August 2014 - mathfun@mailman.xmission.com

[math-fun] Four spheres are mutually tangent
by Bill Gosper 06 Aug '14

06 Aug '14

at the vertices of some octahedron. A fifth and sixth sphere are tangent to all four, a little one inside, a big one engulfing all. Then the sum of the curvatures of the fifth and sixth = the sum of the curvatures of the original four. http://www.tweedledum.com/rwg/Sodddy.htm --rwg

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[math-fun] Grapes and other fruit
by Alex Bellos 06 Aug '14

06 Aug '14

For those who missed it, the New York Times last Sunday had short reviews of five math books that have come out over the last few months: http://www.nytimes.com/2014/08/03/books/review/jordan-ellenbergs-how-not-to… One of them is my own - The Grapes of Math* - which I hope is of interest to math-funsters since I interviewed several of you for it, including Bill Gosper, Tom Rokicki and Adam P Goucher. Thanks to all. * Please read in an American accent. If you are British, the UK title is Alex Through The Looking-Glass.

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Re: [math-fun] analytic function for new Airbnb logo?
by Bill Gosper 06 Aug '14

06 Aug '14

HGB>I wanted an approximation that was entire over the complex plane, so the straight lines and discontinuities would have to be fudged. If I take the image, convert it to black-and-white, and erode the boundaries to a single pixel width, I will get a "continuous" curve that goes around the origin twice. If I consider this curve in (r,theta) coordinates, I can now consider the Fourier transform. I'd like to reset the origin in such a way that the "fundamental" is zero -- i.e., the lowest frequency (which takes 2 revolutions) has a coefficient of zero. (There are other possible locations for the origin: the balancing point, the center of gyration, etc.) That having been done, I could alternatively use (log(r),theta) coordinates, which have some nicer properties, because it is the actual complex log function. Fourier analysis can approximate the discontinuous changes in curvature; hopefully modifying these coefficients a little will remove these discontinuities with a minimum of ringing. <HGB Ringing only happens at true discontinuities (e.g. http://www.tweedledum.com/rwg/gibbs.htm ) A nice example of going around twice, mixing circular arcs and straight segments, is to draw, with a single set of amplitudes and phases (for which I have closed forms) alternately an equilateral triangle and its inscribed circle: gosper.org/Incirc_1.mp4 (The file isn't corrupt--download it.) --rwg The sides of your rounded edge squares are impressively flat. But not as flat as those tan(sin())-sin(tan()) games we played. At 03:48 PM 7/30/2014, Bill Gosper wrote: HGB> My approach would be to take the Fourier transform of the polar plot, rwg>Defined as piecewise linear|circular? and then simplify to a small number of coefficients. Of course, it will take 2 circuits around the origin to complete the figure. The sides are perhaps a little too flat, which will blow up the number of coefficients required. I guess if you transform [0,4pi) to [0,1) you could use some sort of Cheby polynomial approximation. It would be nice to be able to transform it back to p(x,y)=0, where p is a polynomial in x,y. At 02:56 PM 7/29/2014, Whitfield Diffie wrote: HGB>Anyone game for an analytic function to approximate Airbnb's new logo? They should manufacture a paper clip and give it out at trade shows. Whithttp://www.underconsideration.com/brandnew/archives/airbnb_logo_detail.… rwg> You can get some impressively straight segments with theta fcns: gosper.org/halph.pdf (10MB?!) or see http://gosper.org/thetpak.html (scrolled 69%). (.3281... is âˆš(one ninth constant), which I claim should be named Halphen's. It's the largest q for which the curve is convex.)) But a problem with this logo is that it switches curvatures discontinuously. <rwg Another possibility: modulate the arcspeed (slow down for curves, speed up for straightaways) before taking the Fourier.

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[math-fun] Colored edge maze
by James Propp 05 Aug '14

05 Aug '14

I want to amuse my kids by having them help me draw a maze in our asphalt-covered driveway (in three colors of chalk) and then having them try to solve it. The maze I have in mind is one I saw years ago. It's a square grid with colored edges and one is required to traverse the edges in the order red, white, blue, red, white, blue, ... (or whatever). Can anyone give me a link to the design of such a maze? Thanks, Jim

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[math-fun] exp(A)exp(B) = exp(A+B) = exp(B)exp(A)?
by Warren D Smith 05 Aug '14

05 Aug '14

> From: Dan Asimov <dasimov(a)earthlink.net> > It's well known and easy to prove that if NxN matrices A and B commute, then > (*) exp(A)exp(B) = exp(A+B) = exp(B)exp(A) > But ... can happen if A,B do not commute... > I think such examples of complex matrices are easier to come by than of real > ones, so I'm particularly interested in the case where A and B are real. -- 1. I point out, that any complex A,B example using nXn matrices, can immediately be converted to equivalent 2nX2n real matrices. 2. Initially I would suggest seeking upper triangular A,B.

