April 2013 - mathfun@mailman.xmission.com

[math-fun] 3D printing of mathematical objects
by Guy Haworth 08 Apr '13

08 Apr '13

I have not used 3D printers myself but I do know that Marcus du Sautoy at Oxford has had some 3D printing done - so I guess there is some expertise in that area. I believe he had Kepler's nested polyhedral printed as a competition prize - and said this was not something that could have been created any other way. Regards - Guy

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[math-fun] How to solve this relation?
by Mike Stay 08 Apr '13

08 Apr '13

The generating function for Stern's diatomic series http://oeis.org/A002487 satisfies A(x) = (x + 1 + 1/x) A(x^2) Is anything more known about such generating functions than the stuff in R. Stephan's 2003 paper in the bibliography? It was somewhat unsatisfying to have an infinite sum replaced by an infinite product. -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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Re: [math-fun] Tan[Sin[x]]-Sin[Tan[x]] puzzles
by Adam P. Goucher 07 Apr '13

07 Apr '13

When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions); however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/ Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Bill Gosper > Sent: 04/04/13 05:13 AM > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Tan[Sin[x]]-Sin[Tan[x]] puzzles > > From Neil Bickford (under protest): > Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] is ultraflat near 0 and goes > berserk at π/2. > Ultraflat: x^7/30+(29 x^9)/756+(1913 x^11)/75600+O[x]^12 > Berserk: Local minima and maxima cluster at π/2, vibrating between roughly > .5 and 2.5. What limits, if any, do they approach? > Note that by analytic continuation, an arbitrarily short segment of the > flat part > near zero completely predicts the berserk part west of π/2. > Are all the terms of the series approximated above positive? > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Tan[Sin[x]]-Sin[Tan[x]] puzzles
by Bill Gosper 06 Apr '13

06 Apr '13

>From Neil Bickford (under protest): Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] is ultraflat near 0 and goes berserk at π/2. Ultraflat: x^7/30+(29 x^9)/756+(1913 x^11)/75600+O[x]^12 Berserk: Local minima and maxima cluster at π/2, vibrating between roughly .5 and 2.5. What limits, if any, do they approach? Note that by analytic continuation, an arbitrarily short segment of the flat part near zero completely predicts the berserk part west of π/2. Are all the terms of the series approximated above positive?

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[math-fun] 3D printing of mathematical objects ??
by Henry Baker 06 Apr '13

06 Apr '13

I've done some research on how to design objects & 3D print them, but none of these CAD systems seems to be interested in producing mathematical objects. In fact, the whole process of generating these drawings seems exceedingly cumbersome & labor-intensive. It looks like I might have to generate my own objects using low-level file formats. It would appear that the 2 low level formats currently in use are the 'STL' file format, which is basically a long list of flat triangular facets given by triples of triples (X,Y,Z) points, which describe oriented triangles in 3-space. http://en.wikipedia.org/wiki/STL_(file_format) This file is sent to a 'slicer', which produces a 'path program' to direct the path of the printing nozzle for each 2D 'slice' of the 3D object. http://en.wikipedia.org/wiki/G-code If I'm trying to produce a 3D print of a mathematical structure, it might be easiest to simply generate my own G-code file, except for the fact that this G-code file probably contains a lot of 3D-printer-specific information. Has anyone on this list gone to these levels of detail? Also, I haven't been able to find any simple programs to do minimal structural analysis on these models -- e.g., to determine if they can even stand on their own.

