March 2005 - mathfun@mailman.xmission.com

[math-fun] A volunteer!
by Richard Schroeppel 25 Mar '05

25 Mar '05

Michael Kleber <michael.kleber(a)gmail.com> will be taking over as Math-Fun moderator while I'm gone. Be nice. Rich rcs(a)cs.arizona.edu

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[math-fun] moderator outage
by Richard Schroeppel 24 Mar '05

24 Mar '05

I'll be on vacation for a couple of weeks. I'll accept a couple of old-timer volunteers as interim moderators; your duties would be to check the admin web page every few days to delete the spam & approve any real messages from non-subscribers. Drop me (not the list) a note now. Rich rcs(a)cs.arizona.edu

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Re: [math-fun] Galois-actions and star-polytopes
by Richard Schroeppel 24 Mar '05

24 Mar '05

This is from Dylan Thurston. For some reason the mailing list decided it's spam. Rich ------ On Wed, Mar 23, 2005 at 05:57:15AM -0600, James Propp wrote: > The regular pentagon and pentragram can be viewed as Galois conjugates > of one another, under the imbedding in C in which all vertices are fifth > roots of unity. (Alternatively, one can view the vertices as sitting in > RxR, and hit them with simultaneous Galois actions on both coordinates.) >=20 > Is there anything analogous for any higher-dimensional star-polytopes that > would enable one to view them as Galois conjugates of convex polytopes? Yes. For instance, in 3 dimensions, the dodecahedron is conjugate to the great stellated dodecahedron; the icosahedron is conjugate to the great icosahedron; and the small stellated dodecahedron is conjugate to the great dodecahedron. (You can get pictures of all of these at http://www.mathconsult.ch/showroom/unipoly/ ) There's also a version in 4 dimensions; I don't know about higher dimensions. I'll write up why these are conjugate when I have a minute. Peace, Dylan

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[math-fun] Re: Galois-actions and star-polytopes
by James Propp 23 Mar '05

23 Mar '05

Dan writes: >Hmm. Since among Euclidean spaces, only R^1 and R^2 have natural field >structures, the answer to this exact question would seem to be No. I was trying to address this issue when I wrote >(Alternatively, one can view the vertices as sitting in >RxR, and hit them with simultaneous Galois actions on both coordinates.) So for instance you could have a star-polyhedron in R^3, and replace each vertex (x,y,z) with (g(x),g(y),g(z)), where g is some Galois action on the field over Q generated by the coordinates of all the vertices. Jim

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RE: [math-fun] Galois-actions and star-polytopes
by Daniel Asimov 23 Mar '05

23 Mar '05

Jim writes: << The regular pentagon and pentragram can be viewed as Galois conjugates of one another, under the imbedding in C in which all vertices are fifth roots of unity. (Alternatively, one can view the vertices as sitting in RxR, and hit them with simultaneous Galois actions on both coordinates.) Is there anything analogous for any higher-dimensional star-polytopes that would enable one to view them as Galois conjugates of convex polytopes? >> Hmm. Since among Euclidean spaces, only R^1 and R^2 have natural field structures, the answer to this exact question would seem to be No. But R^4 as the quaternions H is, of course, a skew field. Has anyone tried to develop a Galois-like theory for H ? --Dan

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[math-fun] 4CC/4CT
by Henry Baker 23 Mar '05

23 Mar '05

>From what I recall, the 4 color theorem is first reduced to a theorem on planar graphs -- is this right? What are the restrictions on map boundaries? Do fractal boundaries still work?

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[math-fun] Galois-actions and star-polytopes
by James Propp 23 Mar '05

23 Mar '05

The regular pentagon and pentragram can be viewed as Galois conjugates of one another, under the imbedding in C in which all vertices are fifth roots of unity. (Alternatively, one can view the vertices as sitting in RxR, and hit them with simultaneous Galois actions on both coordinates.) Is there anything analogous for any higher-dimensional star-polytopes that would enable one to view them as Galois conjugates of convex polytopes? Jim Propp

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[math-fun] Written TNT pandigital
by Eric Angelini 23 Mar '05

23 Mar '05

Find the shortest and longest doubly-true TNT strings that are pandigital in base 10. Doubly-true means that the amount of S's & O's ("let- ters") is the same as the string's arithmetic result. http://www.ncsu.edu/felder-public/kenny/papers/godel.html The TNT-string below is doubly-true -- but far from being pandigital in base 10: SO.SSO+SSSO.SSSSO [14 "letters"] 1 x 2 + 3 x 4 [14 = 2x7] [Please consider that SSO has 3 letters even though it should be written SS0 instead -- thus 2 S's and 1 zero] Best, É. __________________________________________________ http://angelink.be/?CeRaccourciFaitCinquanteSignes

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[math-fun] collected papers of Ramanujan (on-line with google...!)
by Simon Plouffe 21 Mar '05

21 Mar '05

Hello, This just appeared on google I guess : http://print.google.com/print?id=vX0WOuyYMEYC&prev=http://print.google.com/… apparently the google print service has just began to show some results, I am not certain if we can get a full copy of the book. so the trick to get something is to 1) go to google.com 2) type <book ramanujan> for example 3) see the results. Simon Plouffe

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[math-fun] news article
by Thane Plambeck 18 Mar '05

18 Mar '05

http://www.physorg.com/news3437.html Excerpt * Karl Mahlburg, a young mathematician, has solved a crucial chunk of a puzzle that has haunted number theorists since the math legend Srinivasa Ramanujan scribbled his revolutionary notions into a tattered notebook. "In a nutshell, this [work] is the final chapter in one of the most famous subjects in the story of Ramanujan," says Ken Ono, Mahlburg's graduate advisor and an expert on Ramanujan's work. Ono is a Manasse Professor of Letters and Science in mathematics. * -- Thane Plambeck http://www.plambeck.org/ehome.htm

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