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[math-fun] "Essentially serial" computations for PoW ?
by Warren D Smith 05 Aug '14

05 Aug '14

At end of the earlier thread re bitcoin, I had discussed this question a little. I'd argued that what was wanted was not only that the problem be "essentially serial" but also that it be in NP U co-NP so that you could rapidly prove that you'd done the work, without somebody else having to redo the entire computation in order to believe you (i.e. verification is in P, much faster than finding the proof, which is not in P). Let's take it for granted that P is a *strict* subset of NP U co-NP, since fat chance anybody will be able to prove that. I'd suggested, in what I said at the time was probably a bad suggestion, the same thing Adam Goucher just suggested -- integer factorization. However, the reason this is a bad suggestion, is that the best factorization methods for larger integers (GNFS, quadratic sieve, etc) are very highly parallelizable, in fact that is the only way they've been able to factor large numbers at all. QUESTION: What problems, if any, lie in NP U co-NP, but outside P, and furthermore seem inherently unparallelizable? And the point by Cris Moore about "P-complete" problems being unparallelizable is ok per se, BUT a bad idea for our purpose, because those are in P. We want problems outside of P for proof of work purposes, otherwise the work is too easy. Thing is, I just now thought of an argument that THIS QUESTION HAS NO ANSWER. That is because ANY problem that can be verified in P, but cannot be solved in P, AUTOMATICALLY seems highly parallelizable to solve because you can search for all possible verifications of a putative answer, in parallel. This could be called "THE WHOLE BITCOIN UNDERLYING IDEA IS INHERENTLY IRRECOVERABLY STUPID" THEOREM. What I *can* easily do, is produce a problem which is inherently serial to solve, but once it is solved, the verification can be done parallel. For example, say F() is a strong encryption function. Ask: "how many times do you need to iterate F starting from X, before you reach a word whose first N bits all are 0?" This seems inherently serial and exponentially hard. But, if I solve it, I can publish the intermediate iterates, and then the facts x[k+1]=F(x[k]) and none of the x[k] begin with N zeros, can be verified for all k in parallel. This sucks since there is exponential amount of stuff to publish clogging your memory, but aside from that it works. More generally, the "PCP theorem" shows every problem in NP is, once solved, *very* quickly verifiable, if the solver publishes a (quite long, but polynomially long) proof. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] "Essentially serial" computations for PoW ?
by Henry Baker 05 Aug '14

05 Aug '14

If there are any computational complexity theorists out there, I was wondering if there were any examples of "essentially serial" computations, in which no amount of parallelism (and no amount of memory) can speed it up. I.e., the only speedup comes from a faster CPU clock. Obviously, one can use a Turing Machine to "diagonalize" over all Turing Machines with smaller time & space, but this doesn't say much about access to huge amounts of parallelism. The goal is a Bitcoin-type "proof of work" in which buying more computers won't help at all; hopefully the speedup of special hardware over a random laptop or cellphone would be perhaps 3 orders of magnitude, but not more. Any links/pointers would be welcomed.

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[math-fun] New family of spherical codes and designs, clearly member of God's select list of favorite point sets
by Warren D Smith 04 Aug '14

04 Aug '14

For the standard sphere in D dimensions. Among the 2^D directions (+-1, +-1, +-1, ..., +-1) choose a D^2-element subset forming a "Kerdock code." One exists if D>1 happens to be a power of 4. Also choose these 2D additional directions: the positive & negative coordinate axes. What is so amazing about this set of points on the sphere? 1. The number of points is N=(D+2)*D. 2. The minimum angular separation between points is theta where sec(theta)=sqrt(D). 3. Each point has exactly (D^2)/2 nearest neighbors (at angle exactly theta to it). 4. The points form a 5-design, i.e. averaging F(x) on this point set yields the same result as averaging F over the whole sphere surface, whenever F is a polynomial of degree<=5. 5. A theorem of Delsarte-Goethals-Seidel states that any 5-design must have at least (D+1)*D points, hence ours is optimal up to a subleading term. (I suspect it is optimal, period, and also that it is "rigid.") This nodecount also is optimal up to subleading term for all numerical integration rules (with unequal weights and points off the sphere surface, and perhaps even complex, allowed). This is one of the (nicer & conciser) results of mine in a paper I'm working on about numerical integration... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Fwd: [CASE:1301166] antiprisms
by Bill Gosper 02 Aug '14

