August 2013 - mathfun@mailman.xmission.com

[math-fun] Bit puzzle
by quad 02 Sep '13

02 Sep '13

Here is a little puzzle. I encourage not spelunking around for the answer on the Internet. ;) Given a 32-bit unsigned integer x, and a positive integer n, let a = XOR(x, RIGHTSHIFT(x, n)) or in C syntax a = x ^ (x >> n) find, if one exists, a function f such that f(a) = x. In other words, invert the operation on x, or show otherwise. Cheers, Robert

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[math-fun] Julian's dodecahedron
by Bill Gosper 01 Sep '13

01 Sep '13

is made by replacing ten of the regular pentagons of a dodecahedron with concave equilateral pentagons, preserving the topology: http://gosper.org/dodohedron.png ( !#%&*@! was deleted again. Apologies to any of you who looked for it.) http://en.wikipedia.org/wiki/Dodecahedron glaringly omits it: [...] Topologically distinct dodecahedra include: - Pentagonal dodecahedra: - Regular dodecahedron, 12 pentagonal faces, Ih<http://en.wikipedia.org/wiki/Icosahedral_symmetry>symmetry, order 120 - Pentagonal truncated trapezohedron<http://en.wikipedia.org/wiki/Truncated_trapezohedron>- same topology as regular form, but D5d <http://en.wikipedia.org/wiki/Dihedral_symmetry> symmetry, order 20 - "Pyritohedron <http://en.wikipedia.org/wiki/Pyritohedron>" - same topology as regular form, but Th<http://en.wikipedia.org/wiki/Tetrahedral_symmetry>symmetry, order 12 - Uniform polyhedra: - Decagonal prism <http://en.wikipedia.org/wiki/Decagonal_prism> – 10 squares, 2 decagons, D10h<http://en.wikipedia.org/wiki/Dihedral_symmetry>symmetry, order 40 - Pentagonal antiprism<http://en.wikipedia.org/wiki/Pentagonal_antiprism>– 10 equilateral triangles, 2 pentagons, D5d <http://en.wikipedia.org/wiki/Dihedral_symmetry> symmetry, order 20 - Johnson solids <http://en.wikipedia.org/wiki/Johnson_solid> (regular faced): - Pentagonal cupola <http://en.wikipedia.org/wiki/Pentagonal_cupola>– 5 triangles, 5 squares, 1 pentagon, 1 decagon, C5v <http://en.wikipedia.org/wiki/Cyclic_symmetry> symmetry, order 10 [...] reinforcing the suspicion that Julian was not anticipated. (Perhaps a contributing factor to the news not reaching Wikipedia was its repeated disappearance from my site.) Anyway, each of the three pentagonal dodecahedra has its own article, as well as mention in the above list. Obviously there should be an article on Julian's. (Julian, do you still have the coordinates?) It would be amusing to create the article, and watch the Wikiwardens delete it as Original Research. Do we really need to create Propp's Electronic Journal of Unoriginal Research? Even if those <PEJURative> Wikiwardens relented, we still have the problem of naming this solid! --rwg

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[math-fun] solving, denesting, and reading comprehension
by Bill Gosper 01 Sep '13

01 Sep '13

reg>http://www.math.harvard.edu/~elkies/trinomial.html <http://www.math.harvard.edu/%7Eelkies/trinomial.html> gives a (rational) continuum of irreducible, supposedly solvable sextics: (125 - u)*x^6 + 12*u*(u + 3)^2*x + u*(u - 5)*(u + 3)^2 but my sextic solver fails for u≠0. (≠-3,≠5) Can anybody propose a correction? CG>I have no correction, but I concur with your result. For example, with u = 4 I get 121x^6 + 2352x - 196 with Galois group PSL(2,5): polgalois(121*x^6 + 2352*x - 196) %1 = [60, 1, 1, "L(6) = PSL(2,5) = A_5(6)"] which is not solvable. Charles Greathouse Analyst/Programmer Case Western Reserve University Mea gufa. With more patience than I deserve, Elkies privately explained that his paper claims only that the Galois groups are atypically small, not necessarily small enough to solve. He also had no news re the possibility of a solvable, irreducible septic trinomial. Come to think of it, I don't even recall seeing a solvable, irreducible x^6+ax+b. --rwg

