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[math-fun] Torus tiling question
by D. Asimov 12 May '07

12 May '07

It's probably well known in some quarters that for any integer N > 0 that's the sum of 2 square numbers, the torus T = R^2/Z^2 with the induced metric -- aka the "square torus" -- can be tiled by N congruent square tiles. (Note that such N include all square numbers.) It's a blast to work out how this goes, so I won't spoil your fun. I'm not sure how to prove the converse -- or certain that it's true (though this seems likely). Can someone prove the converse: "If T can be tiled by N congruent squares, then N is the sum of 2 square numbers." . . . . . . or come up with a counterexample ??? --Dan

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[math-fun] Canonical sequence of rational approximants to an irrational
by Daniel Asimov 08 May '07

08 May '07

Given any irrational c > 0, there's a canonical sequence of rational approximants to c, defined as the maximal sequence of rationals with increasing denominators that get increasingly close to c. I calculated the first few terms for pi, e, and sqrt(2). I wasn't surprised to find the denominator sequence for pi was already in OEIS, but not those for e or sqrt(2) . . . and no written references were mentioned. Is there a name for such sequences? Is anything much known about them (e.g., speeds of approximation, growth rates of denominators, relationship to continued fractions) ? (Seems that these sequences could be used to characterize properties of the irrational, though it's not clear whether they could be any more useful than the continued fraction.) Here are the first 19 terms of each sequence: approximants to pi: 3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, 52163/16604, 52518/16717, 52873/16830, 53228/16943, 53583/17056, ... approximants to e: 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540, 15062/5541, 17783/6542, ... approximants to sqrt2: 1/1, 3/2, 4/3, 7/5, 17/12, 24/17, 41/29, 99/70, 140/99, 239/169, 577/408, 816/577, 1393/985, 3363/2378, 4756/3363, 8119/5741, 11482/8119, 19601/13860, 47321/33461, ... --Dan

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[math-fun] Re: Winding line puzzle
by James Propp 04 May '07

04 May '07

>As is well know, for any irrational number c, there exists >an injective immersion of the reals into the square torus, >I_c: R -> T = R^2/Z^2 given by > > I_c(x) = q(x, cx) > >where q: R^2 -> R^2/Z^2 is the quotient mapping. (Such an >image I_c(R) is called a winding line in the torus.) > >The image I_c(R) inherits a the subspace topology from T. > >PUZZLE: Given any irrational numbers c <> d, are the spaces >I_c(R) and I_d(R) (with the subspact topologies) necessarily >homeomorphic? Another great question! Jim

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[math-fun] Winding line puzzle
by Daniel Asimov 04 May '07

04 May '07

As is well know, for any irrational number c, there exists an injective immersion of the reals into the square torus, I_c: R -> T = R^2/Z^2 given by I_c(x) = q(x, cx) where q: R^2 -> R^2/Z^2 is the quotient mapping. (Such an image I_c(R) is called a winding line in the torus.) The image I_c(R) inherits a the subspace topology from T. PUZZLE: Given any irrational numbers c <> d, are the spaces I_c(R) and I_d(R) (with the subspact topologies) necessarily homeomorphic? --Dan

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[math-fun] Correction re simple group of order 168
by Daniel Asimov 04 May '07

04 May '07

In a recent post on the Klein quartic I stated that the simple group of order 168 (call it G) is a subgroup of PSL(2,C) (PSL(2,C) is one way to express the group of all fractional linear transformations over C, aka the group of conformal automorphisms of S^2 =C{^1.) This is incorrect; this group is *not* a subgroup of PSL(2,C). (The finite subgroups of PSL(2,C) are the same as the finite subgroups of SO(3): the cyclic groups C_n, dihedral groups D_n, and rotation groups of the Platonic solids: A_4, S_4,and A_5). G is, however, a subgroup of PSL(3,C) (the group of projective transformations of CP^2, where the Klein quartic naturally lives). (Oddly enough, as I learned from an algebraic geometer friend, the conformal automorphisms of projective varieties in CP^2 of degree <= 4 always extend to CP^2 itself, but this is not necessarily true for degree >= 5.) --Dan