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[math-fun] coloring unit-distance graph in the plane
by Warren D Smith 06 Apr '13

06 Apr '13

As previous posters pointed out, if you want to color the plane so that any 2 points at distance=1 have different colors than you need at least 4 colors (proof:Moser or Golomb's graphs) while 7 colors are known to suffice (proof: tiling by regular hexagons, each monocolored). http://en.wikipedia.org/wiki/Chromatic_number_of_the_plane I now point out, that if your coloring is required to be PERIODIC and I get to choose the period-parallelogram, then one can prove 7 colors are both necessary and sufficient. Reason is, the tiling of the plane by equilateral triangles, is, in fact, viewed modulo an appropriate period parallelogram, the complete graph K7. Here is a picture drawn by Steven Taschuk: http://amotlpaa.nfshost.com/math/k7torus.html This incidentally also shows that torus graphs can require 7 colors (which also is known to suffice). Meanwhile a hex-tiling construction proves sufficiency. The catch is, that the period-parallelogram for the "necessary" is not the same as for the "sufficient"!

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[math-fun] kerPOW
by Bill Gosper 05 Apr '13

05 Apr '13

Plot[Tan[Sqrt[2] Sinh[x]]/Sqrt[2] - Sinh[Tan[Sqrt[2] x]/Sqrt[2]], {x, 0, .956}, PlotRange -> All] --rwg

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[math-fun] a wonderfully generic squiggle
by Bill Gosper 05 Apr '13

05 Apr '13

as might be seen on a scientist's blackboard in a cartoon: Plot[ Sin[Sin[Tan[ArcSinh[x]]]]-ArcSinh[Sin[Tan[Sin[x]]]], {x, 0, 11}] What are the accumulation points of the turning points? --rwg

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[math-fun] Generating function for inverse catalan numbers?
by Bill Gosper 04 Apr '13

04 Apr '13

MikeStay> The ordinary generating function for the Catalan numbers {C_n} is (1-sqrt(1-4z))/2z. How would I go about finding the OGF of {1/C_n} ? -- Mike Stay - metaweta@gmail.comhttp://www.cs.auckland.ac.nz/~mikehttp://reperiendi.wordpress.com Modulo shifting by 1, Out[1157]= -(-1)^n 2^(-1+2 n) Binomial[1/2,n] In[1158]:= Table[%,{n,9}] Out[1158]= {1,1,2,5,14,42,132,429,1430} In[1161]:= Sum[%1157*x^n,{n,\[Infinity]}] Out[1161]= 1/2 (1-Sqrt[1-4 x]) In[1162]:= Sum[x^n/%1157,{n,\[Infinity]}] Out[1162]= x Hypergeometric2F1[1,2,1/2,x/4] In[1163]:= FunctionExpand[%] Out[1163]= x ((2+x/4)/(2 (-1+x/4)^2)+(3 Sqrt[x] ArcCsc[2/Sqrt[x]])/(4 Sqrt[1-x/4] (-1+x/4)^2)) In[1164]:= FullSimplify[%,0<x<1] Out[1164]= (2 x (Sqrt[4-x] (8+x)+12 Sqrt[x] ArcCsc[2/Sqrt[x]]))/(4-x)^(5/2) In[1165]:= Series[%,{x,0,9}] Out[1165]= x+x^2+x^3/2+x^4/5+x^5/14+x^6/42+x^7/132+x^8/429+x^9/1430+O[x]^10 --rwg

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Re: [math-fun] coloring unit-distance graph in the plane
by Warren D Smith 04 Apr '13

04 Apr '13

> From: Cris Moore <moore(a)santafe.edu> > I was thinking along similar lines. Consider the triangular lattice, and > consider all the distances you can get between pairs of lattice points, but > not between two points in the same sublattice (corresponding to the same > color in a 3-coloring of the lattice). These distances separate pairs of > points that we can force to be different colors. > Now, suppose there are two distances in this set with ratio phi (the golden > ratio). I think actually there are no such pairs, which is a nice problem > in algebra... but if there were, then you could build a K_5, which would > give a lower bound of 5 on the chromatic number of the plane. --well, in spirit you may have a good idea (I do not know and feel confused)... but your idea in detail is a bust: Every distance between two points of the eq.tri. lattice is sqrt(integer) and obviously the ratio of two such, cannot be phi, since it must be sqrt(rational) and phi^2=phi+1=irrational.

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