02 Aug '14

Forgot to add: In[379]:= PolyhedronData["Trapezohedron"] During evaluation of In[379]:= PolyhedronData::notdef: PolyhedronData has no value associated with the specified argument(s). >> Out[379]= PolyhedronData["Trapezohedron"] ---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Fri, Aug 1, 2014 at 5:58 PM Subject: Re: [CASE:1301166] antiprisms To: Wolfram Technical Support <support(a)wolfram.com> Dear Xin Xiao, thank for your scandalous revelation that PolyhedronData is pure table-lookup. I had not imagined this limitation on, e.g., In[374]:= PolyhedronData["Prism"] Out[374]= {"Cube", {"Prism", 3}, {"Prism", 5}, {"Prism", 6}, {"Prism", 7}, {"Prism", 8}, {"Prism", 9}, {"Prism", 10}} Why not file a suggestion that those polyhedra which take an "argument" be implemented at run-time, free of arbitrary limits? If computing the general GraphicsComplex is too difficult, I can supply a simple g2gcomp converter ("Abnormal"?). --Bill Gosper On Fri, Aug 1, 2014 at 2:20 PM, Wolfram Technical Support < support(a)wolfram.com> wrote: > Hello Bill, > > Thank you for your email. > > The reason is that PolyhedronData is just fetching things that has already > been built in Mathematica (which can be compared as looking a word in > dictionary), PolyhedronData[{"Antiprism", n}] does not compute that > particular antiprism depending on the value n. You can query what > antiprisms are built-in by the command > > PolyhedronData["Antiprism"] > > I have filed a suggestion for n = 2 (the degenerated case). > > Sincerely, > > Xin Xiao > > Wolfram Technical Support > > > > > > > ------------------------------ > > On 2014-08-01, 03:23, Bill Gosper (billgosper(a)gmail.com) wrote: > > In[366]:= PolyhedronData[{"Antiprism", 12}] > > During evaluation of In[366]:= PolyhedronData::notdef: PolyhedronData has > no value associated with the specified argument(s). >> > > > Out[366]= PolyhedronData[{"Antiprism", 12}] > > GAA! How could you not implement the general case? E.g., > antiprism[n_] := > Block[{R = > Table[{((Sqrt[Sin[(Pi/(2*n))]*Sin[((3*Pi)/(2*n))]])/(2* > > Sin[Pi/n])), ((Csc[Pi/n]*Cos[((2*k*Pi)/n) + Pi/n])/ > 2), ((Csc[Pi/n]*Sin[((2*k*Pi)/n) + Pi/n])/2)}, {k, n}], > L = Table[{-((Sqrt[Sin[(Pi/(2*n))]*Sin[((3*Pi)/(2*n))]])/(2* > Sin[Pi/n])), ((Csc[Pi/n]*Cos[((2*(k + 1)*Pi)/n)])/ > > 2), ((Csc[Pi/n]*Sin[((2*(k + 1)*Pi)/n)])/2)}, {k, n}]}, > Graphics3D[ > Join @@ {{Polygon[L], Polygon[R]}, > Table[Polygon[{L[[k]], R[[k]], R[[Mod[k, n] + 1]]}], {k, n}], > Table[Polygon[{L[[k]], R[[Mod[k, n] + 1]], > > L[[Mod[k, n] + 1]]}], {k, n}]}]] > > (You'll want to change to a GraphicsComplex.) > > The volume formula is > > 1/12 n (1 + 2 Cos[\[Pi]/n]) Csc[\[Pi]/n]^2 Sqrt[Sin[\[Pi]/(2 n)] Sin[(3 > \[Pi])/(2 n)]] > > > In[367]:= PolyhedronData[{"Antiprism", 2}] > > During evaluation of In[367]:= PolyhedronData::notdef: PolyhedronData has > no value associated with the specified argument(s). >> > > Out[367]= PolyhedronData[{"Antiprism", 2}] > > GAAA!! This is the thinking that convinced the ancients that zero > was not a number! > Even Mathworld reluctantly admits it's a tetrahedron. > > > In[337]:= > FullSimplify[Table[%%%, {n, {2, 3, 4, 5, 6, 8}}]] // ToRadicals > > Out[337]= {1/(6 Sqrt[2]), Sqrt[2]/3, 1/3 (2^(1/4) + 2^(3/4)), > 1/6 (5 + 2 Sqrt[5]), Sqrt[2 (1 + Sqrt[3])], (1/(3 (2 - Sqrt[2]))) > > 4 (2 - Sqrt[2 + Sqrt[2]])^(1/4) (1 + Sqrt[2 + Sqrt[2]]) Sqrt[ > Sqrt[2 - Sqrt[2 + Sqrt[2]]] + > Sqrt[(2 + Sqrt[2]) (2 - Sqrt[2 + Sqrt[2]])]]} > > (Note the pretty denesting in the square case.) > --Bill Gosper >