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[math-fun] Solvable trinomials
by elkies＠math.harvard.edu 31 Aug '13

31 Aug '13

... and there are infinitely many solvable trinomials ax^8+bx+c, such as 4x^8+8x+13 (this must be well-known too)... NDE

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[math-fun] The speed of wave motion
by Warren D Smith 30 Aug '13

30 Aug '13

For wave = sin(A*x - B*t) with A,B real constants, x=space, t=time, we all agree the speed of wave motion is |B/A|. But now consider exp(i*A*x - i*B*t) where A,B could be permitted to be complex. What would you claim the "speed of wave motion" is now? "Speed of 'information transmission'?" I'm not sure there is a clear and right answer... but send yours...

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Re: [math-fun] Looking for Javascript Life engine
by scott＠scottkim.com 30 Aug '13

30 Aug '13

Thanks all for the Life Javascript references. Phil, the super-short Life engine is especially charming. Looks like it'll be perfectly easier to roll my own...good to see these examples. -- Scott Original email: ----------------- From: Phil Carmody thefatphil(a)yahoo.co.uk Date: Fri, 30 Aug 2013 06:59:13 +0100 (BST) To: math-fun(a)mailman.xmission.com Subject: Re: [math-fun] Looking for Javascript Life engine From: "scott(a)scottkim.com" <scott(a)scottkim.com> > I'm proposing to Google an interactive Google Doodle to honor Martin > Gardner, which would contain 6 minigames (one for each letter of > GOOGLE), highlighting six things Gardner wrote about. One of those > games of course should be a version of Life. Does anyone know of > or have a Life engine that could be modified for this use? I'd prefer > to open-source the code, but that is not essential. I'm guessing it > should be in Javascript to be browser friendly (the Turing Google > Doodle was open-source Javascript). A noble idea. Here's a tiny modern JS-based life engine, only 376 bytes: http://js1k.com/2013-spring/demo/1478 Phil -- () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani] _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -------------------------------------------------------------------- mail2web.com - Microsoft® Exchange solutions from a leading provider - http://link.mail2web.com/Business/Exchange

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[math-fun] Calculus of variations problem: the one-wire cage.
by Warren D Smith 30 Aug '13

30 Aug '13

> nice. isn't this the path on the edges of a tetrahedron I mentioned? > I'd certainly think that would have the maximum for a 4-segment closed > polygon: 4 consecutive edges of a regular tetrahedron that returns to where > it started. --no. The 4-edge closed polygonal path (all edge lengths 1/4) I mentioned has hinge angle 90 (note 90 is trivially seen best). The thing is made by folding a rhombus with angles 70.522 and 109.478 along its short diagonal at 90 degrees. This is not an edge path on a regular tetrahedron, but some regtet angles worm in there somehow.

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[math-fun] Calculus of variations problem: the one-wire cage.
by Warren D Smith 30 Aug '13

