May 2011 - mathfun@mailman.xmission.com

Re: [math-fun] The Klein quartic
by Dan Asimov 05 May '11

05 May '11

James writes: << Figure 1 of gray.pdf (see link below) seems to be missing. Is it available somewhere on the internet? On Thu, May 5, 2011 at 3:47 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > The Klein quartic "curve" is my very favorite among all > compact surfaces, so if I may be permitted to wax rhapsodic > about it for a bit: > > There is a whole book on it, The Eightfold Way, freely > downloadable from MSRI at > > < http://library.msri.org/books/Book35/contents.html >. >> The article is cited from The Mathematical Intelligencer 4:2 (1982), 59{67. So if this is electronically available to you, I suspect the figure is there. I will also see if I have access to it. --Dan Sometimes the brain has a mind of its own.

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[math-fun] The Klein quartic
by Dan Asimov 05 May '11

05 May '11

The Klein quartic "curve" is my very favorite among all compact surfaces, so if I may be permitted to wax rhapsodic about it for a bit: There is a whole book on it, The Eightfold Way, freely downloadable from MSRI at < http://library.msri.org/books/Book35/contents.html >. This includes Klein's original 1979 paper, translated into English by Silvio Levy. Many chapters are technical, but some are quite readable, especially those by Bill Thurston, Jeremy Gray, and the Fergusons. Klein originally defined this surface as the locus Q in the complex projective plane CP^2 of the equation X Y^3 + Y Z^3 + Z X^3 = 0. As a complex projective curve of genus >= 2, Q is a minimal surface in CP^2. It does not have constant Gaussian curvature, but as a Riemann surface, only conformal equivalence type matters, and every Riemann surface is conformally equivalent to one of constant curvature = -1, 0, or 1. Often "Klein quartic" is used to mean the unique surface K of constant curvature = -1 that is conformally equivalent to Q -- which is what Mike is looking at -- or any metric surface at all that is conformally equivalent to Q. Or more generally the entire conformal equivalence class (aka Riemann surface) of such surfaces. The Klein quartic is the unique Riemann surface S, of the least possible genus (g = 3), that attains the maximum size of its conformal automorphism group Aut(S) allowed by the Hurwitz bound #(Aut(S)) <= 84(g-1) for compact S with g >= 2. Hence Aut(Q) and Aut(K) are each isomorphic to the unique simple group G168 of order 168 (the second-smallest non-trivial simple group). This is also the automorphism group of the smallest non-trivial (combinatorial) projective plane, the Fano plane -- 7 lines and 7 points with each point lying on 3 lines and each line lying on 3 points. (Often depicted as an equilateral triangle and its 3 medians -- plus its incircle, which represents the 7th line.) (Incidentally, in CP^2 a generator of Aut(Q) of order 3 is just the permutation taking X -> Y -> Z -> X. One of order 7 is given by X -> w X, Y -> w^2 Y, Z -> W^4 Z. But it seems far trickier to find one of order 2. See expression 1.3 in the chapter by Noam Elkies in the above book.) G168 is also isomorphic to each projective special linear group PSL(3, Z/2) == PSL(2, Z/7) -- one of only two coincidences among the finite PSL's, the other being PSL(2, Z/4) == PSL(2, Z/5). In fact, the groups Aut(Q) and Aut(K) are identical to the groups Isom+(Q) and Isom+(K) of orientation-preserving isometries of Q and K respectively. (The full isometry groups Isom(Q) and Isom(K) are of order 336. Given that the Aut's == G168, this is easily seen for K from its symmetry; likewise for Q by noticing that (X,Y,Z) |-> (X*,Y*,Z*) is an isometry of Q where * denotes complex conjugate.) --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] claimed proof that zeta(5) is irrational
by Dan Asimov 05 May '11

05 May '11

It's amazing how this issue has resisted progress until 1979. Many finite sets of zeta(odd)'s are now known to contain at least one irrational (e.g., papers by W. Zudilin). But I'd be more interested to know whether zeta(n)/pi^n is rational for n odd, as it is for n even. --Dan Rich writes [of an attempt to show zeta(5) is irrational]: << The paper is only 7 pages, and uses techniques similar to the proof in Hardy & Wright that pi is irrational. It's more complicated of course. There's quite a bit of stuff to check, some quintuple integrals, etc. The author claims he has proof of irrationality of zeta(odd) at the end of the paper. >> Sometimes the brain has a mind of its own.

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[math-fun] Hyperbolic asteroids
by Mike Stay 05 May '11

05 May '11

I'm contemplating writing a version of asteroids on Klein's quartic---a three-holed torus constructed by identifying certain edges of 24 hyperbolic heptahedrons---rather than the usual one-holed torus we get by identifying opposite edges of a square. I've chosen to use Poincare's disk for my model of hyperbolic space, so the path an undisturbed asteroid takes on the screen will be an arc that meets the disk at right angles. I've worked through the algebra for calculating the next position of a particle moving at a constant velocity per timestep, but it's pretty ugly: given a particle's position and velocity vectors, I solve a quadratic to find the center of the circular arc defining the geodesic, then another to find the endpoints where it intersects the unit disc, then a mobius transformation to map the geodesic to the imaginary axis, then a mobius transformation to move it forward along the geodesic according to its velocity, then inverting everything to get back to the original point of view. Does geometric algebra give a nicer description? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] claimed proof that zeta(5) is irrational
by rcs＠xmission.com 05 May '11

05 May '11

The paper is only 7 pages, and uses techniques similar to the proof in Hardy & Wright that pi is irrational. It's more complicated of course. There's quite a bit of stuff to check, some quintuple integrals, etc. The author claims he has proof of irrationality of zeta(odd) at the end of the paper. Rich --------- arXiv:1105.0730 (*cross-listing*) Date: Wed, 4 May 2011 03:36:22 GMT (145kb) Title: $\zeta(5)$ is irrational Authors: Yong-Cheol Kim Categories: math.CA math.NT MSC-class: 11J81, 11M06 \\ We present an elementary proof of the irrationality of $\zeta(5)$ based upon the Dirichlet's approximation theorem and the Prime Number Theorem. \\ ( http://arxiv.org/abs/1105.0730 , 145kb)

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