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[math-fun] Fwd: antiprisms
by Bill Gosper 01 Aug '14

01 Aug '14

Another design dispute with WRI after I lost the argument about weird GCDs. Anybody want to defend the Antiprism status quo? I really hate arbitrary censorship of the "degenerate" digon case. This is all in Mathworld, but I prefer my volume formula. ---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Fri, Aug 1, 2014 at 1:23 AM Subject: antiprisms To: Wolfram Technical Support <support(a)wolfram.com> In[366]:= PolyhedronData[{"Antiprism", 12}] During evaluation of In[366]:= PolyhedronData::notdef: PolyhedronData has no value associated with the specified argument(s). >> Out[366]= PolyhedronData[{"Antiprism", 12}] GAA! How could you not implement the general case? E.g., antiprism[n_] := Block[{R = Table[{((Sqrt[Sin[(Pi/(2*n))]*Sin[((3*Pi)/(2*n))]])/(2* Sin[Pi/n])), ((Csc[Pi/n]*Cos[((2*k*Pi)/n) + Pi/n])/ 2), ((Csc[Pi/n]*Sin[((2*k*Pi)/n) + Pi/n])/2)}, {k, n}], L = Table[{-((Sqrt[Sin[(Pi/(2*n))]*Sin[((3*Pi)/(2*n))]])/(2* Sin[Pi/n])), ((Csc[Pi/n]*Cos[((2*(k + 1)*Pi)/n)])/ 2), ((Csc[Pi/n]*Sin[((2*(k + 1)*Pi)/n)])/2)}, {k, n}]}, Graphics3D[ Join @@ {{Polygon[L], Polygon[R]}, Table[Polygon[{L[[k]], R[[k]], R[[Mod[k, n] + 1]]}], {k, n}], Table[Polygon[{L[[k]], R[[Mod[k, n] + 1]], L[[Mod[k, n] + 1]]}], {k, n}]}]] (You'll want to change to a GraphicsComplex.) The volume formula is n (1 + 2 Cos[Pi/n]) Csc[Pi/n]^2 Sqrt[Sin[Pi/(2 n)] Sin[3 Pi/(2 n)]]/12 In[367]:= PolyhedronData[{"Antiprism", 2}] During evaluation of In[367]:= PolyhedronData::notdef: PolyhedronData has no value associated with the specified argument(s). >> Out[367]= PolyhedronData[{"Antiprism", 2}] GAAA!! This is the thinking that convinced the ancients that zero was not a number! Even Mathworld grudgingly admits it's a tetrahedron. In[337]:= FullSimplify[Table[%%%, {n, {2, 3, 4, 5, 6, 8}}]] // ToRadicals Out[337]= {1/(6 Sqrt[2]), Sqrt[2]/3, 1/3 (2^(1/4) + 2^(3/4)), 1/6 (5 + 2 Sqrt[5]), Sqrt[2 (1 + Sqrt[3])], (1/(3 (2 - Sqrt[2]))) 4 (2 - Sqrt[2 + Sqrt[2]])^(1/4) (1 + Sqrt[2 + Sqrt[2]]) Sqrt[ Sqrt[2 - Sqrt[2 + Sqrt[2]]] + Sqrt[(2 + Sqrt[2]) (2 - Sqrt[2 + Sqrt[2]])]]} (Note the pretty denesting in the square case.) --Bill Gosper

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