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[math-fun] Reverse date factors
by Eric Angelini 02 May '07

02 May '07

Hello Math-Fun, One can produce an integer by concatenating the digits spelling a full date -- example for today, May 2nd, 2007: --> 20070502 This 8-digit integer can be represented like this: abcdefgh. I was wondering if there exist a (close) date fulfilling this: abcdefgh = h*g*f*e*d*c*b*a where as many * as one wishes can be deleted. (but not all of them! -- concatenate accordingly to produce a sound set of factors. For instance: abcdefgh = hgfe*d*c*b*a or abcdefgh = h*g*f*edcba or abcdefgh = hg*f*e*d*c*b*a ... Best, É. (I think the "direct" equality is impossible: abcdefgh = a*b*c*d*e*f*g*h)

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[math-fun] two letters by george mackey sent to then college sophomore stephanie frank singer, ca 1982
by Thane Plambeck 02 May '07

02 May '07

I thought these were interesting http://www.symmetrysinger.com/Mackey/letter1.pdf http://www.symmetrysinger.com/Mackey/letter2.pdf -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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Re: [math-fun] Mathematica 6 launched
by Richard Schroeppel 01 May '07

01 May '07

(This is from wouter.meeussen(a)pandora.be; for some reason the mailing list filter didn't like it. --Rich) ------------------------------ just screened the website, still no ConvexHull3D ? was promissed in Weinstein's ecyclopedia! "... Future versions of Mathematica will support -dimensional convex hulls." hmmm. Wouter. ----- Original Message ----- From: "Ed Pegg Jr" <ed(a)mathpuzzle.com> To: "math-fun" <math-fun(a)mailman.xmission.com> Sent: Tuesday, May 01, 2007 9:01 PM Subject: [math-fun] Mathematica 6 launched > I wrote a column about the just-launched Wolfram Demonstrations Project. It has over 1200 interactive mathematical demonstrations. > > http://www.maa.org/editorial/mathgames/mathgames_05_02_07.html > > And... Mathematica 6 has launched. Lots of new capabilities. > > http://www.wolfram.com/products/mathematica/index.html > > Ed Pegg Jr >

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Re: [math-fun] Topology puzzle
by Michael Reid 01 May '07

01 May '07

oops! yes, that should be the complete bipartite graph K_3,3 , as dan notes. mike > > Determine whether or not the topological space Y x Y > > (the cartesian product of two Y-shaped spaces, each > > the union of 3 closed intervals) can be embedded as > > a subset of 3-space. > > it cannot be embedded. Y x Y is homeomorphic to the cone over > the graph C_3 x C_3 . if there were such an embedding, intersecting > with a small sphere centered at the vertex (the point (y_0, y_0) > in Y x Y , where y_0 is the branch point of Y ) would give an > embedding of C_3 x C_3 in the sphere. this is impossible, because > this graph is non-planar. > >> > > Except for the fact that I would say the graph that Y x Y is a > cone over is K(3,3), that is exactly my conclusion, and my reasoning. > (By the Leibniz rule, > > bd(Y x Y) = bd(Y) x Y u Y x bd(Y), > > (with agreement on bd(Y) x bd(Y)) -- which is homeomorphic to K(3,3).) > > (This is quite clearly valid for Y x Y embedded either smoothly > or as a simplicial complex. There may be some bothersome > technical details to check, however, if the embedding is only > known to be continuous.) > > --Dan

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[math-fun] pi to 500 digits on piano
by Thane Plambeck 01 May '07

01 May '07

I can't decide if it's the F major drone in the accompaniment or the (human determined?) rests that makes this so musical: http://www.tomdukich.com/math%20pi%20piano%20keyboard.html -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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