30 Aug '13

Calculus of variations problem: the one-wire cage. Allan Wechsler: Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? --WDS: This problem actually can be posed in any number of dimensions. In 2D as Salamin said the answer (Dido problem) is a semicircle-arc. The problem also can be posed for closed curves. In 2D for closed curve answer is circle (isoperimetric problem). The problem can also be posed for arbitrary connected "networks" of curve segments (such as a tree) having total length=1. Tilli (cited below) claims answer in 2D again is a semicircle arc. TRY #1: On a spherical earth, go up the greenwich meridian from S pole to N pole (semicircle). Then go down another meridian at longitude=120 back to S pole. Then up another meridian at longitude=240 back to N pole. Smooth the corners by epsilon since Wechler wants differentiable curve. The curve I have described if earth-radius=1/(3*pI), has length=1. Its convex hull is somewhat less than the whole volume of the earth, which would have been Vearth=4/(81*pi^2) = 0.0050035. In fact, I think the volume ratio is the same as the area ratio of an equilateral triangle to its circumcircle, namely sqrt(27) / (4*pi) = 0.413497. If so, then the convex hull volume is Vconvhull = sqrt(3) / (27*pi^3) = 0.0020689. But one can do better by chopping off some length on each of the two ends of this curve (then rescale back to unit total length), it is not clear what optimizes that, and also it seems doubtful the result is optimal. TRY #2a and 2b: Go round a circle in XY plane centered at origin, then line segment up to a point (0,0,Z). Optionally(b) also could put a line segment on the other end to (0,0,-Z). The convex hull is a cone (or intersection of two opposite-facing cones). If radius of circle is R, then curve length is 2*Pi*R + sqrt(R*R+Z*Z) = 1 or with the option(b) taken 2*Pi*R + 2*sqrt(R*R+Z*Z) = 1. The volume is Pi*R*R*Z/3 or if option(b) taken then multiply that by 2. Optimize to find: 2a: R=0.1015659929, Z=0.3472952850, Vcone=0.003751666 2b: R=0.09290346185, Z=0.1862503544, Vdoubcone=0.003366817 [there are actually exact formulas for all those which I omit] 2a is however not optimal because bowing the line segment outward a bit will be superior. TRY #3: segment of helix. Melzak quoting Egervary 1949 says this is best, and the answer is 1 turn of the Helix at pitch 1/sqrt(2). That is (up to scaling) x(t)=cos(t), y(t)=sin(t), z(t)=t/sqrt(2), 0<t<2*pi. Alleged resulting volume is Vhelix = 1/(18*Pi*sqrt(3)) = 0.010209794. TRY #4: Two semicircular arcs hinged at the equator. But in this case it would be better to smooth the corners. TRY #5: For a closed curve in 4D, use a circumf=1/2 circle in XY plane and another in ZT plane. Schoenberg says this is best (and the analogous construction is best for closed curve in every even dimension for 6D you'd use 3 circles of circumf=1/3 each in 3 orthogonal planes; this is a nice generalization of isoperimetric theorem) TRY #6: for closed curve in 3D, Melzak claims the optimum under certain restrictive assumptions is the "Baggins constant" which he evaluates as Vbaggins=0.0031816877. He thinks this has no simple closed form expression. There is 2010 paper by Paolo Tilli on the network problem version: Isoperimetric inequalities for convex hulls and related questions, Trans. Amer. Math. Soc. 362 (2010) 4497-4509 http://www.jointmathematicsmeetings.org/journals/tran/2010-362-09/S0002-994… Tilli says it was posed in a 1991 book "unsolved problems in geometry." Other papers: I.J. Schoenberg: An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces, Acta Math. 91 (1954) 143-164. A.A. Nudel’man: Isoperimetric problems for the convex hulls of polygonal lines and curves in multidimensional spaces, (Russian) Mat. Sbornik (N.S.) 96,138 (1975) 294-313. Michele Gori: On a maximization problem for the convex hull of connected systems of segments, J. of Convex Analysis 14,1 (2007) 49-68. Z.A. Melzak: The isoperimetric problem of the convex hull of a closed space curve, Proc. Amer. Math. Soc. 11 (1960) 265-274. Z.A. Melzak:Numerical evaluation of an isoperimetric constant, Maths of Comput. 22 (1968) 188-190 http://www.ams.org/journals/mcom/1968-22-101/S0025-5718-1968-0223976-4/ Z.A. Melzak: The isoperimetric problem of the convex hull of a closed space curve, Proc. Amer. Math. Soc. 11 (1960) 265-274 http://www.ams.org/journals/proc/1960-011-02/S0002-9939-1960-0116263-0/ E. Egervary: On the smallest convex cover of a simple arc of space-curve, Publ. Math. Debrecen 1 (1949) 65-70.

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[math-fun] Calculus of variations problem: the one-wire cage.
by Allan Wechsler 30 Aug '13

30 Aug '13

Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull?

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[math-fun] correction: carbon bond angle
by Warren D Smith 30 Aug '13

30 Aug '13

Slight correction, carbon bond angle is not 70.522 but rather 180-70.522=109.478.

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