson gives the game away. You never fooled me there, old man! Fred Lunnon On 5/23/10, George W. Hart <george@georgehart.com> wrote:
(Not fun.)
It is with great sadness that I report Martin Gardner died today.
http://blogs.discovermagazine.com/badastronomy/2010/05/22/martin-gardner-191...
When I visited him last October in Norman, OK, he was sharp, alert, and happy to offer a list of great ideas he thought would be good for the Museum of Mathematics. He demonstrated some card tricks, calculator tricks, and a rope illusion. He told a number of interesting stories about his life and said he was starting to work on his autobiography --- a book project which he predicted would take three years to complete. I thought that it took enormous vision to start a three-year project at the age of 95 and am sad that it will never be finished.
George Hart http://momath.org http://georgehart.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (7)
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Alan Schoen -
Andy Latto -
Dan Asimov -
Fred lunnon -
George W. Hart -
Marc LeBrun -
Robert Baillie